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

Optimal. Leaf size=125 \[ a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{\left (a+\frac{b}{x}\right )^{5/2} (2 a d+5 b c)}{5 a}-\frac{1}{3} \left (a+\frac{b}{x}\right )^{3/2} (2 a d+5 b c)-a \sqrt{a+\frac{b}{x}} (2 a d+5 b c)+\frac{c x \left (a+\frac{b}{x}\right )^{7/2}}{a} \]

[Out]

-(a*(5*b*c + 2*a*d)*Sqrt[a + b/x]) - ((5*b*c + 2*a*d)*(a + b/x)^(3/2))/3 - ((5*b
*c + 2*a*d)*(a + b/x)^(5/2))/(5*a) + (c*(a + b/x)^(7/2)*x)/a + a^(3/2)*(5*b*c +
2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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Rubi [A]  time = 0.208579, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{\left (a+\frac{b}{x}\right )^{5/2} (2 a d+5 b c)}{5 a}-\frac{1}{3} \left (a+\frac{b}{x}\right )^{3/2} (2 a d+5 b c)-a \sqrt{a+\frac{b}{x}} (2 a d+5 b c)+\frac{c x \left (a+\frac{b}{x}\right )^{7/2}}{a} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(5*b*c + 2*a*d)*Sqrt[a + b/x]) - ((5*b*c + 2*a*d)*(a + b/x)^(3/2))/3 - ((5*b
*c + 2*a*d)*(a + b/x)^(5/2))/(5*a) + (c*(a + b/x)^(7/2)*x)/a + a^(3/2)*(5*b*c +
2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 17.6048, size = 110, normalized size = 0.88 \[ 2 a^{\frac{3}{2}} \left (a d + \frac{5 b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - a \sqrt{a + \frac{b}{x}} \left (2 a d + 5 b c\right ) - \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (\frac{2 a d}{3} + \frac{5 b c}{3}\right ) + \frac{c x \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{a} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \left (a d + \frac{5 b c}{2}\right )}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(3/2)*(a*d + 5*b*c/2)*atanh(sqrt(a + b/x)/sqrt(a)) - a*sqrt(a + b/x)*(2*a*d
+ 5*b*c) - (a + b/x)**(3/2)*(2*a*d/3 + 5*b*c/3) + c*x*(a + b/x)**(7/2)/a - 2*(a
+ b/x)**(5/2)*(a*d + 5*b*c/2)/(5*a)

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Mathematica [A]  time = 0.252027, size = 105, normalized size = 0.84 \[ \frac{1}{2} a^{3/2} (2 a d+5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )+\frac{\sqrt{a+\frac{b}{x}} \left (a^2 x^2 (15 c x-46 d)-2 a b x (35 c x+11 d)-2 b^2 (5 c x+3 d)\right )}{15 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(-2*b^2*(3*d + 5*c*x) + a^2*x^2*(-46*d + 15*c*x) - 2*a*b*x*(11*d
+ 35*c*x)))/(15*x^2) + (a^(3/2)*(5*b*c + 2*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a
+ b/x]*x])/2

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Maple [B]  time = 0.016, size = 240, normalized size = 1.9 \[{\frac{1}{30\,b{x}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 30\,{a}^{5/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) db{x}^{4}+75\,{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{2}c{x}^{4}+60\,{a}^{3}\sqrt{a{x}^{2}+bx}d{x}^{4}+150\,{a}^{2}\sqrt{a{x}^{2}+bx}cb{x}^{4}-60\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{2}d{x}^{2}-120\,a \left ( a{x}^{2}+bx \right ) ^{3/2}cb{x}^{2}-32\, \left ( a{x}^{2}+bx \right ) ^{3/2}xabd-20\, \left ( a{x}^{2}+bx \right ) ^{3/2}x{b}^{2}c-12\, \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{2}d \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*(c+d/x),x)

[Out]

1/30*((a*x+b)/x)^(1/2)/x^3/b*(30*a^(5/2)*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a
*x+b)/a^(1/2))*d*b*x^4+75*a^(3/2)*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*b^2*c*x^4+60*a^3*(a*x^2+b*x)^(1/2)*d*x^4+150*a^2*(a*x^2+b*x)^(1/2)*c*b*x
^4-60*(a*x^2+b*x)^(3/2)*a^2*d*x^2-120*a*(a*x^2+b*x)^(3/2)*c*b*x^2-32*(a*x^2+b*x)
^(3/2)*x*a*b*d-20*(a*x^2+b*x)^(3/2)*x*b^2*c-12*(a*x^2+b*x)^(3/2)*b^2*d)/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262862, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (5 \, a b c + 2 \, a^{2} d\right )} \sqrt{a} x^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (15 \, a^{2} c x^{3} - 6 \, b^{2} d - 2 \,{\left (35 \, a b c + 23 \, a^{2} d\right )} x^{2} - 2 \,{\left (5 \, b^{2} c + 11 \, a b d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{30 \, x^{2}}, \frac{15 \,{\left (5 \, a b c + 2 \, a^{2} d\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (15 \, a^{2} c x^{3} - 6 \, b^{2} d - 2 \,{\left (35 \, a b c + 23 \, a^{2} d\right )} x^{2} - 2 \,{\left (5 \, b^{2} c + 11 \, a b d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{15 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*(5*a*b*c + 2*a^2*d)*sqrt(a)*x^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)
/x) + b) + 2*(15*a^2*c*x^3 - 6*b^2*d - 2*(35*a*b*c + 23*a^2*d)*x^2 - 2*(5*b^2*c
+ 11*a*b*d)*x)*sqrt((a*x + b)/x))/x^2, 1/15*(15*(5*a*b*c + 2*a^2*d)*sqrt(-a)*x^2
*arctan(sqrt((a*x + b)/x)/sqrt(-a)) + (15*a^2*c*x^3 - 6*b^2*d - 2*(35*a*b*c + 23
*a^2*d)*x^2 - 2*(5*b^2*c + 11*a*b*d)*x)*sqrt((a*x + b)/x))/x^2]

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Sympy [A]  time = 47.6987, size = 561, normalized size = 4.49 \[ \frac{4 a^{\frac{11}{2}} b^{\frac{7}{2}} d x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{9}{2}} d x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{11}{2}} d x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{13}{2}} d \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + 2 a^{\frac{5}{2}} d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} + 5 a^{\frac{3}{2}} b c \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{4 a^{6} b^{3} d x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{4} d x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{2 a^{3} d \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} + a^{2} \sqrt{b} c \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{4 a^{2} \sqrt{b} c \sqrt{x}}{\sqrt{\frac{a x}{b} + 1}} - \frac{2 a^{2} \sqrt{b} d}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} - \frac{4 a b^{\frac{3}{2}} c}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} + 2 a b d \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + b^{2} c \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(11/2)*b**(7/2)*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)) + 2*a**(9/2)*b**(9/2)*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)) - 8*a**(7/2)*b**(11/2)*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)) - 6*a**(5/2)*b**(13/
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)) + 2
*a**(5/2)*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 5*a**(3/2)*b*c*asinh(sqrt(a)*sqrt(x
)/sqrt(b)) - 4*a**6*b**3*d*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**
4*x**(5/2)) - 4*a**5*b**4*d*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b*
*4*x**(5/2)) - 2*a**3*d*sqrt(x)/(sqrt(b)*sqrt(a*x/b + 1)) + a**2*sqrt(b)*c*sqrt(
x)*sqrt(a*x/b + 1) - 4*a**2*sqrt(b)*c*sqrt(x)/sqrt(a*x/b + 1) - 2*a**2*sqrt(b)*d
/(sqrt(x)*sqrt(a*x/b + 1)) - 4*a*b**(3/2)*c/(sqrt(x)*sqrt(a*x/b + 1)) + 2*a*b*d*
Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) + b**2*c*Pi
ecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True))

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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),x, algorithm="giac")

[Out]

Exception raised: TypeError