Optimal. Leaf size=140 \[ \frac{5 a^{7/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{21 b^{9/4} \sqrt{a x+b x^3}}-\frac{10 a \sqrt{a x+b x^3}}{21 b^2}+\frac{2 x^2 \sqrt{a x+b x^3}}{7 b} \]
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Rubi [A] time = 0.13725, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2024, 2011, 329, 220} \[ \frac{5 a^{7/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{a x+b x^3}}-\frac{10 a \sqrt{a x+b x^3}}{21 b^2}+\frac{2 x^2 \sqrt{a x+b x^3}}{7 b} \]
Antiderivative was successfully verified.
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Rule 2024
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{a x+b x^3}} \, dx &=\frac{2 x^2 \sqrt{a x+b x^3}}{7 b}-\frac{(5 a) \int \frac{x^2}{\sqrt{a x+b x^3}} \, dx}{7 b}\\ &=-\frac{10 a \sqrt{a x+b x^3}}{21 b^2}+\frac{2 x^2 \sqrt{a x+b x^3}}{7 b}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt{a x+b x^3}} \, dx}{21 b^2}\\ &=-\frac{10 a \sqrt{a x+b x^3}}{21 b^2}+\frac{2 x^2 \sqrt{a x+b x^3}}{7 b}+\frac{\left (5 a^2 \sqrt{x} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x^2}} \, dx}{21 b^2 \sqrt{a x+b x^3}}\\ &=-\frac{10 a \sqrt{a x+b x^3}}{21 b^2}+\frac{2 x^2 \sqrt{a x+b x^3}}{7 b}+\frac{\left (10 a^2 \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{21 b^2 \sqrt{a x+b x^3}}\\ &=-\frac{10 a \sqrt{a x+b x^3}}{21 b^2}+\frac{2 x^2 \sqrt{a x+b x^3}}{7 b}+\frac{5 a^{7/4} \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 )}{21 b^{9/4} \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0316225, size = 80, normalized size = 0.57 \[ \frac{2 x \left (5 a^2 \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right )-5 a^2-2 a b x^2+3 b^2 x^4\right )}{21 b^2 \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 149, normalized size = 1.1 \begin{align*}{\frac{2\,{x}^{2}}{7\,b}\sqrt{b{x}^{3}+ax}}-{\frac{10\,a}{21\,{b}^{2}}\sqrt{b{x}^{3}+ax}}+{\frac{5\,{a}^{2}}{21\,{b}^{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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{b x^{3} + a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{3} + a x} x^{3}}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{x \left (a + b x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{b x^{3} + a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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