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

Optimal. Leaf size=380 \[ -\frac {5 \sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{96 c^2 x^2}-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{192 a c^2 x}+\frac {5 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )}{64 a c^2}+\frac {5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}-\frac {5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{24 c x^3} \]

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Rubi [A]  time = 0.44, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {97, 149, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {5 \sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{96 c^2 x^2}-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{192 a c^2 x}+\frac {5 d \sqrt {a+b x} \sqrt {c+d x} \left (19 a^2 b c d^2-a^3 d^3+45 a b^2 c^2 d+b^3 c^3\right )}{64 a c^2}+\frac {5 \left (-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (a d+b c) \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}}{4 x^4}-\frac {5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{24 c x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d*(b^3*c^3 + 45*a*b^2*c^2*d + 19*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a*c^2) - (5*(3*b*c
+ a*d)*(b^2*c^2 + 24*a*b*c*d - a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*a*c^2*x) - (5*(3*b^2*c^2 + 14*a*b
*c*d - a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(96*c^2*x^2) - (5*(b*c + a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/
(24*c*x^3) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(4*x^4) + (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 2
0*a^3*b*c*d^3 + a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(3/2)) + 5*b^
(3/2)*d^(3/2)*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[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 154

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

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^5} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {1}{4} \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^4} \, dx\\ &=-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {5}{4} \left (3 b^2 c^2+14 a b c d-a^2 d^2\right )+\frac {5}{2} b d (7 b c+a d) x\right )}{x^3} \, dx}{12 c}\\ &=-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {5}{8} (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right )+\frac {5}{4} b d \left (31 b^2 c^2+18 a b c d-a^2 d^2\right ) x\right )}{x^2 \sqrt {a+b x}} \, dx}{24 c^2}\\ &=-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {\sqrt {c+d x} \left (-\frac {15}{16} \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )+\frac {15}{8} b d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) x\right )}{x \sqrt {a+b x}} \, dx}{24 a c^2}\\ &=\frac {5 d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2}-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {\int \frac {-\frac {15}{16} b c \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )+60 a b^3 c^2 d^2 (b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a b c^2}\\ &=\frac {5 d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2}-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {1}{2} \left (5 b^2 d^2 (b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c}\\ &=\frac {5 d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2}-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\left (5 b d^2 (b c+a d)\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 (5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^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 )}{64 a c}\\ &=\frac {5 d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2}-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{3/2}}+\left (5 b d^2 (b c+a d)\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 {5 d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2}-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (b c+a d) \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 = 4.19, size = 309, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (136 c^2+452 c d x+601 d^2 x^2\right )+a b^2 c x^2 \left (118 c^2+601 c d x-192 d^2 x^2\right )+15 b^3 c^3 x^3\right )}{192 a c x^4}+\frac {5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{3/2}}+\frac {5 d^{3/2} (b c-a d)^{3/2} (a d+b c) \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{(c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 + a*b^2*c*x^2*(118*c^2 + 601*c*d*x - 192*d^2*x^2) + a^2*b*
c*x*(136*c^2 + 452*c*d*x + 601*d^2*x^2) + a^3*(48*c^3 + 136*c^2*d*x + 118*c*d^2*x^2 + 15*d^3*x^3)))/(a*c*x^4)
+ (5*d^(3/2)*(b*c - a*d)^(3/2)*(b*c + a*d)*((b*(c + d*x))/(b*c - a*d))^(3/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[b*c - a*d]])/(c + d*x)^(3/2) + (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^
4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(3/2))

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IntegrateAlgebraic [B]  time = 0.95, size = 1029, normalized size = 2.71 \begin {gather*} \frac {-\frac {15 b d^4 (c+d x)^{9/2} a^7}{(a+b x)^{9/2}}+\frac {15 d^5 (c+d x)^{7/2} a^7}{(a+b x)^{7/2}}-\frac {660 b^2 c d^3 (c+d x)^{9/2} a^6}{(a+b x)^{9/2}}+\frac {395 b c d^4 (c+d x)^{7/2} a^6}{(a+b x)^{7/2}}+\frac {73 c d^5 (c+d x)^{5/2} a^6}{(a+b x)^{5/2}}+\frac {390 b^3 c^2 d^2 (c+d x)^{9/2} a^5}{(a+b x)^{9/2}}+\frac {2030 b^2 c^2 d^3 (c+d x)^{7/2} a^5}{(a+b x)^{7/2}}-\frac {1405 b c^2 d^4 (c+d x)^{5/2} a^5}{(a+b x)^{5/2}}-\frac {55 c^2 d^5 (c+d x)^{3/2} a^5}{(a+b x)^{3/2}}+\frac {300 b^4 c^3 d (c+d x)^{9/2} a^4}{(a+b x)^{9/2}}-\frac {1410 b^3 c^3 d^2 (c+d x)^{7/2} a^4}{(a+b x)^{7/2}}-\frac {1910 b^2 c^3 d^3 (c+d x)^{5/2} a^4}{(a+b x)^{5/2}}+\frac {1085 b c^3 d^4 (c+d x)^{3/2} a^4}{(a+b x)^{3/2}}+\frac {15 c^3 d^5 \sqrt {c+d x} a^4}{\sqrt {a+b x}}-\frac {15 b^5 c^4 (c+d x)^{9/2} a^3}{(a+b x)^{9/2}}-\frac {1085 b^4 c^4 d (c+d x)^{7/2} a^3}{(a+b x)^{7/2}}+\frac {1910 b^3 c^4 d^2 (c+d x)^{5/2} a^3}{(a+b x)^{5/2}}+\frac {1410 b^2 c^4 d^3 (c+d x)^{3/2} a^3}{(a+b x)^{3/2}}-\frac {300 b c^4 d^4 \sqrt {c+d x} a^3}{\sqrt {a+b x}}+\frac {55 b^5 c^5 (c+d x)^{7/2} a^2}{(a+b x)^{7/2}}+\frac {1405 b^4 c^5 d (c+d x)^{5/2} a^2}{(a+b x)^{5/2}}-\frac {2030 b^3 c^5 d^2 (c+d x)^{3/2} a^2}{(a+b x)^{3/2}}-\frac {390 b^2 c^5 d^3 \sqrt {c+d x} a^2}{\sqrt {a+b x}}-\frac {73 b^5 c^6 (c+d x)^{5/2} a}{(a+b x)^{5/2}}-\frac {395 b^4 c^6 d (c+d x)^{3/2} a}{(a+b x)^{3/2}}+\frac {660 b^3 c^6 d^2 \sqrt {c+d x} a}{\sqrt {a+b x}}-\frac {15 b^5 c^7 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {15 b^4 c^7 d \sqrt {c+d x}}{\sqrt {a+b x}}}{192 a c \left (c-\frac {a (c+d x)}{a+b x}\right )^4 \left (\frac {b (c+d x)}{a+b x}-d\right )}+\frac {5 \left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{3/2} c^{3/2}}+5 \left (c d^{3/2} b^{5/2}+a d^{5/2} b^{3/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((15*b^4*c^7*d*Sqrt[c + d*x])/Sqrt[a + b*x] + (660*a*b^3*c^6*d^2*Sqrt[c + d*x])/Sqrt[a + b*x] - (390*a^2*b^2*c
^5*d^3*Sqrt[c + d*x])/Sqrt[a + b*x] - (300*a^3*b*c^4*d^4*Sqrt[c + d*x])/Sqrt[a + b*x] + (15*a^4*c^3*d^5*Sqrt[c
+ d*x])/Sqrt[a + b*x] - (15*b^5*c^7*(c + d*x)^(3/2))/(a + b*x)^(3/2) - (395*a*b^4*c^6*d*(c + d*x)^(3/2))/(a +
b*x)^(3/2) - (2030*a^2*b^3*c^5*d^2*(c + d*x)^(3/2))/(a + b*x)^(3/2) + (1410*a^3*b^2*c^4*d^3*(c + d*x)^(3/2))/
(a + b*x)^(3/2) + (1085*a^4*b*c^3*d^4*(c + d*x)^(3/2))/(a + b*x)^(3/2) - (55*a^5*c^2*d^5*(c + d*x)^(3/2))/(a +
b*x)^(3/2) - (73*a*b^5*c^6*(c + d*x)^(5/2))/(a + b*x)^(5/2) + (1405*a^2*b^4*c^5*d*(c + d*x)^(5/2))/(a + b*x)^
(5/2) + (1910*a^3*b^3*c^4*d^2*(c + d*x)^(5/2))/(a + b*x)^(5/2) - (1910*a^4*b^2*c^3*d^3*(c + d*x)^(5/2))/(a + b
*x)^(5/2) - (1405*a^5*b*c^2*d^4*(c + d*x)^(5/2))/(a + b*x)^(5/2) + (73*a^6*c*d^5*(c + d*x)^(5/2))/(a + b*x)^(5
/2) + (55*a^2*b^5*c^5*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (1085*a^3*b^4*c^4*d*(c + d*x)^(7/2))/(a + b*x)^(7/2)
- (1410*a^4*b^3*c^3*d^2*(c + d*x)^(7/2))/(a + b*x)^(7/2) + (2030*a^5*b^2*c^2*d^3*(c + d*x)^(7/2))/(a + b*x)^(7
/2) + (395*a^6*b*c*d^4*(c + d*x)^(7/2))/(a + b*x)^(7/2) + (15*a^7*d^5*(c + d*x)^(7/2))/(a + b*x)^(7/2) - (15*a
^3*b^5*c^4*(c + d*x)^(9/2))/(a + b*x)^(9/2) + (300*a^4*b^4*c^3*d*(c + d*x)^(9/2))/(a + b*x)^(9/2) + (390*a^5*b
^3*c^2*d^2*(c + d*x)^(9/2))/(a + b*x)^(9/2) - (660*a^6*b^2*c*d^3*(c + d*x)^(9/2))/(a + b*x)^(9/2) - (15*a^7*b*
d^4*(c + d*x)^(9/2))/(a + b*x)^(9/2))/(192*a*c*(c - (a*(c + d*x))/(a + b*x))^4*(-d + (b*(c + d*x))/(a + b*x)))
+ (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x
])/(Sqrt[c]*Sqrt[a + b*x])])/(64*a^(3/2)*c^(3/2)) + 5*(b^(5/2)*c*d^(3/2) + a*b^(3/2)*d^(5/2))*ArcTanh[(Sqrt[b]
*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])]

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fricas [A]  time = 28.12, size = 1633, normalized size = 4.30

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^5,x, algorithm="fricas")

[Out]

[1/768*(960*(a^2*b^2*c^3*d + a^3*b*c^2*d^2)*sqrt(b*d)*x^4*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*(b^4*c^4 - 20*a*
b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*c)*x^4*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*(192*a^2*b^2*c^2*d^2*x^4 - 48*a^4*c^4 - (15*a*b^3*c^4 + 601*a^2*b^2*c^3*d + 601*a^3*b*c^2*d^2 + 15*a^4
*c*d^3)*x^3 - 2*(59*a^2*b^2*c^4 + 226*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 136*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(
b*x + a)*sqrt(d*x + c))/(a^2*c^2*x^4), -1/768*(1920*(a^2*b^2*c^3*d + a^3*b*c^2*d^2)*sqrt(-b*d)*x^4*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*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*c)*x^4*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*(192*a^2*b^2*c^2*d^2*x^4 - 48*a^4*c^4 - (15*a*b^3*c^4 + 601*a^2*b^2*c^3*d + 601*a
^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(59*a^2*b^2*c^4 + 226*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 136*(a^3*b*c^4
+ a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^2*x^4), -1/384*(15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*
c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a*c)*x^4*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)) - 480*(a^2*b^2*c^3*d + a^3*b*c^2*d^2)*sqrt(b*d
)*x^4*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) - 2*(192*a^2*b^2*c^2*d^2*x^4 - 48*a^4*c^4 - (15*a*b^3*c^4 + 601*a^2*b^2*c
^3*d + 601*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(59*a^2*b^2*c^4 + 226*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 136
*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^2*x^4), -1/384*(15*(b^4*c^4 - 20*a*b^3*c^3*d -
90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a*c)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*s
qrt(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)) + 960*(a^2*b^2*c^3*d + a^3*b*c^2*d
^2)*sqrt(-b*d)*x^4*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)) - 2*(192*a^2*b^2*c^2*d^2*x^4 - 48*a^4*c^4 - (15*a*b^3*c^4 + 601*a^2*b^2*c^3*d +
601*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(59*a^2*b^2*c^4 + 226*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 136*(a^3*b
*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^2*x^4)]

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giac [B]  time = 11.97, size = 3937, normalized size = 10.36

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^5,x, algorithm="giac")

[Out]

1/192*(192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*b*d^2*abs(b) - 480*(sqrt(b*d)*b^2*c*d*abs(b) + sq
rt(b*d)*a*b*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*(sqrt(b*d)
*b^5*c^4*abs(b) - 20*sqrt(b*d)*a*b^4*c^3*d*abs(b) - 90*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) - 20*sqrt(b*d)*a^3*b^2
*c*d^3*abs(b) + sqrt(b*d)*a^4*b*d^4*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*b*c) - 2*(15*sqrt(b*d)*b^19*c^11*abs(b) +
481*sqrt(b*d)*a*b^18*c^10*d*abs(b) - 3787*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) + 11195*sqrt(b*d)*a^3*b^16*c^8*d^3
*abs(b) - 15898*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) + 7994*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) + 7994*sqrt(b*d)*a^
6*b^13*c^5*d^6*abs(b) - 15898*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 11195*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) - 37
87*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 481*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) + 15*sqrt(b*d)*a^11*b^8*d^11*abs(b
) - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) - 3446*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d*abs(b) + 15371*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^15*c^8*d^2*abs(b) - 18056*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^14*c^7*d^3*abs(b) - 8354*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^4*b^13*c^6*d^4*abs(b) + 29180*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^12*c^5*d^5*abs(b) - 8354*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^11*c^4*d^6*abs(b) - 18056*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7*abs(b) + 15371*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^8*b^9*c^2*d^8*abs(b) - 3446*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^8*c*d^9*abs(b) - 105*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^7*d^10*abs(b) + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^15*c^9*abs(b) + 10867*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^14*c^8*d*abs(b) - 19124*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^13*c^7*d^2*abs(b) - 10980*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^12*c^6*d^3*abs(b) + 18922*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^11*c^5*d^4*abs(b) + 18922*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^10*c^4*d^5*abs(b) - 10980*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^9*c^3*d^6*abs(b) - 19124*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^8*c^2*d^7*abs(b) + 10867*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^7*c*d^8*abs(b) + 315*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^6*d^9*abs(b) - 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^13*c^8*a
bs(b) - 19480*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^12*c^7*d*abs(b)
- 4460*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^11*c^6*d^2*abs(b) + 1
9288*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^10*c^5*d^3*abs(b) + 103
54*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^9*c^4*d^4*abs(b) + 19288*
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^8*c^3*d^5*abs(b) - 4460*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^7*c^2*d^6*abs(b) - 19480*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^6*c*d^7*abs(b) - 525*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^5*d^8*abs(b) + 525*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7*abs(b) + 21335*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^10*c^6*d*abs(b) + 33325*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^9*c^5*d^2*abs(b) + 23663*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^8*c^4*d^3*abs(b) + 23663*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^7*c^3*d^4*abs(b) + 33325*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^6*c^2*d^5*abs(b) + 21335*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^5*c*d^6*abs(b) + 525*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^4*d^7*abs(b) - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^10*b^9*c^6*abs(b) - 14206*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^8*c^5*d*abs(b) - 33013*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^
7*c^4*d^2*abs(b) - 38052*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^6*
c^3*d^3*abs(b) - 33013*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^5*c^
2*d^4*abs(b) - 14206*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^4*c*d^
5*abs(b) - 315*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^3*d^6*abs(b)
+ 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^7*c^5*abs(b) + 5301*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^6*c^4*d*abs(b) + 13794*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^5*c^3*d^2*abs(b) + 13794*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^4*c^2*d^3*abs(b) + 5301*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^3*c*d^4*abs(b) + 105*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^2*d^5*abs(b) - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*b^5*c^4*abs(b) - 852*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^4*c^3*d*abs(b) - 2106*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^3*c^2*d^2*abs(b) - 852*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^2*c*d^3*abs(b) - 15*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*d^4*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)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^4*a*c))/b

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maple [B]  time = 0.02, size = 962, normalized size = 2.53 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \sqrt {b d}\, a^{4} d^{4} x^{4} \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 )-300 \sqrt {b d}\, a^{3} b c \,d^{3} x^{4} \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 )-1350 \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2} x^{4} \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 )+960 \sqrt {a c}\, a^{2} b^{2} c \,d^{3} x^{4} \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 )-300 \sqrt {b d}\, a \,b^{3} c^{3} d \,x^{4} \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 )+960 \sqrt {a c}\, a \,b^{3} c^{2} d^{2} x^{4} \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 )+15 \sqrt {b d}\, b^{4} c^{4} x^{4} \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 )+384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{4}-30 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{3} x^{3}-1202 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}-1202 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{3} x^{3}-236 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-904 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-236 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{2} d x -272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a c \,x^{4}} \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^5,x)

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c*(15*(b*d)^(1/2)*a^4*d^4*x^4*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)-300*(b*d)^(1/2)*a^3*b*c*d^3*x^4*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)-1350*(b*d)^(1/2)*a^2*b^2*c^2*d^2*x^4*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)-300*(b*d)^(1/2)*a*b^3*c^3*d*x^4*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)+15*(b*d)^(1/2)*b^4*c^4*x^4*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)+960*(a*c)^(1/2)*a^2*b^2*c*d^3*x^4*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))+960*(a*c)^(1/2)*a*b^3*c^2*d^2*x^4*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))+384*(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^2*c*d^2-30*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*d^3*x^3-1202*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c
)^(1/2)*a^2*b*c*d^2*x^3-1202*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2*c^2*d*x^3-30*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*b^3*c^3*x^3-236*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*
(a*c)^(1/2)*a^3*c*d^2*x^2-904*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2-236*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2*c^3*x^2-272*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1
/2)*(a*c)^(1/2)*a^3*c^2*d*x-272*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*c^3*x-96*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c^3)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^4/(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^5,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^5} \,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^5,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^5, 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^{5}}\, 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**5,x)

[Out]

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

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