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

Optimal. Leaf size=164 \[ -\frac{d \left (a+\frac{b}{x}\right )^{3/2} \left (\frac{3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{35 b^2}-3 c^2 \sqrt{a+\frac{b}{x}} (2 a d+b c)+3 \sqrt{a} c^2 (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^3-\frac{9}{7} d \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 \]

[Out]

-3*c^2*(b*c + 2*a*d)*Sqrt[a + b/x] - (9*d*(a + b/x)^(3/2)*(c + d/x)^2)/7 - (d*(a
+ b/x)^(3/2)*(2*(13*b*c - a*d)*(5*b*c + 2*a*d) + (3*b*d*(19*b*c + 2*a*d))/x))/(
35*b^2) + (a + b/x)^(3/2)*(c + d/x)^3*x + 3*Sqrt[a]*c^2*(b*c + 2*a*d)*ArcTanh[Sq
rt[a + b/x]/Sqrt[a]]

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Rubi [A]  time = 0.44844, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{d \left (a+\frac{b}{x}\right )^{3/2} \left (\frac{3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{35 b^2}-3 c^2 \sqrt{a+\frac{b}{x}} (2 a d+b c)+3 \sqrt{a} c^2 (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^3-\frac{9}{7} d \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 \]

Antiderivative was successfully verified.

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

[Out]

-3*c^2*(b*c + 2*a*d)*Sqrt[a + b/x] - (9*d*(a + b/x)^(3/2)*(c + d/x)^2)/7 - (d*(a
+ b/x)^(3/2)*(2*(13*b*c - a*d)*(5*b*c + 2*a*d) + (3*b*d*(19*b*c + 2*a*d))/x))/(
35*b^2) + (a + b/x)^(3/2)*(c + d/x)^3*x + 3*Sqrt[a]*c^2*(b*c + 2*a*d)*ArcTanh[Sq
rt[a + b/x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 48.4643, size = 151, normalized size = 0.92 \[ 3 \sqrt{a} c^{2} \left (2 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 3 c^{2} \sqrt{a + \frac{b}{x}} \left (2 a d + b c\right ) - \frac{9 d \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (c + \frac{d}{x}\right )^{2}}{7} + x \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (c + \frac{d}{x}\right )^{3} + \frac{8 d \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (- \frac{9 b d \left (2 a d + 19 b c\right )}{8 x} + \left (\frac{3 a d}{4} - \frac{39 b c}{4}\right ) \left (2 a d + 5 b c\right )\right )}{105 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(a)*c**2*(2*a*d + b*c)*atanh(sqrt(a + b/x)/sqrt(a)) - 3*c**2*sqrt(a + b/x)
*(2*a*d + b*c) - 9*d*(a + b/x)**(3/2)*(c + d/x)**2/7 + x*(a + b/x)**(3/2)*(c + d
/x)**3 + 8*d*(a + b/x)**(3/2)*(-9*b*d*(2*a*d + 19*b*c)/(8*x) + (3*a*d/4 - 39*b*c
/4)*(2*a*d + 5*b*c))/(105*b**2)

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Mathematica [A]  time = 0.259178, size = 169, normalized size = 1.03 \[ \frac{\sqrt{a+\frac{b}{x}} \left (4 a^3 d^3 x^3-2 a^2 b d^2 x^2 (21 c x+d)+a b^2 x \left (35 c^3 x^3-280 c^2 d x^2-84 c d^2 x-16 d^3\right )-2 b^3 \left (35 c^3 x^3+35 c^2 d x^2+21 c d^2 x+5 d^3\right )\right )}{35 b^2 x^3}+\frac{3}{2} \sqrt{a} c^2 (2 a d+b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(4*a^3*d^3*x^3 - 2*a^2*b*d^2*x^2*(d + 21*c*x) + a*b^2*x*(-16*d^3
- 84*c*d^2*x - 280*c^2*d*x^2 + 35*c^3*x^3) - 2*b^3*(5*d^3 + 21*c*d^2*x + 35*c^2*
d*x^2 + 35*c^3*x^3)))/(35*b^2*x^3) + (3*Sqrt[a]*c^2*(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 = 0.02, size = 332, normalized size = 2. \[{\frac{1}{70\,{x}^{4}{b}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 210\,{c}^{2}{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) d{b}^{2}{x}^{5}+105\,\sqrt{a}{c}^{3}{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{5}+420\,{c}^{2}{a}^{2}\sqrt{a{x}^{2}+bx}db{x}^{5}+210\,a{c}^{3}\sqrt{a{x}^{2}+bx}{b}^{2}{x}^{5}-420\,{c}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}adb{x}^{3}-140\,{c}^{3} \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{2}{x}^{3}+8\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}{a}^{2}{d}^{3}-84\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}abc{d}^{2}-140\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}{b}^{2}{c}^{2}d-12\, \left ( a{x}^{2}+bx \right ) ^{3/2}xab{d}^{3}-84\, \left ( a{x}^{2}+bx \right ) ^{3/2}x{b}^{2}c{d}^{2}-20\, \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{2}{d}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*(c+d/x)^3,x)

[Out]

1/70*((a*x+b)/x)^(1/2)/x^4/b^2*(210*c^2*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^2*x^5+105*a^(1/2)*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^5+420*c^2*a^2*(a*x^2+b*x)^(1/2)*d*b*x^5+210*a*c^3*(a
*x^2+b*x)^(1/2)*b^2*x^5-420*c^2*(a*x^2+b*x)^(3/2)*a*d*b*x^3-140*c^3*(a*x^2+b*x)^
(3/2)*b^2*x^3+8*(a*x^2+b*x)^(3/2)*x^2*a^2*d^3-84*(a*x^2+b*x)^(3/2)*x^2*a*b*c*d^2
-140*(a*x^2+b*x)^(3/2)*x^2*b^2*c^2*d-12*(a*x^2+b*x)^(3/2)*x*a*b*d^3-84*(a*x^2+b*
x)^(3/2)*x*b^2*c*d^2-20*(a*x^2+b*x)^(3/2)*b^2*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)^(3/2)*(c + d/x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255313, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt{a} x^{3} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \,{\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \,{\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{70 \, b^{2} x^{3}}, \frac{105 \,{\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt{-a} x^{3} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \,{\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \,{\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{35 \, b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/70*(105*(b^3*c^3 + 2*a*b^2*c^2*d)*sqrt(a)*x^3*log(2*a*x + 2*sqrt(a)*x*sqrt((a
*x + b)/x) + b) + 2*(35*a*b^2*c^3*x^4 - 10*b^3*d^3 - 2*(35*b^3*c^3 + 140*a*b^2*c
^2*d + 21*a^2*b*c*d^2 - 2*a^3*d^3)*x^3 - 2*(35*b^3*c^2*d + 42*a*b^2*c*d^2 + a^2*
b*d^3)*x^2 - 2*(21*b^3*c*d^2 + 8*a*b^2*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^3), 1/3
5*(105*(b^3*c^3 + 2*a*b^2*c^2*d)*sqrt(-a)*x^3*arctan(sqrt((a*x + b)/x)/sqrt(-a))
+ (35*a*b^2*c^3*x^4 - 10*b^3*d^3 - 2*(35*b^3*c^3 + 140*a*b^2*c^2*d + 21*a^2*b*c
*d^2 - 2*a^3*d^3)*x^3 - 2*(35*b^3*c^2*d + 42*a*b^2*c*d^2 + a^2*b*d^3)*x^2 - 2*(2
1*b^3*c*d^2 + 8*a*b^2*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^3)]

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Sympy [A]  time = 41.5849, size = 1862, normalized size = 11.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**(19/2)*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)) - 40*a**(17/2)*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)) - 30*a**(15/2)*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)) - 40*a**(13/2)*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**(13/2)*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)) - 100*a*
*(11/2)*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)) + 12*a**(11/2)*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**(11/2)*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)) - 96*a**(9/2)*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)) + 6*a**(9/2)*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**(9/2)*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)) - 30*a**(7/2)*b**(23/2)*d**3*sqrt(a*x/b + 1)/(1
05*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**(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**(7/2)*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)
) - 18*a**(5/2)*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)) + 6*a**(3/2)*c**2*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 3*
sqrt(a)*b*c**3*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 16*a**10*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)) + 48*a**9*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)) + 48*a**8*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)) + 16*a**7*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**7*b*d**3*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*
x**(5/2)) - 12*a**6*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**6*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**5*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)) - 6*a**2*c**2*d*sqrt(x)/(sqrt(b)*sqrt(a*x/b + 1)) +
a*sqrt(b)*c**3*sqrt(x)*sqrt(a*x/b + 1) - 2*a*sqrt(b)*c**3*sqrt(x)/sqrt(a*x/b + 1
) - 6*a*sqrt(b)*c**2*d/(sqrt(x)*sqrt(a*x/b + 1)) + 3*a*c*d**2*Piecewise((-sqrt(a
)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) - 2*b**(3/2)*c**3/(sqrt(x)*sq
rt(a*x/b + 1)) + 3*b*c**2*d*Piecewise((-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)^(3/2)*(c + d/x)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError