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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]