Optimal. Leaf size=497 \[ -\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt [4]{3} \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt [3]{b} x \left (a+b x^3\right )}{a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{a x} \]
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Rubi [A] time = 0.472092, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2025, 2032, 329, 308, 225, 1881} \[ -\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt [4]{3} \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt [3]{b} x \left (a+b x^3\right )}{a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{a x} \]
Antiderivative was successfully verified.
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Rule 2025
Rule 2032
Rule 329
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a x+b x^4}} \, dx &=-\frac{2 \sqrt{a x+b x^4}}{a x}+\frac{(2 b) \int \frac{x^2}{\sqrt{a x+b x^4}} \, dx}{a}\\ &=-\frac{2 \sqrt{a x+b x^4}}{a x}+\frac{\left (2 b \sqrt{x} \sqrt{a+b x^3}\right ) \int \frac{x^{3/2}}{\sqrt{a+b x^3}} \, dx}{a \sqrt{a x+b x^4}}\\ &=-\frac{2 \sqrt{a x+b x^4}}{a x}+\frac{\left (4 b \sqrt{x} \sqrt{a+b x^3}\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b x^6}} \, dx,x,\sqrt{x}\right )}{a \sqrt{a x+b x^4}}\\ &=-\frac{2 \sqrt{a x+b x^4}}{a x}-\frac{\left (2 \sqrt [3]{b} \sqrt{x} \sqrt{a+b x^3}\right ) \operatorname{Subst}\left (\int \frac{\left (-1+\sqrt{3}\right ) a^{2/3}-2 b^{2/3} x^4}{\sqrt{a+b x^6}} \, dx,x,\sqrt{x}\right )}{a \sqrt{a x+b x^4}}-\frac{\left (2 \left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt{x} \sqrt{a+b x^3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^6}} \, dx,x,\sqrt{x}\right )}{\sqrt [3]{a} \sqrt{a x+b x^4}}\\ &=\frac{2 \left (1+\sqrt{3}\right ) \sqrt [3]{b} x \left (a+b x^3\right )}{a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{a x}-\frac{2 \sqrt [4]{3} \sqrt [3]{b} 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 (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\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 )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} 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 (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\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 )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0114298, size = 48, normalized size = 0.1 \[ -\frac{2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};-\frac{b x^3}{a}\right )}{\sqrt{x \left (a+b x^3\right )}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 1083, normalized size = 2.2 \begin{align*} \text{result too large to display} \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{1}{\sqrt{b x^{4} + a x} 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^{4} + a x}}{b x^{5} + a x^{2}}, 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{1}{x \sqrt{x \left (a + b x^{3}\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{1}{\sqrt{b x^{4} + a x} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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