Optimal. Leaf size=154 \[ \frac{\left (a+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{3/2}} \]
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Rubi [A] time = 0.135235, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {760, 133} \[ \frac{\left (a+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 760
Rule 133
Rubi steps
\begin{align*} \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx &=\frac{\left (a+c x^2\right )^{3/2} \operatorname{Subst}\left (\int x^m \left (1-\frac{x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \, dx,x,d+e x\right )}{e \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2}}\\ &=\frac{(d+e x)^{1+m} \left (a+c x^2\right )^{3/2} F_1\left (1+m;-\frac{3}{2},-\frac{3}{2};2+m;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (1+m) \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2}}\\ \end{align*}
Mathematica [F] time = 0.0809725, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.558, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+a \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} + a\right )}^{\frac{3}{2}}{\left (e x + d\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} + a\right )}^{\frac{3}{2}}{\left (e x + d\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 (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{m}\, 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} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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