Optimal. Leaf size=167 \[ \frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
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Rubi [A] time = 0.0912727, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {712, 68} \[ \frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
Antiderivative was successfully verified.
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Rule 712
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^n}{a+c x^2} \, dx &=\int \left (\frac{\sqrt{-a} (d+e x)^n}{2 a \left (\sqrt{-a}-\sqrt{c} x\right )}+\frac{\sqrt{-a} (d+e x)^n}{2 a \left (\sqrt{-a}+\sqrt{c} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{2 \sqrt{-a}}-\frac{\int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{2 \sqrt{-a}}\\ &=\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{-a} \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{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0744895, size = 145, normalized size = 0.87 \[ \frac{(d+e x)^{n+1} \left (\frac{\, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{\sqrt{c} d-\sqrt{-a} e}-\frac{\, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{\sqrt{-a} e+\sqrt{c} d}\right )}{2 \sqrt{-a} (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.739, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}}{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}}{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}}{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{\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}}{c x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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