Optimal. Leaf size=238 \[ \frac{2 \sqrt{a x^2+b x^5}}{5 b}-\frac{4 \sqrt{2+\sqrt{3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}} \]
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Rubi [A] time = 0.273866, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 \sqrt{a x^2+b x^5}}{5 b}-\frac{4 \sqrt{2+\sqrt{3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}} \]
Antiderivative was successfully verified.
[In] Int[x^4/Sqrt[a*x^2 + b*x^5],x]
[Out]
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Rubi in Sympy [A] time = 21.1946, size = 216, normalized size = 0.91 \[ - \frac{4 \cdot 3^{\frac{3}{4}} a \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x^{2} + b x^{5}} F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{15 b^{\frac{4}{3}} x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (a + b x^{3}\right )} + \frac{2 \sqrt{a x^{2} + b x^{5}}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**5+a*x**2)**(1/2),x)
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Mathematica [C] time = 0.509121, size = 165, normalized size = 0.69 \[ \frac{-6 \sqrt [3]{-b} x^2 \left (a+b x^3\right )+4 i 3^{3/4} a^{4/3} x \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{15 (-b)^{4/3} \sqrt{x^2 \left (a+b x^3\right )}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^4/Sqrt[a*x^2 + b*x^5],x]
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Maple [A] time = 0.055, size = 248, normalized size = 1. \[{\frac{2\,x}{15\,{b}^{2}} \left ( ia\sqrt{3}\sqrt [3]{-a{b}^{2}}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-a{b}^{2}}-2\,bx-\sqrt [3]{-a{b}^{2}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{-2\,{\frac{-bx+\sqrt [3]{-a{b}^{2}}}{\sqrt [3]{-a{b}^{2}} \left ( i\sqrt{3}-3 \right ) }}}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-a{b}^{2}}+2\,bx+\sqrt [3]{-a{b}^{2}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-a{b}^{2}}-2\,bx-\sqrt [3]{-a{b}^{2}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) +3\,{b}^{2}{x}^{4}+3\,abx \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^5+a*x^2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(b*x^5 + a*x^2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{b x^{5} + a x^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(b*x^5 + a*x^2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**5+a*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(b*x^5 + a*x^2),x, algorithm="giac")
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