Optimal. Leaf size=71 \[ -\frac{2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 c^2 x} \]
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Rubi [A] time = 0.0294293, antiderivative size = 71, 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 = {674, 67, 65} \[ -\frac{2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 c^2 x} \]
Antiderivative was successfully verified.
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Rule 674
Rule 67
Rule 65
Rubi steps
\begin{align*} \int (d x)^m \left (b x+c x^2\right )^{3/2} \, dx &=\frac{\left (x^{-\frac{3}{2}-m} (d x)^m \left (b x+c x^2\right )^{3/2}\right ) \int x^{\frac{3}{2}+m} (b+c x)^{3/2} \, dx}{(b+c x)^{3/2}}\\ &=-\frac{\left (b \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m \left (b x+c x^2\right )^{3/2}\right ) \int \left (-\frac{c x}{b}\right )^{\frac{3}{2}+m} (b+c x)^{3/2} \, dx}{c x (b+c x)^{3/2}}\\ &=-\frac{2 b \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{3/2} \, _2F_1\left (\frac{5}{2},-\frac{3}{2}-m;\frac{7}{2};1+\frac{c x}{b}\right )}{5 c^2 x}\\ \end{align*}
Mathematica [A] time = 0.116832, size = 60, normalized size = 0.85 \[ -\frac{2 (x (b+c x))^{5/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{5}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}\, 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}{2}} \left (d x\right )^{m}\,{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}{2}} \left (d x\right )^{m}, 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 (d x\right )^{m} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}\, 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}{2}} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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