Optimal. Leaf size=84 \[ -\frac{2 x^3 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}-\frac{2 B x}{c^2 \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.208617, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 x^3 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}-\frac{2 B x}{c^2 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x))/(b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 12.7894, size = 76, normalized size = 0.9 \[ - \frac{2 B x}{c^{2} \sqrt{b x + c x^{2}}} + \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} + \frac{2 x^{3} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x+A)/(c*x**2+b*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.12108, size = 94, normalized size = 1.12 \[ \frac{x \left (2 \sqrt{c} x \left (A c^2 x-3 b^2 B-4 b B c x\right )+6 b B \sqrt{x} (b+c x)^{3/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{3 b c^{5/2} (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x))/(b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.01, size = 206, normalized size = 2.5 \[ -{\frac{A{x}^{2}}{c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{Abx}{3\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Ax}{3\,bc}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{A}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{{x}^{3}B}{3\,c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{Bb{x}^{2}}{2\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{2}Bx}{6\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,Bx}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{Bb}{6\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{B\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x+A)/(c*x^2+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((B*x + A)*x^3/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.315349, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B b c x + B b^{2}\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (3 \, B b^{2} x +{\left (4 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt{c}}{3 \,{\left (b c^{3} x + b^{2} c^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}, \frac{2 \,{\left (3 \,{\left (B b c x + B b^{2}\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (3 \, B b^{2} x +{\left (4 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt{-c}\right )}}{3 \,{\left (b c^{3} x + b^{2} c^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^3/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x+A)/(c*x**2+b*x)**(5/2),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((B*x + A)*x^3/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]