Optimal. Leaf size=163 \[ -\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
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Rubi [A] time = 0.0942188, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {831, 68} \[ -\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
Antiderivative was successfully verified.
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Rule 831
Rule 68
Rubi steps
\begin{align*} \int \frac{x (d+e x)^n}{a+c x^2} \, dx &=\int \left (-\frac{(d+e x)^n}{2 \sqrt{c} \left (\sqrt{-a}-\sqrt{c} x\right )}+\frac{(d+e x)^n}{2 \sqrt{c} \left (\sqrt{-a}+\sqrt{c} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{2 \sqrt{c}}+\frac{\int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{2 \sqrt{c}}\\ &=-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{c} \left (\sqrt{c} d-\sqrt{-a} e\right ) (1+n)}-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{c} \left (\sqrt{c} d+\sqrt{-a} e\right ) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0671825, size = 151, normalized size = 0.93 \[ -\frac{(d+e x)^{n+1} \left (\left (\sqrt{-a} e+\sqrt{c} d\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )+\left (\sqrt{c} d-\sqrt{-a} e\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )\right )}{2 \sqrt{c} (n+1) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.768, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}x}{c{x}^{2}+a}}\, 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 (e x + d\right )}^{n} x}{c x^{2} + a}\,{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 (e x + d\right )}^{n} x}{c x^{2} + a}, 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 \frac{x \left (d + e x\right )^{n}}{a + c x^{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 \frac{{\left (e x + d\right )}^{n} x}{c x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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