Optimal. Leaf size=781 \[ \frac{\sqrt [6]{2} 3^{3/4} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{7 c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}+\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}-\frac{3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{7 \sqrt [3]{2} c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{14 \sqrt [3]{2} c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
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Rubi [A] time = 0.947094, antiderivative size = 781, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {622, 619, 195, 235, 304, 219, 1879} \[ \frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}-\frac{3 (b+2 c x) \left (b x+c x^2\right )^{2/3}}{7 \sqrt [3]{2} c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{\sqrt [6]{2} 3^{3/4} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \left (b x+c x^2\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{14 \sqrt [3]{2} c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 622
Rule 619
Rule 195
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \left (b x+c x^2\right )^{2/3} \, dx &=\frac{\left (b x+c x^2\right )^{2/3} \int \left (-\frac{c x}{b}-\frac{c^2 x^2}{b^2}\right )^{2/3} \, dx}{\left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=-\frac{\left (b^2 \left (b x+c x^2\right )^{2/3}\right ) \operatorname{Subst}\left (\int \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 )}{4 \sqrt [3]{2} c^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}-\frac{\left (b^2 \left (b x+c x^2\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-\frac{b^2 x^2}{c^2}}} \, dx,x,-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right )}{7 \sqrt [3]{2} c^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}+\frac{\left (3 \left (b x+c x^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{x}{\sqrt{-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac{c x \left (1+\frac{c x}{b}\right )}{b}}\right )}{14 \sqrt [3]{2} \left (-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right ) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}-\frac{\left (3 \left (b x+c x^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{3}-x}{\sqrt{-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac{c x \left (1+\frac{c x}{b}\right )}{b}}\right )}{14 \sqrt [3]{2} \left (-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right ) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}+\frac{\left (3 \sqrt{2+\sqrt{3}} \left (b x+c x^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 )}{7\ 2^{5/6} \left (-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right ) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}\\ &=\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (b x+c x^2\right )^{2/3}}{14 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3}}+\frac{3 b^2 \left (b x+c x^2\right )^{2/3} \sqrt{-1-\frac{4 c x}{b}-\frac{4 c^2 x^2}{b^2}} \sqrt{-1-\frac{4 c x (b+c x)}{b^2}}}{7 \sqrt [3]{2} c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-\sqrt{3}-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right )}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \left (b x+c x^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}} E\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 )}{14 \sqrt [3]{2} c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^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}}}+\frac{\sqrt [6]{2} 3^{3/4} b^2 \left (b x+c x^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 )}{7 c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^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 = 0.0094444, size = 45, normalized size = 0.06 \[ \frac{3 x (x (b+c x))^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{5}{3};\frac{8}{3};-\frac{c x}{b}\right )}{5 \left (\frac{c x}{b}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.61, 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{\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 ({\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 \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{\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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