3.238 \(\int (a+\frac{b}{x})^{5/2} (c+\frac{d}{x})^3 \, dx\)

Optimal. Leaf size=198 \[ -\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (-10 a^2 d^2+135 a b c d+469 b^2 c^2\right )+\frac{5 b d (10 a d+89 b c)}{x}\right )}{315 b^2}+a^{3/2} c^2 (6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{1}{3} c^2 \left (a+\frac{b}{x}\right )^{3/2} (6 a d+5 b c)-a c^2 \sqrt{a+\frac{b}{x}} (6 a d+5 b c)+x \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2 \]

[Out]

-(a*c^2*(5*b*c + 6*a*d)*Sqrt[a + b/x]) - (c^2*(5*b*c + 6*a*d)*(a + b/x)^(3/2))/3 - (11*d*(a + b/x)^(5/2)*(c +
d/x)^2)/9 - (d*(a + b/x)^(5/2)*(2*(469*b^2*c^2 + 135*a*b*c*d - 10*a^2*d^2) + (5*b*d*(89*b*c + 10*a*d))/x))/(31
5*b^2) + (a + b/x)^(5/2)*(c + d/x)^3*x + a^(3/2)*c^2*(5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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Rubi [A]  time = 0.158298, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {375, 97, 153, 147, 50, 63, 208} \[ -\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (-10 a^2 d^2+135 a b c d+469 b^2 c^2\right )+\frac{5 b d (10 a d+89 b c)}{x}\right )}{315 b^2}+a^{3/2} c^2 (6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{1}{3} c^2 \left (a+\frac{b}{x}\right )^{3/2} (6 a d+5 b c)-a c^2 \sqrt{a+\frac{b}{x}} (6 a d+5 b c)+x \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*(c + d/x)^3,x]

[Out]

-(a*c^2*(5*b*c + 6*a*d)*Sqrt[a + b/x]) - (c^2*(5*b*c + 6*a*d)*(a + b/x)^(3/2))/3 - (11*d*(a + b/x)^(5/2)*(c +
d/x)^2)/9 - (d*(a + b/x)^(5/2)*(2*(469*b^2*c^2 + 135*a*b*c*d - 10*a^2*d^2) + (5*b*d*(89*b*c + 10*a*d))/x))/(31
5*b^2) + (a + b/x)^(5/2)*(c + d/x)^3*x + a^(3/2)*c^2*(5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

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 153

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] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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 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]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2} (c+d x)^3}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (c+d x)^2 \left (\frac{1}{2} (5 b c+6 a d)+\frac{11 b d x}{2}\right )}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2+\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x-\frac{2 \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (c+d x) \left (\frac{9}{4} b c (5 b c+6 a d)+\frac{1}{4} b d (89 b c+10 a d) x\right )}{x} \, dx,x,\frac{1}{x}\right )}{9 b}\\ &=-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2-\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (469 b^2 c^2+135 a b c d-10 a^2 d^2\right )+\frac{5 b d (89 b c+10 a d)}{x}\right )}{315 b^2}+\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x-\frac{1}{2} \left (c^2 (5 b c+6 a d)\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3} c^2 (5 b c+6 a d) \left (a+\frac{b}{x}\right )^{3/2}-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2-\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (469 b^2 c^2+135 a b c d-10 a^2 d^2\right )+\frac{5 b d (89 b c+10 a d)}{x}\right )}{315 b^2}+\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x-\frac{1}{2} \left (a c^2 (5 b c+6 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-a c^2 (5 b c+6 a d) \sqrt{a+\frac{b}{x}}-\frac{1}{3} c^2 (5 b c+6 a d) \left (a+\frac{b}{x}\right )^{3/2}-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2-\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (469 b^2 c^2+135 a b c d-10 a^2 d^2\right )+\frac{5 b d (89 b c+10 a d)}{x}\right )}{315 b^2}+\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x-\frac{1}{2} \left (a^2 c^2 (5 b c+6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-a c^2 (5 b c+6 a d) \sqrt{a+\frac{b}{x}}-\frac{1}{3} c^2 (5 b c+6 a d) \left (a+\frac{b}{x}\right )^{3/2}-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2-\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (469 b^2 c^2+135 a b c d-10 a^2 d^2\right )+\frac{5 b d (89 b c+10 a d)}{x}\right )}{315 b^2}+\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x-\frac{\left (a^2 c^2 (5 b c+6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{b}\\ &=-a c^2 (5 b c+6 a d) \sqrt{a+\frac{b}{x}}-\frac{1}{3} c^2 (5 b c+6 a d) \left (a+\frac{b}{x}\right )^{3/2}-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2-\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (469 b^2 c^2+135 a b c d-10 a^2 d^2\right )+\frac{5 b d (89 b c+10 a d)}{x}\right )}{315 b^2}+\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3 x+a^{3/2} c^2 (5 b c+6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [A]  time = 0.212277, size = 201, normalized size = 1.02 \[ \frac{\sqrt{a+\frac{b}{x}} \left (-3 a^2 b^2 x^2 \left (966 c^2 d x^2-105 c^3 x^3+270 c d^2 x+50 d^3\right )-10 a^3 b d^2 x^3 (27 c x+d)+20 a^4 d^3 x^4-2 a b^3 x \left (693 c^2 d x^2+735 c^3 x^3+405 c d^2 x+95 d^3\right )-2 b^4 \left (189 c^2 d x^2+105 c^3 x^3+135 c d^2 x+35 d^3\right )\right )}{315 b^2 x^4}+a^{3/2} c^2 (6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*(c + d/x)^3,x]

[Out]

(Sqrt[a + b/x]*(20*a^4*d^3*x^4 - 10*a^3*b*d^2*x^3*(d + 27*c*x) - 3*a^2*b^2*x^2*(50*d^3 + 270*c*d^2*x + 966*c^2
*d*x^2 - 105*c^3*x^3) - 2*b^4*(35*d^3 + 135*c*d^2*x + 189*c^2*d*x^2 + 105*c^3*x^3) - 2*a*b^3*x*(95*d^3 + 405*c
*d^2*x + 693*c^2*d*x^2 + 735*c^3*x^3)))/(315*b^2*x^4) + a^(3/2)*c^2*(5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt
[a]]

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Maple [B]  time = 0.013, size = 457, normalized size = 2.3 \begin{align*}{\frac{1}{630\,{b}^{2}{x}^{5}}\sqrt{{\frac{ax+b}{x}}} \left ( 3780\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{6}b{c}^{2}d+3150\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{6}{b}^{2}{c}^{3}+40\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}{x}^{3}{d}^{3}-3780\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{x}^{4}b{c}^{2}d-540\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{x}^{3}bc{d}^{2}-2520\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}{x}^{4}{b}^{2}{c}^{3}+1890\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{6}{a}^{3}{b}^{2}{c}^{2}d+1575\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{6}{a}^{2}{b}^{3}{c}^{3}-60\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{x}^{2}b{d}^{3}-2016\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}{x}^{3}{b}^{2}{c}^{2}d-1080\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}{x}^{2}{b}^{2}c{d}^{2}-420\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}{x}^{3}{b}^{3}{c}^{3}-240\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}x{b}^{2}{d}^{3}-756\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}{x}^{2}{b}^{3}{c}^{2}d-540\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}x{b}^{3}c{d}^{2}-140\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)*(c+d/x)^3,x)

[Out]

1/630*((a*x+b)/x)^(1/2)*(3780*(a*x^2+b*x)^(1/2)*a^(7/2)*x^6*b*c^2*d+3150*(a*x^2+b*x)^(1/2)*a^(5/2)*x^6*b^2*c^3
+40*(a*x^2+b*x)^(3/2)*a^(7/2)*x^3*d^3-3780*(a*x^2+b*x)^(3/2)*a^(5/2)*x^4*b*c^2*d-540*(a*x^2+b*x)^(3/2)*a^(5/2)
*x^3*b*c*d^2-2520*(a*x^2+b*x)^(3/2)*a^(3/2)*x^4*b^2*c^3+1890*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1
/2))*x^6*a^3*b^2*c^2*d+1575*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^6*a^2*b^3*c^3-60*(a*x^2+b*
x)^(3/2)*a^(5/2)*x^2*b*d^3-2016*(a*x^2+b*x)^(3/2)*a^(3/2)*x^3*b^2*c^2*d-1080*(a*x^2+b*x)^(3/2)*a^(3/2)*x^2*b^2
*c*d^2-420*(a*x^2+b*x)^(3/2)*a^(1/2)*x^3*b^3*c^3-240*(a*x^2+b*x)^(3/2)*a^(3/2)*x*b^2*d^3-756*(a*x^2+b*x)^(3/2)
*a^(1/2)*x^2*b^3*c^2*d-540*(a*x^2+b*x)^(3/2)*a^(1/2)*x*b^3*c*d^2-140*(a*x^2+b*x)^(3/2)*a^(1/2)*b^3*d^3)/x^5/b^
2/((a*x+b)*x)^(1/2)/a^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*(c+d/x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37091, size = 1115, normalized size = 5.63 \begin{align*} \left [\frac{315 \,{\left (5 \, a b^{3} c^{3} + 6 \, a^{2} b^{2} c^{2} d\right )} \sqrt{a} x^{4} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (315 \, a^{2} b^{2} c^{3} x^{5} - 70 \, b^{4} d^{3} - 2 \,{\left (735 \, a b^{3} c^{3} + 1449 \, a^{2} b^{2} c^{2} d + 135 \, a^{3} b c d^{2} - 10 \, a^{4} d^{3}\right )} x^{4} - 2 \,{\left (105 \, b^{4} c^{3} + 693 \, a b^{3} c^{2} d + 405 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{3} - 6 \,{\left (63 \, b^{4} c^{2} d + 135 \, a b^{3} c d^{2} + 25 \, a^{2} b^{2} d^{3}\right )} x^{2} - 10 \,{\left (27 \, b^{4} c d^{2} + 19 \, a b^{3} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{630 \, b^{2} x^{4}}, -\frac{315 \,{\left (5 \, a b^{3} c^{3} + 6 \, a^{2} b^{2} c^{2} d\right )} \sqrt{-a} x^{4} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (315 \, a^{2} b^{2} c^{3} x^{5} - 70 \, b^{4} d^{3} - 2 \,{\left (735 \, a b^{3} c^{3} + 1449 \, a^{2} b^{2} c^{2} d + 135 \, a^{3} b c d^{2} - 10 \, a^{4} d^{3}\right )} x^{4} - 2 \,{\left (105 \, b^{4} c^{3} + 693 \, a b^{3} c^{2} d + 405 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{3} - 6 \,{\left (63 \, b^{4} c^{2} d + 135 \, a b^{3} c d^{2} + 25 \, a^{2} b^{2} d^{3}\right )} x^{2} - 10 \,{\left (27 \, b^{4} c d^{2} + 19 \, a b^{3} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{315 \, b^{2} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*(c+d/x)^3,x, algorithm="fricas")

[Out]

[1/630*(315*(5*a*b^3*c^3 + 6*a^2*b^2*c^2*d)*sqrt(a)*x^4*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(31
5*a^2*b^2*c^3*x^5 - 70*b^4*d^3 - 2*(735*a*b^3*c^3 + 1449*a^2*b^2*c^2*d + 135*a^3*b*c*d^2 - 10*a^4*d^3)*x^4 - 2
*(105*b^4*c^3 + 693*a*b^3*c^2*d + 405*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^3 - 6*(63*b^4*c^2*d + 135*a*b^3*c*d^2 + 2
5*a^2*b^2*d^3)*x^2 - 10*(27*b^4*c*d^2 + 19*a*b^3*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^4), -1/315*(315*(5*a*b^3*c^
3 + 6*a^2*b^2*c^2*d)*sqrt(-a)*x^4*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (315*a^2*b^2*c^3*x^5 - 70*b^4*d^3 - 2
*(735*a*b^3*c^3 + 1449*a^2*b^2*c^2*d + 135*a^3*b*c*d^2 - 10*a^4*d^3)*x^4 - 2*(105*b^4*c^3 + 693*a*b^3*c^2*d +
405*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^3 - 6*(63*b^4*c^2*d + 135*a*b^3*c*d^2 + 25*a^2*b^2*d^3)*x^2 - 10*(27*b^4*c*
d^2 + 19*a*b^3*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^4)]

________________________________________________________________________________________

Sympy [A]  time = 91.14, size = 5513, normalized size = 27.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(5/2)*(c+d/x)**3,x)

[Out]

32*a**(29/2)*b**(27/2)*d**3*x**10*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19
/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*
a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 176*a**(27/2)*b**(29/2)*d**3*x**9*sqrt(a*x/b + 1)/(
315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/
2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(
9/2)) + 396*a**(25/2)*b**(31/2)*d**3*x**8*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**1
6*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2)
+ 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 462*a**(23/2)*b**(33/2)*d**3*x**7*sqrt(a*x/
b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300
*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b*
*21*x**(9/2)) + 210*a**(21/2)*b**(35/2)*d**3*x**6*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19
/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x
**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 32*a**(21/2)*b**(11/2)*d**3*x**6*sq
rt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*
a**(7/2)*b**10*x**(7/2)) - 378*a**(19/2)*b**(37/2)*d**3*x**5*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) +
1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13
/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 48*a**(19/2)*b**(13/2)*c
*d**2*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x*
*(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 80*a**(19/2)*b**(13/2)*d**3*x**5*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x
**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 1134*a**
(17/2)*b**(39/2)*d**3*x**4*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4
725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/
2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 120*a**(17/2)*b**(15/2)*c*d**2*x**5*sqrt(a*x/b + 1)/(105*a
**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7
/2)) - 60*a**(17/2)*b**(15/2)*d**3*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**
(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 1494*a**(15/2)*b**(41/2)*d**3*x**3*sqrt(a
*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6
300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)
*b**21*x**(9/2)) - 90*a**(15/2)*b**(17/2)*c*d**2*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(
11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 80*a**(15/2)*b**(17/2)*d**3
*x**3*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2
) + 105*a**(7/2)*b**10*x**(7/2)) + 4*a**(15/2)*b**(3/2)*d**3*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) +
15*a**(5/2)*b**4*x**(5/2)) - 1098*a**(13/2)*b**(43/2)*d**3*x**2*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2
) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a*
*(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 120*a**(13/2)*b**(19
/2)*c*d**2*x**3*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b*
*9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 200*a**(13/2)*b**(19/2)*d**3*x**2*sqrt(a*x/b + 1)/(105*a**(13/2)*
b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 24
*a**(13/2)*b**(5/2)*c*d**2*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 2*a*
*(13/2)*b**(5/2)*d**3*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 430*a**(1
1/2)*b**(45/2)*d**3*x*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a
**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b*
*20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 300*a**(11/2)*b**(21/2)*c*d**2*x**2*sqrt(a*x/b + 1)/(105*a**(13
/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2))
- 192*a**(11/2)*b**(21/2)*d**3*x*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2)
+ 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 12*a**(11/2)*b**(7/2)*c**2*d*x**3*sqrt(a*x/b + 1
)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 12*a**(11/2)*b**(7/2)*c*d**2*x**2*sqrt(a*x/b + 1)/
(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**(11/2)*b**(7/2)*d**3*x*sqrt(a*x/b + 1)/(15*a**(
7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 70*a**(9/2)*b**(47/2)*d**3*sqrt(a*x/b + 1)/(315*a**(21/2)*b*
*15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/
2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 288*a**(
9/2)*b**(23/2)*c*d**2*x*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**
(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 60*a**(9/2)*b**(23/2)*d**3*sqrt(a*x/b + 1)/(105*a**(13/2)
*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 6
*a**(9/2)*b**(9/2)*c**2*d*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 48*a*
*(9/2)*b**(9/2)*c*d**2*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**(9/2)*
b**(9/2)*d**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 90*a**(7/2)*b**(25/2)*
c*d**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/
2) + 105*a**(7/2)*b**10*x**(7/2)) - 24*a**(7/2)*b**(11/2)*c**2*d*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2)
+ 15*a**(5/2)*b**4*x**(5/2)) - 36*a**(7/2)*b**(11/2)*c*d**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a*
*(5/2)*b**4*x**(5/2)) - 18*a**(5/2)*b**(13/2)*c**2*d*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*
b**4*x**(5/2)) + a**(3/2)*b*c**3*asinh(sqrt(a)*sqrt(x)/sqrt(b)) - 32*a**15*b**13*d**3*x**(21/2)/(315*a**(21/2)
*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(
15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 192*a
**14*b**14*d**3*x**(19/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**
17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2
) + 315*a**(9/2)*b**21*x**(9/2)) - 480*a**13*b**15*d**3*x**(17/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19
/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x
**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 640*a**12*b**16*d**3*x**(15/2)/(315
*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*
b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2
)) - 480*a**11*b**17*d**3*x**(13/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**
(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**2
0*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 32*a**11*b**5*d**3*x**(13/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*
a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 192*a**10*b**18*d**3*x*
*(11/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 630
0*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b
**21*x**(9/2)) + 48*a**10*b**6*c*d**2*x**(13/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) +
315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 96*a**10*b**6*d**3*x**(11/2)/(105*a**(13/2)*b**7*
x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 32*a**9
*b**19*d**3*x**(9/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x*
*(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 3
15*a**(9/2)*b**21*x**(9/2)) + 144*a**9*b**7*c*d**2*x**(11/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**
8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 96*a**9*b**7*d**3*x**(9/2)/(105*a**(
13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)
) + 144*a**8*b**8*c*d**2*x**(9/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*
b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 32*a**8*b**8*d**3*x**(7/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*
a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 4*a**8*b*d**3*x**(7/2)/
(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 48*a**7*b**9*c*d**2*x**(7/2)/(105*a**(13/2)*b**7*x**
(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 24*a**7*b*
*2*c*d**2*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**7*b**2*d**3*x**(5/2)/(15*a**
(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 12*a**6*b**3*c**2*d*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 1
5*a**(5/2)*b**4*x**(5/2)) - 24*a**6*b**3*c*d**2*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2
)) - 12*a**5*b**4*c**2*d*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**3*c**2*d*atan
(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + a**2*sqrt(b)*c**3*sqrt(x)*sqrt(a*x/b + 1) - 4*a**2*b*c**3*atan(sqrt(a + b/
x)/sqrt(-a))/sqrt(-a) - 6*a**2*c**2*d*sqrt(a + b/x) + 3*a**2*c*d**2*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a +
b/x)**(3/2)/(3*b), True)) - 4*a*b*c**3*sqrt(a + b/x) + 6*a*b*c**2*d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a
+ b/x)**(3/2)/(3*b), True)) + b**2*c**3*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True))

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*(c+d/x)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError