Optimal. Leaf size=73 \[ \frac{2 b^2 (b+c x) \left (b x+c x^2\right )^{5/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right )}{7 c^3 x^2} \]
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Rubi [A] time = 0.030488, antiderivative size = 73, 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} \[ \frac{2 b^2 (b+c x) \left (b x+c x^2\right )^{5/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right )}{7 c^3 x^2} \]
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 )^{5/2} \, dx &=\frac{\left (x^{-\frac{5}{2}-m} (d x)^m \left (b x+c x^2\right )^{5/2}\right ) \int x^{\frac{5}{2}+m} (b+c x)^{5/2} \, dx}{(b+c x)^{5/2}}\\ &=\frac{\left (b^2 \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m \left (b x+c x^2\right )^{5/2}\right ) \int \left (-\frac{c x}{b}\right )^{\frac{5}{2}+m} (b+c x)^{5/2} \, dx}{c^2 x^2 (b+c x)^{5/2}}\\ &=\frac{2 b^2 \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{5/2} \, _2F_1\left (\frac{7}{2},-\frac{5}{2}-m;\frac{9}{2};1+\frac{c x}{b}\right )}{7 c^3 x^2}\\ \end{align*}
Mathematica [A] time = 0.122992, size = 70, normalized size = 0.96 \[ \frac{2 b^2 (b+c x)^3 \sqrt{x (b+c x)} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right )}{7 c^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.429, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{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{5}{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^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x} \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{5}{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{5}{2}} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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