∖int∖left(a + b x∖right)5∕2∖left(c + d x∖right)3∖,dx

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

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

[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 + 1

35*a*b*c*d - 10*a^2*d^2) + (5*b*d*(89*b*c + 10*a*d))/x))/(315*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 = 0.499922, 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 \[ a^{3/2} c^2 (6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{d \left (a+\frac{b}{x}\right )^{5/2} \left (2 \left (-10 a^2 d^2+135 a b c d+469 b^2 c^2\right )+\frac{5 b d (10 a d+89 b c)}{x}\right )}{315 b^2}-\frac{1}{3} c^2 \left (a+\frac{b}{x}\right )^{3/2} (6 a d+5 b c)-a c^2 \sqrt{a+\frac{b}{x}} (6 a d+5 b c)+x \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3-\frac{11}{9} d \left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2 \]

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 + 1

35*a*b*c*d - 10*a^2*d^2) + (5*b*d*(89*b*c + 10*a*d))/x))/(315*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 in Sympy [A] time = 54.8412, size = 184, normalized size = 0.93 \[ a^{\frac{3}{2}} c^{2} \left (6 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - a c^{2} \sqrt{a + \frac{b}{x}} \left (6 a d + 5 b c\right ) - \frac{c^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (6 a d + 5 b c\right )}{3} - \frac{11 d \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \left (c + \frac{d}{x}\right )^{2}}{9} + x \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \left (c + \frac{d}{x}\right )^{3} + \frac{8 d \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \left (\frac{5 a^{2} d^{2}}{2} - \frac{135 a b c d}{4} - \frac{469 b^{2} c^{2}}{4} - \frac{5 b d \left (10 a d + 89 b c\right )}{8 x}\right )}{315 b^{2}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*c**2*(6*a*d + 5*b*c)*atanh(sqrt(a + b/x)/sqrt(a)) - a*c**2*sqrt(a + b/x

)*(6*a*d + 5*b*c) - c**2*(a + b/x)**(3/2)*(6*a*d + 5*b*c)/3 - 11*d*(a + b/x)**(5

/2)*(c + d/x)**2/9 + x*(a + b/x)**(5/2)*(c + d/x)**3 + 8*d*(a + b/x)**(5/2)*(5*a

**2*d**2/2 - 135*a*b*c*d/4 - 469*b**2*c**2/4 - 5*b*d*(10*a*d + 89*b*c)/(8*x))/(3

15*b**2)

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Mathematica [A] time = 0.354917, size = 212, normalized size = 1.07 \[ \frac{1}{2} a^{3/2} c^2 (6 a d+5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )+\frac{\sqrt{a+\frac{b}{x}} \left (20 a^4 d^3 x^4-10 a^3 b d^2 x^3 (27 c x+d)-3 a^2 b^2 x^2 \left (-105 c^3 x^3+966 c^2 d x^2+270 c d^2 x+50 d^3\right )-2 a b^3 x \left (735 c^3 x^3+693 c^2 d x^2+405 c d^2 x+95 d^3\right )-2 b^4 \left (105 c^3 x^3+189 c^2 d x^2+135 c d^2 x+35 d^3\right )\right )}{315 b^2 x^4} \]

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)*Log[b + 2*a*x +

2*Sqrt[a]*Sqrt[a + b/x]*x])/2

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Maple [B] time = 0.022, size = 434, normalized size = 2.2 \[{\frac{1}{630\,{x}^{5}{b}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 1890\,{a}^{5/2}{c}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) d{x}^{6}{b}^{2}+1575\,{a}^{3/2}{c}^{3}{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{6}+3780\,{a}^{3}{c}^{2}\sqrt{a{x}^{2}+bx}d{x}^{6}b+3150\,{a}^{2}{c}^{3}\sqrt{a{x}^{2}+bx}{x}^{6}{b}^{2}-3780\,{a}^{2}{c}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}d{x}^{4}b-2520\,a{c}^{3} \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{4}{b}^{2}+40\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{3}{a}^{3}{d}^{3}-540\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{3}{a}^{2}bc{d}^{2}-2016\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{3}a{b}^{2}{c}^{2}d-420\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{3}{b}^{3}{c}^{3}-60\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}{a}^{2}b{d}^{3}-1080\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}a{b}^{2}c{d}^{2}-756\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}{b}^{3}{c}^{2}d-240\, \left ( a{x}^{2}+bx \right ) ^{3/2}xa{b}^{2}{d}^{3}-540\, \left ( a{x}^{2}+bx \right ) ^{3/2}x{b}^{3}c{d}^{2}-140\, \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

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)/x^5/b^2*(1890*a^(5/2)*c^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^

(1/2)+2*a*x+b)/a^(1/2))*d*x^6*b^2+1575*a^(3/2)*c^3*b^3*ln(1/2*(2*(a*x^2+b*x)^(1/

2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^6+3780*a^3*c^2*(a*x^2+b*x)^(1/2)*d*x^6*b+3150*a^2

*c^3*(a*x^2+b*x)^(1/2)*x^6*b^2-3780*a^2*c^2*(a*x^2+b*x)^(3/2)*d*x^4*b-2520*a*c^3

*(a*x^2+b*x)^(3/2)*x^4*b^2+40*(a*x^2+b*x)^(3/2)*x^3*a^3*d^3-540*(a*x^2+b*x)^(3/2

)*x^3*a^2*b*c*d^2-2016*(a*x^2+b*x)^(3/2)*x^3*a*b^2*c^2*d-420*(a*x^2+b*x)^(3/2)*x

^3*b^3*c^3-60*(a*x^2+b*x)^(3/2)*x^2*a^2*b*d^3-1080*(a*x^2+b*x)^(3/2)*x^2*a*b^2*c

*d^2-756*(a*x^2+b*x)^(3/2)*x^2*b^3*c^2*d-240*(a*x^2+b*x)^(3/2)*x*a*b^2*d^3-540*(

a*x^2+b*x)^(3/2)*x*b^3*c*d^2-140*(a*x^2+b*x)^(3/2)*b^3*d^3)/(x*(a*x+b))^(1/2)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.261345, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (5 \, a b^{3} c^{3} + 6 \, a^{2} b^{2} c^{2} d\right )} \sqrt{a} x^{4} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (315 \, a^{2} b^{2} c^{3} x^{5} - 70 \, b^{4} d^{3} - 2 \,{\left (735 \, a b^{3} c^{3} + 1449 \, a^{2} b^{2} c^{2} d + 135 \, a^{3} b c d^{2} - 10 \, a^{4} d^{3}\right )} x^{4} - 2 \,{\left (105 \, b^{4} c^{3} + 693 \, a b^{3} c^{2} d + 405 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{3} - 6 \,{\left (63 \, b^{4} c^{2} d + 135 \, a b^{3} c d^{2} + 25 \, a^{2} b^{2} d^{3}\right )} x^{2} - 10 \,{\left (27 \, b^{4} c d^{2} + 19 \, a b^{3} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{630 \, b^{2} x^{4}}, \frac{315 \,{\left (5 \, a b^{3} c^{3} + 6 \, a^{2} b^{2} c^{2} d\right )} \sqrt{-a} x^{4} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (315 \, a^{2} b^{2} c^{3} x^{5} - 70 \, b^{4} d^{3} - 2 \,{\left (735 \, a b^{3} c^{3} + 1449 \, a^{2} b^{2} c^{2} d + 135 \, a^{3} b c d^{2} - 10 \, a^{4} d^{3}\right )} x^{4} - 2 \,{\left (105 \, b^{4} c^{3} + 693 \, a b^{3} c^{2} d + 405 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{3} - 6 \,{\left (63 \, b^{4} c^{2} d + 135 \, a b^{3} c d^{2} + 25 \, a^{2} b^{2} d^{3}\right )} x^{2} - 10 \,{\left (27 \, b^{4} c d^{2} + 19 \, a b^{3} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{315 \, b^{2} x^{4}}\right ] \]

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*(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), 1/315*(315*(5*a*b^3*c^3 + 6*a^2*b^2*c^2*d)*sqrt(-a)*x^4*arctan(

sqrt((a*x + b)/x)/sqrt(-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 + 6

93*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)]

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Sympy [A] time = 82.497, size = 5557, normalized size = 28.07 \[ \text{result too large to display} \]

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**1

9*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 39

6*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**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)) + 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) + 1

890*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*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)) - 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) + 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**(

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**(2

1/2)*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)) - 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**(1

3/2) + 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*sqr

t(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) + 1

5*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*sqr

t(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))

- 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)) + 6*a**(5/2)*c**2*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 5*a**(3/2)*b*c**3*asin

h(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) + 63

00*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) + 63

00*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**11*b**5*d**3*x**(13/2)/(10

5*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) + 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**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) + 1

890*a**(11/2)*b**20*x**(11/2) + 315*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)/(10

5*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) + 1

05*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) + 15*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*sqrt(x)/(sq

rt(b)*sqrt(a*x/b + 1)) + a**2*sqrt(b)*c**3*sqrt(x)*sqrt(a*x/b + 1) - 4*a**2*sqrt

(b)*c**3*sqrt(x)/sqrt(a*x/b + 1) - 6*a**2*sqrt(b)*c**2*d/(sqrt(x)*sqrt(a*x/b + 1

)) + 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**(3/2)*c**3/(sqrt(x)*sqrt(a*x/b + 1)) + 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/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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

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