3.7.72 \(\int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx\) [672]

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

[Out]

-1/8*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/c/x^4-1/5*(b*x+a)^(5/2)*(d*x+c)^(5/2)/x^5-1/128*(a*d+b*c)*(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)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(
1/2))/a^(5/2)/c^(5/2)+2*b^(5/2)*d^(5/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))-1/192*(3*a^3*d^3-
19*a^2*b*c*d^2+109*a*b^2*c^2*d+3*b^3*c^3)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/c^2/x^2-1/48*(-3*a^2*d^2+16*a*b*c*d+3*
b^2*c^2)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/c^2/x^3+1/128*(-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)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2/x

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Rubi [A]
time = 0.29, 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 = {99, 154, 163, 65, 223, 212, 95, 214} \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 (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} \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 65

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 95

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 99

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 154

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] && ILtQ[m, -1] && GtQ[n, 0]

Rule 163

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 212

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

Rule 214

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

Rule 223

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 ) \text {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 ) \text {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 ) \text {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 = 1.05, size = 330, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-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 \left (372 c^2+1289 c d x+1877 d^2 x^2\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 )+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 )\right )}{1920 a^2 c^2 x^5}-\frac {\left (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\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}}+2 b^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \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) - ((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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(984\) vs. \(2(350)=700\).
time = 0.08, size = 985, normalized size = 2.43

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{5} d^{5} x^{5} \sqrt {b d}-375 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b c \,d^{4} x^{5} \sqrt {b d}+2250 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{3} x^{5} \sqrt {b d}+2250 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d^{2} x^{5} \sqrt {b d}-375 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} d \,x^{5} \sqrt {b d}+45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{5} c^{5} x^{5} \sqrt {b d}-3840 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{2} d^{3} x^{5} \sqrt {a c}-90 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}+720 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}+7508 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}+720 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}-90 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+60 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}+5156 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}+5156 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}+60 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}+5792 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}+1488 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x +2016 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{4} x +768 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{4}\right )}{3840 a^{2} c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{5} \sqrt {b d}\, \sqrt {a c}}\)	\(985\)

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,method=_RETURNVERBOSE)

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

________________________________________________________________________________________

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 detail

________________________________________________________________________________________

Fricas [A]
time = 13.93, size = 1849, normalized size = 4.55 \begin {gather*} \left [\frac {3840 \, \sqrt {b d} a^{3} b^{2} c^{3} d^{2} x^{5} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 15 \, {\left (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}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (384 \, a^{5} c^{5} - {\left (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}\right )} x^{4} + 2 \, {\left (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}\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c^{5} + 362 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 1008 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{3} c^{3} x^{5}}, -\frac {7680 \, \sqrt {-b d} a^{3} b^{2} c^{3} d^{2} x^{5} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 15 \, {\left (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}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (384 \, a^{5} c^{5} - {\left (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}\right )} x^{4} + 2 \, {\left (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}\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c^{5} + 362 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 1008 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{3} c^{3} x^{5}}, \frac {1920 \, \sqrt {b d} a^{3} b^{2} c^{3} d^{2} x^{5} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 15 \, {\left (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}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (384 \, a^{5} c^{5} - {\left (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}\right )} x^{4} + 2 \, {\left (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}\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c^{5} + 362 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 1008 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{3} c^{3} x^{5}}, -\frac {3840 \, \sqrt {-b d} a^{3} b^{2} c^{3} d^{2} x^{5} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 15 \, {\left (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}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (384 \, a^{5} c^{5} - {\left (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}\right )} x^{4} + 2 \, {\left (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}\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c^{5} + 362 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 1008 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{3} c^{3} x^{5}}\right ] \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="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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5983 vs. \(2 (350) = 700\).
time = 4.17, size = 5983, normalized size = 14.74 \begin {gather*} \text {Too large to display} \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="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(...

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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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