Optimal. Leaf size=270 \[ -\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.0857186, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {731, 727} \[ -\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 731
Rule 727
Rubi steps
\begin{align*} \int (d+e x)^{-3-2 p} \left (a+c x^2\right )^p \, dx &=-\frac{e (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (1+p)}+\frac{(c d) \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{c d^2+a e^2}\\ &=-\frac{e (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (1+p)}-\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (-\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (1+2 p)}\\ \end{align*}
Mathematica [A] time = 49.557, size = 368, normalized size = 1.36 \[ \frac{2^{-2 p-3} \text{Gamma}\left (-p-\frac{1}{2}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 (p+1)} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (1-\frac{d+e x}{e \sqrt{-\frac{a}{c}}+d}\right )^{p+1} \left (\text{Gamma}(1-2 p) \text{Gamma}(-p) \left (e \sqrt{-\frac{a}{c}}+d\right ) \left (e \left (2 p \sqrt{-\frac{a}{c}}+\sqrt{-\frac{a}{c}}+x\right )+2 d (p+1)\right ) \, _2F_1\left (1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d+\sqrt{-\frac{a}{c}} e\right ) \left (x+\sqrt{-\frac{a}{c}}\right )}\right )+\frac{2 e \text{Gamma}(1-p) \text{Gamma}(-2 p) \left (c x \sqrt{-\frac{a}{c}}+a\right ) (d+e x) \, _2F_1\left (2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d+\sqrt{-\frac{a}{c}} e\right ) \left (x+\sqrt{-\frac{a}{c}}\right )}\right )}{c \left (\sqrt{-\frac{a}{c}}+x\right )}\right )}{\sqrt{\pi } e (p+1) \text{Gamma}(1-2 p) \text{Gamma}(-2 p) \left (e \sqrt{-\frac{a}{c}}+d\right )^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.614, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-3-2\,p} \left ( c{x}^{2}+a \right ) ^{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{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{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 )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}, 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{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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