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