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

Optimal. Leaf size=126 \[ \frac{c^2 x \left (a+\frac{b}{x}\right )^{5/2}}{a}-\frac{c \left (a+\frac{b}{x}\right )^{3/2} (4 a d+3 b c)}{3 a}-c \sqrt{a+\frac{b}{x}} (4 a d+3 b c)+\sqrt{a} c (4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

[Out]

-(c*(3*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(3*b*c + 4*a*d)*(a + b/x)^(3/2))/(3*a) -
(2*d^2*(a + b/x)^(5/2))/(5*b) + (c^2*(a + b/x)^(5/2)*x)/a + Sqrt[a]*c*(3*b*c +
4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

_______________________________________________________________________________________

Rubi [A]  time = 0.249053, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{c^2 x \left (a+\frac{b}{x}\right )^{5/2}}{a}-\frac{c \left (a+\frac{b}{x}\right )^{3/2} (4 a d+3 b c)}{3 a}-c \sqrt{a+\frac{b}{x}} (4 a d+3 b c)+\sqrt{a} c (4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

-(c*(3*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(3*b*c + 4*a*d)*(a + b/x)^(3/2))/(3*a) -
(2*d^2*(a + b/x)^(5/2))/(5*b) + (c^2*(a + b/x)^(5/2)*x)/a + Sqrt[a]*c*(3*b*c +
4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.4613, size = 107, normalized size = 0.85 \[ \sqrt{a} c \left (4 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - c \sqrt{a + \frac{b}{x}} \left (4 a d + 3 b c\right ) - \frac{2 d^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b} + \frac{c^{2} x \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{a} - \frac{c \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (4 a d + 3 b c\right )}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*c*(4*a*d + 3*b*c)*atanh(sqrt(a + b/x)/sqrt(a)) - c*sqrt(a + b/x)*(4*a*d
+ 3*b*c) - 2*d**2*(a + b/x)**(5/2)/(5*b) + c**2*x*(a + b/x)**(5/2)/a - c*(a + b/
x)**(3/2)*(4*a*d + 3*b*c)/(3*a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.238831, size = 115, normalized size = 0.91 \[ \frac{1}{15} \sqrt{a+\frac{b}{x}} \left (-\frac{6 a^2 d^2}{b}-\frac{4 d (3 a d+5 b c)}{x}+15 a c^2 x-80 a c d-30 b c^2-\frac{6 b d^2}{x^2}\right )+\frac{1}{2} \sqrt{a} c (4 a d+3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(-30*b*c^2 - 80*a*c*d - (6*a^2*d^2)/b - (6*b*d^2)/x^2 - (4*d*(5*b
*c + 3*a*d))/x + 15*a*c^2*x))/15 + (Sqrt[a]*c*(3*b*c + 4*a*d)*Log[b + 2*a*x + 2*
Sqrt[a]*Sqrt[a + b/x]*x])/2

_______________________________________________________________________________________

Maple [B]  time = 0.019, size = 244, normalized size = 1.9 \[{\frac{1}{30\,b{x}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 60\,c{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) db{x}^{4}+45\,\sqrt{a}{c}^{2}{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{4}+120\,c{a}^{2}\sqrt{a{x}^{2}+bx}d{x}^{4}+90\,a{c}^{2}\sqrt{a{x}^{2}+bx}b{x}^{4}-120\,c \left ( a{x}^{2}+bx \right ) ^{3/2}ad{x}^{2}-60\, \left ( a{x}^{2}+bx \right ) ^{3/2}{c}^{2}b{x}^{2}-12\, \left ( a{x}^{2}+bx \right ) ^{3/2}xa{d}^{2}-40\, \left ( a{x}^{2}+bx \right ) ^{3/2}xbcd-12\, \left ( a{x}^{2}+bx \right ) ^{3/2}b{d}^{2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*(c+d/x)^2,x)

[Out]

1/30*((a*x+b)/x)^(1/2)/x^3/b*(60*c*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*b*x^4+45*a^(1/2)*c^2*b^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+
2*a*x+b)/a^(1/2))*x^4+120*c*a^2*(a*x^2+b*x)^(1/2)*d*x^4+90*a*c^2*(a*x^2+b*x)^(1/
2)*b*x^4-120*c*(a*x^2+b*x)^(3/2)*a*d*x^2-60*(a*x^2+b*x)^(3/2)*c^2*b*x^2-12*(a*x^
2+b*x)^(3/2)*x*a*d^2-40*(a*x^2+b*x)^(3/2)*x*b*c*d-12*(a*x^2+b*x)^(3/2)*b*d^2)/(x
*(a*x+b))^(1/2)

_______________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.258953, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt{a} x^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \,{\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{30 \, b x^{2}}, \frac{15 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \,{\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{15 \, b x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(a)*x^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x
+ b)/x) + b) + 2*(15*a*b*c^2*x^3 - 6*b^2*d^2 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^
2*d^2)*x^2 - 4*(5*b^2*c*d + 3*a*b*d^2)*x)*sqrt((a*x + b)/x))/(b*x^2), 1/15*(15*(
3*b^2*c^2 + 4*a*b*c*d)*sqrt(-a)*x^2*arctan(sqrt((a*x + b)/x)/sqrt(-a)) + (15*a*b
*c^2*x^3 - 6*b^2*d^2 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^2*d^2)*x^2 - 4*(5*b^2*c*
d + 3*a*b*d^2)*x)*sqrt((a*x + b)/x))/(b*x^2)]

_______________________________________________________________________________________

Sympy [A]  time = 26.8339, size = 576, normalized size = 4.57 \[ \frac{4 a^{\frac{11}{2}} b^{\frac{5}{2}} d^{2} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{7}{2}} d^{2} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{9}{2}} d^{2} x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} d^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + 4 a^{\frac{3}{2}} c d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} + 3 \sqrt{a} b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{4 a^{6} b^{2} d^{2} x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{3} d^{2} x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{2} c d \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} + a \sqrt{b} c^{2} \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{2 a \sqrt{b} c^{2} \sqrt{x}}{\sqrt{\frac{a x}{b} + 1}} - \frac{4 a \sqrt{b} c d}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} + a d^{2} \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) - \frac{2 b^{\frac{3}{2}} c^{2}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} + 2 b c d \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(11/2)*b**(5/2)*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**(9/2)*b**(7/2)*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**(7/2)*b**(9/2)*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**(5/2)
*b**(11/2)*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)) + 4*a**(3/2)*c*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 3*sqrt(a)*b*c**2*asinh
(sqrt(a)*sqrt(x)/sqrt(b)) - 4*a**6*b**2*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**5*b**3*d**2*x**(5/2)/(15*a**(7/2)*b**3*x**(
7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**2*c*d*sqrt(x)/(sqrt(b)*sqrt(a*x/b + 1))
+ a*sqrt(b)*c**2*sqrt(x)*sqrt(a*x/b + 1) - 2*a*sqrt(b)*c**2*sqrt(x)/sqrt(a*x/b
+ 1) - 4*a*sqrt(b)*c*d/(sqrt(x)*sqrt(a*x/b + 1)) + a*d**2*Piecewise((-sqrt(a)/x,
Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) - 2*b**(3/2)*c**2/(sqrt(x)*sqrt(a
*x/b + 1)) + 2*b*c*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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError