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

Optimal. Leaf size=100 \[ -\frac{\left (a+\frac{b}{x}\right )^{3/2} (2 a d+3 b c)}{3 a}-\sqrt{a+\frac{b}{x}} (2 a d+3 b c)+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{c x \left (a+\frac{b}{x}\right )^{5/2}}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.181951, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{\left (a+\frac{b}{x}\right )^{3/2} (2 a d+3 b c)}{3 a}-\sqrt{a+\frac{b}{x}} (2 a d+3 b c)+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{c x \left (a+\frac{b}{x}\right )^{5/2}}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7568, size = 87, normalized size = 0.87 \[ 2 \sqrt{a} \left (a d + \frac{3 b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - \sqrt{a + \frac{b}{x}} \left (2 a d + 3 b c\right ) + \frac{c x \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{a} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (a d + \frac{3 b c}{2}\right )}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.162674, size = 84, normalized size = 0.84 \[ \frac{\sqrt{a+\frac{b}{x}} (a x (3 c x-8 d)-2 b (3 c x+d))}{3 x}+\frac{1}{2} \sqrt{a} (2 a d+3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(a*x*(-8*d + 3*c*x) - 2*b*(d + 3*c*x)))/(3*x) + (Sqrt[a]*(3*b*c +
2*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/2

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 194, normalized size = 1.9 \[{\frac{1}{6\,b{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 6\,{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) d{x}^{3}b+9\,\sqrt{a}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{2}c{x}^{3}+12\,{a}^{2}\sqrt{a{x}^{2}+bx}d{x}^{3}+18\,a\sqrt{a{x}^{2}+bx}c{x}^{3}b-12\, \left ( a{x}^{2}+bx \right ) ^{3/2}adx-12\, \left ( a{x}^{2}+bx \right ) ^{3/2}cxb-4\,d \left ( a{x}^{2}+bx \right ) ^{3/2}b \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*(c+d/x),x)

[Out]

1/6*((a*x+b)/x)^(1/2)/x^2*(6*a^(3/2)*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b
)/a^(1/2))*d*x^3*b+9*a^(1/2)*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^3+12*a^2*(a*x^2+b*x)^(1/2)*d*x^3+18*a*(a*x^2+b*x)^(1/2)*c*x^3*b-12*(a
*x^2+b*x)^(3/2)*a*d*x-12*(a*x^2+b*x)^(3/2)*c*x*b-4*d*(a*x^2+b*x)^(3/2)*b)/(x*(a*
x+b))^(1/2)/b

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25885, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (3 \, b c + 2 \, a d\right )} \sqrt{a} x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (3 \, a c x^{2} - 2 \, b d - 2 \,{\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{6 \, x}, \frac{3 \,{\left (3 \, b c + 2 \, a d\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (3 \, a c x^{2} - 2 \, b d - 2 \,{\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 32.5682, size = 204, normalized size = 2.04 \[ 2 a^{\frac{3}{2}} d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} + 3 \sqrt{a} b c \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{2 a^{2} d \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} + a \sqrt{b} c \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{2 a \sqrt{b} c \sqrt{x}}{\sqrt{\frac{a x}{b} + 1}} - \frac{2 a \sqrt{b} d}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} - \frac{2 b^{\frac{3}{2}} c}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} + b d \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(3/2)*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 3*sqrt(a)*b*c*asinh(sqrt(a)*sqrt(x
)/sqrt(b)) - 2*a**2*d*sqrt(x)/(sqrt(b)*sqrt(a*x/b + 1)) + a*sqrt(b)*c*sqrt(x)*sq
rt(a*x/b + 1) - 2*a*sqrt(b)*c*sqrt(x)/sqrt(a*x/b + 1) - 2*a*sqrt(b)*d/(sqrt(x)*s
qrt(a*x/b + 1)) - 2*b**(3/2)*c/(sqrt(x)*sqrt(a*x/b + 1)) + b*d*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)^(3/2)*(c + d/x),x, algorithm="giac")

[Out]

Exception raised: TypeError