Optimal. Leaf size=243 \[ -\frac{\sqrt{2+\sqrt{3}} b^{2/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}-\frac{\sqrt{a x^2+b x^5}}{2 a x^3} \]
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Rubi [A] time = 0.128627, antiderivative size = 243, 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, Rules used = {2025, 2032, 218} \[ -\frac{\sqrt{2+\sqrt{3}} b^{2/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}-\frac{\sqrt{a x^2+b x^5}}{2 a x^3} \]
Antiderivative was successfully verified.
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Rule 2025
Rule 2032
Rule 218
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{a x^2+b x^5}} \, dx &=-\frac{\sqrt{a x^2+b x^5}}{2 a x^3}-\frac{b \int \frac{x}{\sqrt{a x^2+b x^5}} \, dx}{4 a}\\ &=-\frac{\sqrt{a x^2+b x^5}}{2 a x^3}-\frac{\left (b x \sqrt{a+b x^3}\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{4 a \sqrt{a x^2+b x^5}}\\ &=-\frac{\sqrt{a x^2+b x^5}}{2 a x^3}-\frac{\sqrt{2+\sqrt{3}} b^{2/3} 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{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} a \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}}\\ \end{align*}
Mathematica [C] time = 0.0130886, size = 55, normalized size = 0.23 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{2}{3},\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x \sqrt{x^2 \left (a+b x^3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 248, normalized size = 1. \begin{align*}{\frac{1}{12\,ax} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}\sqrt{-2\,{\frac{-bx+\sqrt [3]{-{b}^{2}a}}{\sqrt [3]{-{b}^{2}a} \left ( i\sqrt{3}-3 \right ) }}}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}+2\,bx+\sqrt [3]{-{b}^{2}a} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{3}}{6}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ){x}^{2}-6\,b{x}^{3}-6\,a \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \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^{5} + a x^{2}} x^{2}}\,{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^{5} + a x^{2}}}{b x^{7} + a x^{4}}, 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^{2} \sqrt{x^{2} \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^{5} + a x^{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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