Optimal. Leaf size=49 \[ \frac{d x \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^{m+1} \, _2F_1\left (1,m+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a+b}\right )}{a+b} \]
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Rubi [A] time = 0.0167575, antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1972, 246, 245} \[ x \left (\frac{c x^2}{a+b}+1\right )^{-m} \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right ) \]
Antiderivative was successfully verified.
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Rule 1972
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (\frac{a+b+c x^2}{d}\right )^m \, dx &=\int \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^m \, dx\\ &=\left (\left (1+\frac{c x^2}{a+b}\right )^{-m} \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^m\right ) \int \left (1+\frac{c x^2}{a+b}\right )^m \, dx\\ &=x \left (1+\frac{c x^2}{a+b}\right )^{-m} \left (\frac{a+b}{d}+\frac{c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right )\\ \end{align*}
Mathematica [A] time = 0.0120852, size = 53, normalized size = 1.08 \[ x \left (\frac{c x^2}{a+b}+1\right )^{-m} \left (\frac{a+b+c x^2}{d}\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{c{x}^{2}+a+b}{d}} \right ) ^{m}\, 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 (\frac{c x^{2} + a + b}{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 (\frac{c x^{2} + a + b}{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 (\frac{a + b + c x^{2}}{d}\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 (\frac{c x^{2} + a + b}{d}\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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