Optimal. Leaf size=103 \[ -\frac{(5 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b c-2 a d}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{5 b c-2 a d}{3 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c x}{a \left (a+\frac{b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.198709, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{(5 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b c-2 a d}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{5 b c-2 a d}{3 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c x}{a \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d/x)/(a + b/x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 16.3615, size = 90, normalized size = 0.87 \[ \frac{c x}{a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{2 \left (a d - \frac{5 b c}{2}\right )}{3 a^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{2 \left (a d - \frac{5 b c}{2}\right )}{a^{3} \sqrt{a + \frac{b}{x}}} + \frac{2 \left (a d - \frac{5 b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d/x)/(a+b/x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.140561, size = 102, normalized size = 0.99 \[ \frac{(2 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{7/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (a^2 x (3 c x-8 d)+a b (20 c x-6 d)+15 b^2 c\right )}{3 a^3 (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d/x)/(a + b/x)^(5/2),x]
[Out]
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Maple [B] time = 0.019, size = 548, normalized size = 5.3 \[ -{\frac{x}{6\,b \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 12\,{a}^{15/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}d-30\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}bc-12\,{a}^{13/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xd+36\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}bd+24\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xbc-90\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}{b}^{2}c-8\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}bd+36\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}d+20\,{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}{b}^{2}c-90\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }x{b}^{3}c+12\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}d-30\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{4}c-6\,{a}^{7}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}bd+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{6}{b}^{2}c-18\,{a}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{b}^{2}d+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{5}{b}^{3}c-18\,{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{3}d+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{4}c-6\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{4}d+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{5}c \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d/x)/(a+b/x)^(5/2),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)/(a + b/x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241657, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{2} c - 2 \, a b d +{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (3 \, a^{2} c x^{2} + 15 \, b^{2} c - 6 \, a b d + 4 \,{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{a}}{6 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{3 \,{\left (5 \, b^{2} c - 2 \, a b d +{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} c x^{2} + 15 \, b^{2} c - 6 \, a b d + 4 \,{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{-a}}{3 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c + d/x)/(a + b/x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 39.1088, size = 1479, normalized size = 14.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d/x)/(a+b/x)**(5/2),x)
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GIAC/XCAS [A] time = 0.255601, size = 185, normalized size = 1.8 \[ -\frac{1}{3} \, b{\left (\frac{3 \, c \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3}} - \frac{3 \,{\left (5 \, b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b} - \frac{2 \,{\left (a b c - a^{2} d + \frac{6 \,{\left (a x + b\right )} b c}{x} - \frac{3 \,{\left (a x + b\right )} a d}{x}\right )} x}{{\left (a x + b\right )} a^{3} b \sqrt{\frac{a x + b}{x}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c + d/x)/(a + b/x)^(5/2),x, algorithm="giac")
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