3.7.47 \(\int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=406 \[ -\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{48 c^2 x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{192 a c^2 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 a^2 c^2 x}-\frac {(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 c x^4} \]

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Rubi [A]  time = 0.42, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {97, 149, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{48 c^2 x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-19 a^2 b c d^2+3 a^3 d^3+109 a b^2 c^2 d+3 b^3 c^3\right )}{192 a c^2 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 a^2 c^2 x}-\frac {(a d+b c) \left (178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 c x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6,x]

[Out]

((3*b^4*c^4 - 22*a*b^3*c^3*d - 128*a^2*b^2*c^2*d^2 + 22*a^3*b*c*d^3 - 3*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(128*a^2*c^2*x) - ((3*b^3*c^3 + 109*a*b^2*c^2*d - 19*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(
192*a*c^2*x^2) - ((3*b^2*c^2 + 16*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(48*c^2*x^3) - ((b*c + a
*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*c*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*x^5) - ((b*c + a*d)*(3*b
^4*c^4 - 28*a*b^3*c^3*d + 178*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(5/2)) + 2*b^(5/2)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqr
t[c + d*x])]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\frac {1}{5} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac {5}{2} (b c+a d)+5 b d x\right )}{x^5} \, dx\\ &=-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {5}{4} \left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right )+20 b^2 c d x\right )}{x^4} \, dx}{20 c}\\ &=-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {5}{8} \left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right )+60 b^3 c^2 d x\right )}{x^3 \sqrt {a+b x}} \, dx}{60 c^2}\\ &=-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\frac {\int \frac {\sqrt {c+d x} \left (-\frac {15}{16} \left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right )+120 a b^3 c^2 d^2 x\right )}{x^2 \sqrt {a+b x}} \, dx}{120 a c^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^2 x}-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\frac {\int \frac {\frac {15}{32} (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )+120 a^2 b^3 c^2 d^3 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^2 c^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^2 x}-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\left (b^3 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^2 c^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^2 x}-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}+\left (2 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )+\frac {\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^2 c^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^2 x}-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac {(b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{5/2}}+\left (2 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d-128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^2 x}-\frac {\left (3 b^3 c^3+109 a b^2 c^2 d-19 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x^2}-\frac {\left (3 b^2 c^2+16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 c^2 x^3}-\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac {(b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\\ \end {align*}

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Mathematica [A]  time = 4.49, size = 365, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )+2 a^3 b c x \left (504 c^3+1448 c^2 d x+1289 c d^2 x^2+180 d^3 x^3\right )+2 a^2 b^2 c^2 x^2 \left (372 c^2+1289 c d x+1877 d^2 x^2\right )+30 a b^3 c^3 x^3 (c+12 d x)-45 b^4 c^4 x^4\right )}{1920 a^2 c^2 x^5}-\frac {\left (3 a^5 d^5-25 a^4 b c d^4+150 a^3 b^2 c^2 d^3+150 a^2 b^3 c^3 d^2-25 a b^4 c^4 d+3 b^5 c^5\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{5/2}}+\frac {2 d^{5/2} (b c-a d)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{(c+d x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6,x]

[Out]

-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*b^4*c^4*x^4 + 30*a*b^3*c^3*x^3*(c + 12*d*x) + 2*a^2*b^2*c^2*x^2*(372
*c^2 + 1289*c*d*x + 1877*d^2*x^2) + 2*a^3*b*c*x*(504*c^3 + 1448*c^2*d*x + 1289*c*d^2*x^2 + 180*d^3*x^3) + 3*a^
4*(128*c^4 + 336*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(a^2*c^2*x^5) + (2*d^(5/2)*(b*c - a*
d)^(5/2)*((b*(c + d*x))/(b*c - a*d))^(5/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c + d*x)^(5/2) -
((3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*ArcTan
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(5/2))

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IntegrateAlgebraic [B]  time = 1.00, size = 1078, normalized size = 2.66 \begin {gather*} 2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) d^{5/2}+\frac {-\frac {45 d^5 (c+d x)^{9/2} a^9}{(a+b x)^{9/2}}+\frac {375 b c d^4 (c+d x)^{9/2} a^8}{(a+b x)^{9/2}}+\frac {210 c d^5 (c+d x)^{7/2} a^8}{(a+b x)^{7/2}}+\frac {1590 b^2 c^2 d^3 (c+d x)^{9/2} a^7}{(a+b x)^{9/2}}-\frac {470 b c^2 d^4 (c+d x)^{7/2} a^7}{(a+b x)^{7/2}}+\frac {384 c^2 d^5 (c+d x)^{5/2} a^7}{(a+b x)^{5/2}}-\frac {2250 b^3 c^3 d^2 (c+d x)^{9/2} a^6}{(a+b x)^{9/2}}-\frac {8700 b^2 c^3 d^3 (c+d x)^{7/2} a^6}{(a+b x)^{7/2}}-\frac {3200 b c^3 d^4 (c+d x)^{5/2} a^6}{(a+b x)^{5/2}}-\frac {210 c^3 d^5 (c+d x)^{3/2} a^6}{(a+b x)^{3/2}}+\frac {375 b^4 c^4 d (c+d x)^{9/2} a^5}{(a+b x)^{9/2}}+\frac {10500 b^3 c^4 d^2 (c+d x)^{7/2} a^5}{(a+b x)^{7/2}}+\frac {19200 b^2 c^4 d^3 (c+d x)^{5/2} a^5}{(a+b x)^{5/2}}+\frac {1750 b c^4 d^4 (c+d x)^{3/2} a^5}{(a+b x)^{3/2}}+\frac {45 c^4 d^5 \sqrt {c+d x} a^5}{\sqrt {a+b x}}-\frac {45 b^5 c^5 (c+d x)^{9/2} a^4}{(a+b x)^{9/2}}-\frac {1750 b^4 c^5 d (c+d x)^{7/2} a^4}{(a+b x)^{7/2}}-\frac {19200 b^3 c^5 d^2 (c+d x)^{5/2} a^4}{(a+b x)^{5/2}}-\frac {10500 b^2 c^5 d^3 (c+d x)^{3/2} a^4}{(a+b x)^{3/2}}-\frac {375 b c^5 d^4 \sqrt {c+d x} a^4}{\sqrt {a+b x}}+\frac {210 b^5 c^6 (c+d x)^{7/2} a^3}{(a+b x)^{7/2}}+\frac {3200 b^4 c^6 d (c+d x)^{5/2} a^3}{(a+b x)^{5/2}}+\frac {8700 b^3 c^6 d^2 (c+d x)^{3/2} a^3}{(a+b x)^{3/2}}+\frac {2250 b^2 c^6 d^3 \sqrt {c+d x} a^3}{\sqrt {a+b x}}-\frac {384 b^5 c^7 (c+d x)^{5/2} a^2}{(a+b x)^{5/2}}+\frac {470 b^4 c^7 d (c+d x)^{3/2} a^2}{(a+b x)^{3/2}}-\frac {1590 b^3 c^7 d^2 \sqrt {c+d x} a^2}{\sqrt {a+b x}}-\frac {210 b^5 c^8 (c+d x)^{3/2} a}{(a+b x)^{3/2}}-\frac {375 b^4 c^8 d \sqrt {c+d x} a}{\sqrt {a+b x}}+\frac {45 b^5 c^9 \sqrt {c+d x}}{\sqrt {a+b x}}}{1920 a^2 c^2 \left (c-\frac {a (c+d x)}{a+b x}\right )^5}+\frac {\left (-3 b^5 c^5+25 a b^4 d c^4-150 a^2 b^3 d^2 c^3-150 a^3 b^2 d^3 c^2+25 a^4 b d^4 c-3 a^5 d^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{5/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6,x]

[Out]

((45*b^5*c^9*Sqrt[c + d*x])/Sqrt[a + b*x] - (375*a*b^4*c^8*d*Sqrt[c + d*x])/Sqrt[a + b*x] - (1590*a^2*b^3*c^7*
d^2*Sqrt[c + d*x])/Sqrt[a + b*x] + (2250*a^3*b^2*c^6*d^3*Sqrt[c + d*x])/Sqrt[a + b*x] - (375*a^4*b*c^5*d^4*Sqr
t[c + d*x])/Sqrt[a + b*x] + (45*a^5*c^4*d^5*Sqrt[c + d*x])/Sqrt[a + b*x] - (210*a*b^5*c^8*(c + d*x)^(3/2))/(a
+ b*x)^(3/2) + (470*a^2*b^4*c^7*d*(c + d*x)^(3/2))/(a + b*x)^(3/2) + (8700*a^3*b^3*c^6*d^2*(c + d*x)^(3/2))/(a
+ b*x)^(3/2) - (10500*a^4*b^2*c^5*d^3*(c + d*x)^(3/2))/(a + b*x)^(3/2) + (1750*a^5*b*c^4*d^4*(c + d*x)^(3/2))
/(a + b*x)^(3/2) - (210*a^6*c^3*d^5*(c + d*x)^(3/2))/(a + b*x)^(3/2) - (384*a^2*b^5*c^7*(c + d*x)^(5/2))/(a +
b*x)^(5/2) + (3200*a^3*b^4*c^6*d*(c + d*x)^(5/2))/(a + b*x)^(5/2) - (19200*a^4*b^3*c^5*d^2*(c + d*x)^(5/2))/(a
+ b*x)^(5/2) + (19200*a^5*b^2*c^4*d^3*(c + d*x)^(5/2))/(a + b*x)^(5/2) - (3200*a^6*b*c^3*d^4*(c + d*x)^(5/2))
/(a + b*x)^(5/2) + (384*a^7*c^2*d^5*(c + d*x)^(5/2))/(a + b*x)^(5/2) + (210*a^3*b^5*c^6*(c + d*x)^(7/2))/(a +
b*x)^(7/2) - (1750*a^4*b^4*c^5*d*(c + d*x)^(7/2))/(a + b*x)^(7/2) + (10500*a^5*b^3*c^4*d^2*(c + d*x)^(7/2))/(a
+ b*x)^(7/2) - (8700*a^6*b^2*c^3*d^3*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (470*a^7*b*c^2*d^4*(c + d*x)^(7/2))/(
a + b*x)^(7/2) + (210*a^8*c*d^5*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (45*a^4*b^5*c^5*(c + d*x)^(9/2))/(a + b*x)^
(9/2) + (375*a^5*b^4*c^4*d*(c + d*x)^(9/2))/(a + b*x)^(9/2) - (2250*a^6*b^3*c^3*d^2*(c + d*x)^(9/2))/(a + b*x)
^(9/2) + (1590*a^7*b^2*c^2*d^3*(c + d*x)^(9/2))/(a + b*x)^(9/2) + (375*a^8*b*c*d^4*(c + d*x)^(9/2))/(a + b*x)^
(9/2) - (45*a^9*d^5*(c + d*x)^(9/2))/(a + b*x)^(9/2))/(1920*a^2*c^2*(c - (a*(c + d*x))/(a + b*x))^5) + ((-3*b^
5*c^5 + 25*a*b^4*c^4*d - 150*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d^3 + 25*a^4*b*c*d^4 - 3*a^5*d^5)*ArcTanh[(Sqrt
[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(5/2)*c^(5/2)) + 2*b^(5/2)*d^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c
+ d*x])/(Sqrt[d]*Sqrt[a + b*x])]

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fricas [A]  time = 60.26, size = 1849, normalized size = 4.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(3840*sqrt(b*d)*a^3*b^2*c^3*d^2*x^5*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b
*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 15*(3*b^5*c^5 - 25*a*b^4*c^4*d +
150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(a*c)*x^5*log((8*a^2*c^2 + (b^2*c^
2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 +
a^2*c*d)*x)/x^2) - 4*(384*a^5*c^5 - (45*a*b^4*c^5 - 360*a^2*b^3*c^4*d - 3754*a^3*b^2*c^3*d^2 - 360*a^4*b*c^2*d
^3 + 45*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c^5 + 1289*a^3*b^2*c^4*d + 1289*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 + 8
*(93*a^3*b^2*c^5 + 362*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 + 1008*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(
d*x + c))/(a^3*c^3*x^5), -1/7680*(7680*sqrt(-b*d)*a^3*b^2*c^3*d^2*x^5*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b
*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 15*(3*b^5*c^5 - 25*a*b^4*c^
4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(a*c)*x^5*log((8*a^2*c^2 + (
b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*
c^2 + a^2*c*d)*x)/x^2) + 4*(384*a^5*c^5 - (45*a*b^4*c^5 - 360*a^2*b^3*c^4*d - 3754*a^3*b^2*c^3*d^2 - 360*a^4*b
*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c^5 + 1289*a^3*b^2*c^4*d + 1289*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x
^3 + 8*(93*a^3*b^2*c^5 + 362*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 + 1008*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)
*sqrt(d*x + c))/(a^3*c^3*x^5), 1/3840*(1920*sqrt(b*d)*a^3*b^2*c^3*d^2*x^5*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*
c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 15*
(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-a*
c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b
*c^2 + a^2*c*d)*x)) - 2*(384*a^5*c^5 - (45*a*b^4*c^5 - 360*a^2*b^3*c^4*d - 3754*a^3*b^2*c^3*d^2 - 360*a^4*b*c^
2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c^5 + 1289*a^3*b^2*c^4*d + 1289*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3
+ 8*(93*a^3*b^2*c^5 + 362*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 + 1008*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sq
rt(d*x + c))/(a^3*c^3*x^5), -1/3840*(3840*sqrt(-b*d)*a^3*b^2*c^3*d^2*x^5*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt
(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 15*(3*b^5*c^5 - 25*a*b^4
*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*
a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) +
2*(384*a^5*c^5 - (45*a*b^4*c^5 - 360*a^2*b^3*c^4*d - 3754*a^3*b^2*c^3*d^2 - 360*a^4*b*c^2*d^3 + 45*a^5*c*d^4)
*x^4 + 2*(15*a^2*b^3*c^5 + 1289*a^3*b^2*c^4*d + 1289*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 + 8*(93*a^3*b^2*c^5 +
362*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 + 1008*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^3
*x^5)]

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giac [B]  time = 15.05, size = 5983, normalized size = 14.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^6,x, algorithm="giac")

[Out]

-1/1920*(1920*sqrt(b*d)*b^2*d^2*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)
+ 15*(3*sqrt(b*d)*b^6*c^5*abs(b) - 25*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 150*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) + 15
0*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) - 25*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) + 3*sqrt(b*d)*a^5*b*d^5*abs(b))*arctan(
-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(
sqrt(-a*b*c*d)*a^2*b*c^2) - 2*(45*sqrt(b*d)*b^24*c^14*abs(b) - 810*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 1871*sqrt(
b*d)*a^2*b^22*c^12*d^2*abs(b) + 15580*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) - 112635*sqrt(b*d)*a^4*b^20*c^10*d^4*
abs(b) + 346890*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) - 642945*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) + 784008*sqrt(b*d
)*a^7*b^17*c^7*d^7*abs(b) - 642945*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) + 346890*sqrt(b*d)*a^9*b^15*c^5*d^9*abs(b
) - 112635*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) + 15580*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) + 1871*sqrt(b*d)*a^
12*b^12*c^2*d^12*abs(b) - 810*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) + 45*sqrt(b*d)*a^14*b^10*d^14*abs(b) - 405*sqr
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^22*c^13*abs(b) + 6015*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^21*c^12*d*abs(b) - 1670*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^20*c^11*d^2*abs(b) - 122710*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^19*c^10*d^3*abs(b) + 456425*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^18*c^9*d^4*abs(b) - 698035*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^17*c^8*d^5*abs(b) + 360380*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^16*c^7*d^6*abs(b) + 360380*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^15*c^6*d^7*abs(b) - 698035*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^14*c^5*d^8*abs(b) + 456425*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^13*c^4*d^9*abs(b) - 122710*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^12*c^3*d^10*abs(b) - 1670*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^11*b^11*c^2*d^11*abs(b) + 6015*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^12*b^10*c*d^12*abs(b) - 405*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^13*b^9*d^13*abs(b) + 1620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^20*c^12*abs(b) - 19800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^4*a*b^19*c^11*d*abs(b) - 43560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^4*a^2*b^18*c^10*d^2*abs(b) + 389000*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^4*a^3*b^17*c^9*d^3*abs(b) - 642900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^4*a^4*b^16*c^8*d^4*abs(b) + 204240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^4*a^5*b^15*c^7*d^5*abs(b) + 222800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^6*b^14*c^6*d^6*abs(b) + 204240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^7*b^13*c^5*d^7*abs(b) - 642900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^4*a^8*b^12*c^4*d^8*abs(b) + 389000*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^4*a^9*b^11*c^3*d^9*abs(b) - 43560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^4*a^10*b^10*c^2*d^10*abs(b) - 19800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^4*a^11*b^9*c*d^11*abs(b) + 1620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^12*b^8*d^12*abs(b) - 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^6*b^18*c^11*abs(b) + 38220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*
b^17*c^10*d*abs(b) + 194780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^
16*c^9*d^2*abs(b) - 575220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^1
5*c^8*d^3*abs(b) + 231480*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^14
*c^7*d^4*abs(b) + 114520*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^13*
c^6*d^5*abs(b) + 114520*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^12*c
^5*d^6*abs(b) + 231480*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^11*c^
4*d^7*abs(b) - 575220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^10*c^3
*d^8*abs(b) + 194780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^9*b^9*c^2*d
^9*abs(b) + 38220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^10*b^8*c*d^10*
abs(b) - 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^11*b^7*d^11*abs(b)
+ 5670*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^16*c^10*abs(b) - 48300*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^15*c^9*d*abs(b) - 402290*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^14*c^8*d^2*abs(b) + 272560*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^3*b^13*c^7*d^3*abs(b) + 109900*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^4*b^12*c^6*d^4*abs(b) + 124920*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^5*b^11*c^5*d^5*abs(b) + 109900*sqrt(b*d)*
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^10*c^4*d^6*abs(b) + 272560*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^7*b^9*c^3*d^7*abs(b) - 402290*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^8*b^8*c^2*d^8*abs(b) - 48300*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^9*b^7*c*d^9*abs(b) + 5670*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^10*b^6*d^10*abs(b) - 5670*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^14*c^9*abs(b) + 42210*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^13*c^8*d*abs(b) + 495504*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^12*c^7*d^2*abs(b) + 327040*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^11*c^6*d^3*abs(b) + 263220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^4*b^10*c^5*d^4*abs(b) + 263220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^5*b^9*c^4*d^5*abs(b) + 327040*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^8*c^3*d^6*abs(b) + 495504*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^10*a^7*b^7*c^2*d^7*abs(b) + 42210*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^10*a^8*b^6*c*d^8*abs(b) - 5670*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^10*a^9*b^5*d^9*abs(b) + 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^12*b^12*c^8*abs(b) - 26040*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^12*a*b^11*c^7*d*abs(b) - 393760*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^12*a^2*b^10*c^6*d^2*abs(b) - 567880*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^12*a^3*b^9*c^5*d^3*abs(b) - 592200*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^12*a^4*b^8*c^4*d^4*abs(b) - 567880*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^12*a^5*b^7*c^3*d^5*abs(b) - 393760*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^12*a^6*b^6*c^2*d^6*abs(b) - 26040*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^1
2*a^7*b^5*c*d^7*abs(b) + 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^8
*b^4*d^8*abs(b) - 1620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*b^10*c^7*a
bs(b) + 11100*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a*b^9*c^6*d*abs(b)
+ 203620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^8*c^5*d^2*abs(b) +
350100*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^3*b^7*c^4*d^3*abs(b) +
350100*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^4*b^6*c^3*d^4*abs(b) + 2
03620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^5*b^5*c^2*d^5*abs(b) + 11
100*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^6*b^4*c*d^6*abs(b) - 1620*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^7*b^3*d^7*abs(b) + 405*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*b^8*c^6*abs(b) - 2970*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a*b^7*c^5*d*abs(b) - 63765*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^2*b^6*c^4*d^2*abs(b) - 97740*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^3*b^5*c^3*d^3*abs(b) - 63765*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^4*b^4*c^2*d^4*abs(b) - 2970*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^5*b^3*c*d^5*abs(b) + 405*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^16*a^6*b^2*d^6*abs(b) - 45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^18*b^6*c^5*abs(b) + 375*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^18*a*b^5*c^4*d*abs(b) + 9270*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^18*a^2*b^4*c^3*d^2*abs(b) + 9270*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^18*a^3*b^3*c^2*d^3*abs(b) + 375*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
18*a^4*b^2*c*d^4*abs(b) - 45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^5*
b*d^5*abs(b))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^5*a^2*c^2))/b

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maple [B]  time = 0.02, size = 1146, normalized size = 2.82 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (45 \sqrt {b d}\, a^{5} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-375 \sqrt {b d}\, a^{4} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+2250 \sqrt {b d}\, a^{3} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+2250 \sqrt {b d}\, a^{2} b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-3840 \sqrt {a c}\, a^{2} b^{3} c^{2} d^{3} x^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-375 \sqrt {b d}\, a \,b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+45 \sqrt {b d}\, b^{5} c^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-90 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} d^{4} x^{4}+720 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}+7508 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+720 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}-90 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+60 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}+5156 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+5156 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}+60 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+5792 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+1488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c^{3} d x +2016 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} c^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2*(45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2))/x)*x^5*a^5*d^5*(b*d)^(1/2)-375*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*
x^5*a^4*b*c*d^4*(b*d)^(1/2)+2250*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a
^3*b^2*c^2*d^3*(b*d)^(1/2)+2250*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a^
2*b^3*c^3*d^2*(b*d)^(1/2)-375*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*a*b^
4*c^4*d*(b*d)^(1/2)+45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)*x^5*b^5*c^5*(b*
d)^(1/2)-3840*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*x^5*a^2*b^3*
c^2*d^3*(a*c)^(1/2)-90*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^4*a^4*d^4+720*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^4*a^3*b*c*d^3+7508*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c
)^(1/2)*x^4*a^2*b^2*c^2*d^2+720*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^4*a*b^3*c^3*d-90*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^4*b^4*c^4+60*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)*(a*c)^(1/2)*x^3*a^4*c*d^3+5156*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^3*a^3*b*c^2*d^2+515
6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^3*a^2*b^2*c^3*d+60*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*(b*d)^(1/2)*(a*c)^(1/2)*x^3*a*b^3*c^4+1488*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^2*a^4*c^
2*d^2+5792*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^2*a^3*b*c^3*d+1488*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x^2*a^2*b^2*c^4+2016*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)
*x*a^4*c^3*d+2016*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x*a^3*b*c^4+768*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^4*c^4)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(b*d)^(1/2)/(a*c)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**6,x)

[Out]

Integral((a + b*x)**(5/2)*(c + d*x)**(5/2)/x**6, x)

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