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

Optimal. Leaf size=132 \[ -\frac{3 c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{(b c-2 a d) \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )-\frac{a b d^2 (2 a d+b c)}{x}}{a^2 b^2 \sqrt{a+\frac{b}{x}}}+\frac{c x \left (c+\frac{d}{x}\right )^2}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.326522, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{3 c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{(b c-2 a d) \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )-\frac{a b d^2 (2 a d+b c)}{x}}{a^2 b^2 \sqrt{a+\frac{b}{x}}}+\frac{c x \left (c+\frac{d}{x}\right )^2}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.2067, size = 122, normalized size = 0.92 \[ \frac{c x \left (c + \frac{d}{x}\right )^{2}}{a \sqrt{a + \frac{b}{x}}} - \frac{4 \left (\frac{a b d^{2} \left (2 a d + b c\right )}{4 x} + \left (\frac{a d}{2} - \frac{b c}{4}\right ) \left (2 a^{2} d^{2} - 2 a b c d + 3 b^{2} c^{2}\right )\right )}{a^{2} b^{2} \sqrt{a + \frac{b}{x}}} + \frac{3 c^{2} \left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.179282, size = 123, normalized size = 0.93 \[ \frac{3 c^2 (2 a d-b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{5/2}}+\frac{\sqrt{a+\frac{b}{x}} \left (-4 a^3 d^3 x-2 a^2 b d^2 (d-3 c x)+a b^2 c^2 x (c x-6 d)+3 b^3 c^3 x\right )}{a^2 b^2 (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.025, size = 976, normalized size = 7.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^3/(a+b/x)^(3/2),x)

[Out]

-1/2*((a*x+b)/x)^(1/2)/x/a^(9/2)*(3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)
/a^(1/2))*x^4*a^4*b^4*c^3-6*a^(5/2)*(x*(a*x+b))^(1/2)*x^2*b^5*c^3+6*ln(1/2*(2*(x
*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^3*b^5*c^3+3*ln(1/2*(2*(x*(a*x+b)
)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^2*b^6*c^3-6*a^(9/2)*(x*(a*x+b))^(1/2)*x^
4*b^3*c^3+4*a^(7/2)*(x*(a*x+b))^(3/2)*c^3*x^2*b^3-12*a^(7/2)*(x*(a*x+b))^(1/2)*x
^3*b^4*c^3+8*a^(11/2)*(a*x^2+b*x)^(3/2)*x*b*d^3-3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^
(1/2)+2*a*x+b)/a^(1/2))*a^4*x^2*b^4*c*d^2-6*a^(9/2)*(a*x^2+b*x)^(1/2)*x^2*b^3*c*
d^2-6*a^(9/2)*(x*(a*x+b))^(1/2)*x^2*b^3*c*d^2+12*a^(7/2)*(x*(a*x+b))^(1/2)*x^2*b
^4*c^2*d-6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*x^2*b^5*c^2
*d-12*a^(11/2)*(x*(a*x+b))^(1/2)*x^3*b^2*c*d^2+24*a^(9/2)*(x*(a*x+b))^(1/2)*x^3*
b^3*c^2*d+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*x^2*b^4*c*
d^2-12*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*x^3*b^4*c^2*d+4
*a^(13/2)*(a*x^2+b*x)^(3/2)*x^2*d^3-4*a^(13/2)*(x*(a*x+b))^(3/2)*d^3*x^2+4*a^(9/
2)*(a*x^2+b*x)^(3/2)*b^2*d^3-12*a^(9/2)*(x*(a*x+b))^(3/2)*c^2*d*x^2*b^2+3*ln(1/2
*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*x^4*b^2*c*d^2-3*ln(1/2*(2*(a
*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*x^4*b^2*c*d^2+12*a^(11/2)*(x*(a*x+
b))^(3/2)*c*d^2*x^2*b-12*a^(11/2)*(a*x^2+b*x)^(1/2)*x^3*b^2*c*d^2-6*a^(13/2)*(a*
x^2+b*x)^(1/2)*x^4*b*c*d^2-6*a^(13/2)*(x*(a*x+b))^(1/2)*x^4*b*c*d^2+12*a^(11/2)*
(x*(a*x+b))^(1/2)*x^4*b^2*c^2*d+6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*a^5*x^3*b^3*c*d^2-6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2)
)*a^5*x^4*b^3*c^2*d-6*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*
x^3*b^3*c*d^2)/(x*(a*x+b))^(1/2)/b^3/(a*x+b)^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((c + d/x)^3/(a + b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256538, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d\right )} x \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (a b^{2} c^{3} x^{2} - 2 \, a^{2} b d^{3} +{\left (3 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x\right )} \sqrt{a}}{2 \, a^{\frac{5}{2}} b^{2} x \sqrt{\frac{a x + b}{x}}}, \frac{3 \,{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d\right )} x \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (a b^{2} c^{3} x^{2} - 2 \, a^{2} b d^{3} +{\left (3 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x\right )} \sqrt{-a}}{\sqrt{-a} a^{2} b^{2} x \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x + d\right )^{3}}{x^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*x + d)**3/(x**3*(a + b/x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.252264, size = 297, normalized size = 2.25 \[ -b{\left (\frac{2 \, d^{3} \sqrt{\frac{a x + b}{x}}}{b^{3}} - \frac{3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b} - \frac{2 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3} - \frac{3 \,{\left (a x + b\right )} b^{3} c^{3}}{x} + \frac{6 \,{\left (a x + b\right )} a b^{2} c^{2} d}{x} - \frac{6 \,{\left (a x + b\right )} a^{2} b c d^{2}}{x} + \frac{2 \,{\left (a x + b\right )} a^{3} d^{3}}{x}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )} a^{2} b^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(2*d^3*sqrt((a*x + b)/x)/b^3 - 3*(b*c^3 - 2*a*c^2*d)*arctan(sqrt((a*x + b)/x)
/sqrt(-a))/(sqrt(-a)*a^2*b) - (2*a*b^3*c^3 - 6*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - 2
*a^4*d^3 - 3*(a*x + b)*b^3*c^3/x + 6*(a*x + b)*a*b^2*c^2*d/x - 6*(a*x + b)*a^2*b
*c*d^2/x + 2*(a*x + b)*a^3*d^3/x)/((a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x +
b)/x)/x)*a^2*b^3))