Optimal. Leaf size=59 \[ \frac{2 \sqrt{2} b \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right ),2\right )}{c \left (b x+c x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0234089, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {622, 619, 232} \[ \frac{2 \sqrt{2} b \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{c \left (b x+c x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 622
Rule 619
Rule 232
Rubi steps
\begin{align*} \int \frac{1}{\left (b x+c x^2\right )^{3/4}} \, dx &=\frac{\left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} \int \frac{1}{\left (-\frac{c x}{b}-\frac{c^2 x^2}{b^2}\right )^{3/4}} \, dx}{\left (b x+c x^2\right )^{3/4}}\\ &=-\frac{\left (\sqrt{2} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b^2 x^2}{c^2}\right )^{3/4}} \, dx,x,-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt{2} b \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (1+\frac{2 c x}{b}\right )\right |2\right )}{c \left (b x+c x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0102803, size = 43, normalized size = 0.73 \[ \frac{4 x \left (\frac{c x}{b}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{c x}{b}\right )}{(x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.678, 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 \frac{1}{{\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 (\frac{1}{{\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 \frac{1}{\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 \frac{1}{{\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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