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