Optimal. Leaf size=134 \[ \frac{4 a^{3/4} b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a x+b x^3}}-\frac{2 \left (a x+b x^3\right )^{3/2}}{3 x^3}+\frac{4}{3} b \sqrt{a x+b x^3} \]
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Rubi [A] time = 0.249916, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{4 a^{3/4} b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a x+b x^3}}-\frac{2 \left (a x+b x^3\right )^{3/2}}{3 x^3}+\frac{4}{3} b \sqrt{a x+b x^3} \]
Antiderivative was successfully verified.
[In] Int[(a*x + b*x^3)^(3/2)/x^4,x]
[Out]
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Rubi in Sympy [A] time = 23.1007, size = 129, normalized size = 0.96 \[ \frac{4 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{x} \left (a + b x^{2}\right )} + \frac{4 b \sqrt{a x + b x^{3}}}{3} - \frac{2 \left (a x + b x^{3}\right )^{\frac{3}{2}}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a*x)**(3/2)/x**4,x)
[Out]
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Mathematica [C] time = 0.285941, size = 107, normalized size = 0.8 \[ \frac{2 \left (-a^2+\frac{4 i a b x^{5/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}+b^2 x^4\right )}{3 x \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x + b*x^3)^(3/2)/x^4,x]
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Maple [A] time = 0.027, size = 139, normalized size = 1. \[ -{\frac{2\,a}{3\,{x}^{2}}\sqrt{b{x}^{3}+ax}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+ax}}+{\frac{4\,a}{3}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a*x)^(3/2)/x^4,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x)^(3/2)/x^4,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x)^(3/2)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a*x)**(3/2)/x**4,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x)^(3/2)/x^4,x, algorithm="giac")
[Out]