Optimal. Leaf size=322 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
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Rubi [A] time = 0.394286, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {622, 619, 236, 219} \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 622
Rule 619
Rule 236
Rule 219
Rubi steps
\begin{align*} \int \frac{1}{\left (b x+c x^2\right )^{2/3}} \, dx &=\frac{\left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \int \frac{1}{\left (-\frac{c x}{b}-\frac{c^2 x^2}{b^2}\right )^{2/3}} \, dx}{\left (b x+c x^2\right )^{2/3}}\\ &=-\frac{\left (\sqrt [3]{2} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b^2 x^2}{c^2}\right )^{2/3}} \, dx,x,-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{2/3}}\\ &=\frac{\left (3 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt{-1-\frac{4 c x}{b}-\frac{4 c^2 x^2}{b^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac{c x \left (1+\frac{c x}{b}\right )}{b}}\right )}{2^{2/3} \left (-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right ) \left (b x+c x^2\right )^{2/3}}\\ &=\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt{-1-\frac{4 c x}{b}-\frac{4 c^2 x^2}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{1+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt{3}-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{1-\sqrt{3}-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt{-1-\frac{4 c x (b+c x)}{b^2}} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (1-\sqrt{3}-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0112151, size = 43, normalized size = 0.13 \[ \frac{3 x \left (\frac{c x}{b}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{c x}{b}\right )}{(x (b+c x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.651, size = 0, normalized size = 0. \begin{align*} \int \left ( c{x}^{2}+bx \right ) ^{-{\frac{2}{3}}}\, dx \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}{{\left (c x^{2} + b x\right )}^{\frac{2}{3}}}\,{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{1}{{\left (c x^{2} + b x\right )}^{\frac{2}{3}}}, 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}{\left (b x + c x^{2}\right )^{\frac{2}{3}}}\, 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}{{\left (c x^{2} + b x\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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