Optimal. Leaf size=160 \[ \frac{\left (a+c x^2\right )^p (d+e x)^{1-2 p} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (1-2 p)} \]
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Rubi [A] time = 0.0852414, antiderivative size = 160, 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} \[ \frac{\left (a+c x^2\right )^p (d+e x)^{1-2 p} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (1-2 p)} \]
Antiderivative was successfully verified.
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Rule 760
Rule 133
Rubi steps
\begin{align*} \int (d+e x)^{-2 p} \left (a+c x^2\right )^p \, dx &=\frac{\left (\left (a+c x^2\right )^p \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p}\right ) \operatorname{Subst}\left (\int x^{-2 p} \left (1-\frac{x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^p \left (1-\frac{x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x)^{1-2 p} \left (a+c x^2\right )^p \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (1-2 p)}\\ \end{align*}
Mathematica [A] time = 0.126451, size = 166, normalized size = 1.04 \[ -\frac{\left (a+c x^2\right )^p (d+e x)^{1-2 p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{e (2 p-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.583, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{2\,p}}}\, 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{{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}\,{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{{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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