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

Optimal. Leaf size=152 \[ a^{3/2} c (4 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{c^2 x \left (a+\frac{b}{x}\right )^{7/2}}{a}-\frac{c \left (a+\frac{b}{x}\right )^{5/2} (4 a d+5 b c)}{5 a}-\frac{1}{3} c \left (a+\frac{b}{x}\right )^{3/2} (4 a d+5 b c)-a c \sqrt{a+\frac{b}{x}} (4 a d+5 b c)-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]

[Out]

-(a*c*(5*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(5*b*c + 4*a*d)*(a + b/x)^(3/2))/3 - (
c*(5*b*c + 4*a*d)*(a + b/x)^(5/2))/(5*a) - (2*d^2*(a + b/x)^(7/2))/(7*b) + (c^2*
(a + b/x)^(7/2)*x)/a + a^(3/2)*c*(5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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Rubi [A]  time = 0.299007, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ a^{3/2} c (4 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{c^2 x \left (a+\frac{b}{x}\right )^{7/2}}{a}-\frac{c \left (a+\frac{b}{x}\right )^{5/2} (4 a d+5 b c)}{5 a}-\frac{1}{3} c \left (a+\frac{b}{x}\right )^{3/2} (4 a d+5 b c)-a c \sqrt{a+\frac{b}{x}} (4 a d+5 b c)-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

-(a*c*(5*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(5*b*c + 4*a*d)*(a + b/x)^(3/2))/3 - (
c*(5*b*c + 4*a*d)*(a + b/x)^(5/2))/(5*a) - (2*d^2*(a + b/x)^(7/2))/(7*b) + (c^2*
(a + b/x)^(7/2)*x)/a + a^(3/2)*c*(5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 26.7544, size = 131, normalized size = 0.86 \[ a^{\frac{3}{2}} c \left (4 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - a c \sqrt{a + \frac{b}{x}} \left (4 a d + 5 b c\right ) - \frac{c \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (4 a d + 5 b c\right )}{3} - \frac{2 d^{2} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b} + \frac{c^{2} x \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{a} - \frac{c \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \left (4 a d + 5 b c\right )}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*c*(4*a*d + 5*b*c)*atanh(sqrt(a + b/x)/sqrt(a)) - a*c*sqrt(a + b/x)*(4*a
*d + 5*b*c) - c*(a + b/x)**(3/2)*(4*a*d + 5*b*c)/3 - 2*d**2*(a + b/x)**(7/2)/(7*
b) + c**2*x*(a + b/x)**(7/2)/a - c*(a + b/x)**(5/2)*(4*a*d + 5*b*c)/(5*a)

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Mathematica [A]  time = 0.249973, size = 156, normalized size = 1.03 \[ \frac{1}{2} a^{3/2} c (4 a d+5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )+\frac{\sqrt{a+\frac{b}{x}} \left (-30 a^3 d^2 x^3+a^2 b x^2 \left (105 c^2 x^2-644 c d x-90 d^2\right )-2 a b^2 x \left (245 c^2 x^2+154 c d x+45 d^2\right )-2 b^3 \left (35 c^2 x^2+42 c d x+15 d^2\right )\right )}{105 b x^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(-30*a^3*d^2*x^3 - 2*b^3*(15*d^2 + 42*c*d*x + 35*c^2*x^2) + a^2*b
*x^2*(-90*d^2 - 644*c*d*x + 105*c^2*x^2) - 2*a*b^2*x*(45*d^2 + 154*c*d*x + 245*c
^2*x^2)))/(105*b*x^3) + (a^(3/2)*c*(5*b*c + 4*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqr
t[a + b/x]*x])/2

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Maple [B]  time = 0.02, size = 318, normalized size = 2.1 \[{\frac{1}{210\,b{x}^{4}}\sqrt{{\frac{ax+b}{x}}} \left ( 420\,{a}^{5/2}c\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) db{x}^{5}+525\,{a}^{3/2}{c}^{2}{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{5}+840\,{a}^{3}c\sqrt{a{x}^{2}+bx}d{x}^{5}+1050\,{a}^{2}{c}^{2}\sqrt{a{x}^{2}+bx}b{x}^{5}-840\,{a}^{2}c \left ( a{x}^{2}+bx \right ) ^{3/2}d{x}^{3}-840\,a{c}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}b{x}^{3}-60\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}{a}^{2}{d}^{2}-448\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}abcd-140\, \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}{b}^{2}{c}^{2}-120\, \left ( a{x}^{2}+bx \right ) ^{3/2}xab{d}^{2}-168\, \left ( a{x}^{2}+bx \right ) ^{3/2}x{b}^{2}cd-60\, \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*(c+d/x)^2,x)

[Out]

1/210*((a*x+b)/x)^(1/2)/x^4/b*(420*a^(5/2)*c*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)
+2*a*x+b)/a^(1/2))*d*b*x^5+525*a^(3/2)*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^5+840*a^3*c*(a*x^2+b*x)^(1/2)*d*x^5+1050*a^2*c^2*(a*x^2+b
*x)^(1/2)*b*x^5-840*a^2*c*(a*x^2+b*x)^(3/2)*d*x^3-840*a*c^2*(a*x^2+b*x)^(3/2)*b*
x^3-60*(a*x^2+b*x)^(3/2)*x^2*a^2*d^2-448*(a*x^2+b*x)^(3/2)*x^2*a*b*c*d-140*(a*x^
2+b*x)^(3/2)*x^2*b^2*c^2-120*(a*x^2+b*x)^(3/2)*x*a*b*d^2-168*(a*x^2+b*x)^(3/2)*x
*b^2*c*d-60*(a*x^2+b*x)^(3/2)*b^2*d^2)/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252331, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (5 \, a b^{2} c^{2} + 4 \, a^{2} b c d\right )} \sqrt{a} x^{3} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (105 \, a^{2} b c^{2} x^{4} - 30 \, b^{3} d^{2} - 2 \,{\left (245 \, a b^{2} c^{2} + 322 \, a^{2} b c d + 15 \, a^{3} d^{2}\right )} x^{3} - 2 \,{\left (35 \, b^{3} c^{2} + 154 \, a b^{2} c d + 45 \, a^{2} b d^{2}\right )} x^{2} - 6 \,{\left (14 \, b^{3} c d + 15 \, a b^{2} d^{2}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{210 \, b x^{3}}, \frac{105 \,{\left (5 \, a b^{2} c^{2} + 4 \, a^{2} b c d\right )} \sqrt{-a} x^{3} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (105 \, a^{2} b c^{2} x^{4} - 30 \, b^{3} d^{2} - 2 \,{\left (245 \, a b^{2} c^{2} + 322 \, a^{2} b c d + 15 \, a^{3} d^{2}\right )} x^{3} - 2 \,{\left (35 \, b^{3} c^{2} + 154 \, a b^{2} c d + 45 \, a^{2} b d^{2}\right )} x^{2} - 6 \,{\left (14 \, b^{3} c d + 15 \, a b^{2} d^{2}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{105 \, b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/210*(105*(5*a*b^2*c^2 + 4*a^2*b*c*d)*sqrt(a)*x^3*log(2*a*x + 2*sqrt(a)*x*sqrt
((a*x + b)/x) + b) + 2*(105*a^2*b*c^2*x^4 - 30*b^3*d^2 - 2*(245*a*b^2*c^2 + 322*
a^2*b*c*d + 15*a^3*d^2)*x^3 - 2*(35*b^3*c^2 + 154*a*b^2*c*d + 45*a^2*b*d^2)*x^2
- 6*(14*b^3*c*d + 15*a*b^2*d^2)*x)*sqrt((a*x + b)/x))/(b*x^3), 1/105*(105*(5*a*b
^2*c^2 + 4*a^2*b*c*d)*sqrt(-a)*x^3*arctan(sqrt((a*x + b)/x)/sqrt(-a)) + (105*a^2
*b*c^2*x^4 - 30*b^3*d^2 - 2*(245*a*b^2*c^2 + 322*a^2*b*c*d + 15*a^3*d^2)*x^3 - 2
*(35*b^3*c^2 + 154*a*b^2*c*d + 45*a^2*b*d^2)*x^2 - 6*(14*b^3*c*d + 15*a*b^2*d^2)
*x)*sqrt((a*x + b)/x))/(b*x^3)]

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Sympy [A]  time = 48.817, size = 1884, normalized size = 12.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**(19/2)*b**(13/2)*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)) - 40*a**(17/2)*b**(15/2)*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)) - 30*a**(15/2)*b**(17/2)*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)) - 40*a**(13/2)*b**(19/2)*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)) + 8*a**(13/2)*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)) - 100*a*
*(11/2)*b**(21/2)*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)) + 8*a**(11/2)*b**(7/2)*c*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)) + 4*a**(11/2)*b**(7/2)*d**2*x**2*sqrt(a*x/b + 1)/(1
5*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 96*a**(9/2)*b**(23/2)*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)) + 4*a**(9/2)*b**(9
/2)*c*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)) - 16*a**(9/2)*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)) - 30*a**(7/2)*b**(25/2)*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)) - 16*a**(7/2)*b**(11/2)*c*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)) - 12*a**(7/2)*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)) - 12
*a**(5/2)*b**(13/2)*c*d*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**(5/2)*c*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 5*a**(3/2)*b*c
**2*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 16*a**10*b**6*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) + 10
5*a**(7/2)*b**10*x**(7/2)) + 48*a**9*b**7*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)) + 48*a**8*b**8*d**2*x**(9/2)/(105*a**(13/2)*b**7*x**(13/2) + 3
15*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**9*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)) -
8*a**7*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)
) - 8*a**6*b**3*c*d*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5
/2)) - 8*a**6*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)) - 8*a**5*b**4*c*d*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**
4*x**(5/2)) - 4*a**3*c*d*sqrt(x)/(sqrt(b)*sqrt(a*x/b + 1)) + a**2*sqrt(b)*c**2*s
qrt(x)*sqrt(a*x/b + 1) - 4*a**2*sqrt(b)*c**2*sqrt(x)/sqrt(a*x/b + 1) - 4*a**2*sq
rt(b)*c*d/(sqrt(x)*sqrt(a*x/b + 1)) + a**2*d**2*Piecewise((-sqrt(a)/x, Eq(b, 0))
, (-2*(a + b/x)**(3/2)/(3*b), True)) - 4*a*b**(3/2)*c**2/(sqrt(x)*sqrt(a*x/b + 1
)) + 4*a*b*c*d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), Tru
e)) + b**2*c**2*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), Tr
ue))

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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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError