Optimal. Leaf size=207 \[ \frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 a (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )}{a d (n+1)} \]
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Rubi [A] time = 0.183317, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {961, 65, 831, 68} \[ \frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 a (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )}{a d (n+1)} \]
Antiderivative was successfully verified.
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Rule 961
Rule 65
Rule 831
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^n}{x \left (a+c x^2\right )} \, dx &=\int \left (\frac{(d+e x)^n}{a x}-\frac{c x (d+e x)^n}{a \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{(d+e x)^n}{x} \, dx}{a}-\frac{c \int \frac{x (d+e x)^n}{a+c x^2} \, dx}{a}\\ &=-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{e x}{d}\right )}{a d (1+n)}-\frac{c \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}{a}\\ &=-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{e x}{d}\right )}{a d (1+n)}+\frac{\sqrt{c} \int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{2 a}-\frac{\sqrt{c} \int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{2 a}\\ &=\frac{\sqrt{c} (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 a \left (\sqrt{c} d-\sqrt{-a} e\right ) (1+n)}+\frac{\sqrt{c} (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 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;1+\frac{e x}{d}\right )}{a d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.130955, size = 189, normalized size = 0.91 \[ \frac{(d+e x)^{n+1} \left (-2 \left (a e^2+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )+\left (\sqrt{-a} \sqrt{c} d e+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )+\left (c d^2-\sqrt{-a} \sqrt{c} d e\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )\right )}{2 a d (n+1) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.742, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}}{x \left ( c{x}^{2}+a \right ) }}\, 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}}{{\left (c x^{2} + a\right )} x}\,{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^{3} + a x}, 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}}{x \left (a + c x^{2}\right )}\, 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}}{{\left (c x^{2} + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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