Optimal. Leaf size=90 \[ \frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{3 b^3 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{10 \sqrt{2} c^2 \sqrt [4]{b x+c x^2}} \]
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Rubi [A] time = 0.0339669, antiderivative size = 90, 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 = {612, 622, 619, 228} \[ \frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{3 b^3 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{10 \sqrt{2} c^2 \sqrt [4]{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 612
Rule 622
Rule 619
Rule 228
Rubi steps
\begin{align*} \int \left (b x+c x^2\right )^{3/4} \, dx &=\frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{\left (3 b^2\right ) \int \frac{1}{\sqrt [4]{b x+c x^2}} \, dx}{20 c}\\ &=\frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{\left (3 b^2 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}}\right ) \int \frac{1}{\sqrt [4]{-\frac{c x}{b}-\frac{c^2 x^2}{b^2}}} \, dx}{20 c \sqrt [4]{b x+c x^2}}\\ &=\frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}+\frac{\left (3 b^4 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{b^2 x^2}{c^2}}} \, dx,x,-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right )}{20 \sqrt{2} c^3 \sqrt [4]{b x+c x^2}}\\ &=\frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{3 b^3 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (1+\frac{2 c x}{b}\right )\right |2\right )}{10 \sqrt{2} c^2 \sqrt [4]{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0127553, size = 45, normalized size = 0.5 \[ \frac{4 x (x (b+c x))^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{7}{4};\frac{11}{4};-\frac{c x}{b}\right )}{7 \left (\frac{c x}{b}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.656, size = 0, normalized size = 0. \begin{align*} \int \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{4}}}\, 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{3}{4}}\,{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{3}{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 \left (b x + c x^{2}\right )^{\frac{3}{4}}\, 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{3}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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