Optimal. Leaf size=489 \[ -\frac{\sqrt{c} e n \left (\sqrt{-a} \sqrt{c} d+a e\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )}{a^2 d (n+1)}+\frac{\sqrt{c} e n \left (\sqrt{-a} e+\sqrt{c} d\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{c (d-e x) (d+e x)^{n+1}}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.604872, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {961, 65, 823, 831, 68} \[ -\frac{\sqrt{c} e n \left (\sqrt{-a} \sqrt{c} d+a e\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )}{a^2 d (n+1)}+\frac{\sqrt{c} e n \left (\sqrt{-a} e+\sqrt{c} d\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{c (d-e x) (d+e x)^{n+1}}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 961
Rule 65
Rule 823
Rule 831
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^n}{x \left (a+c x^2\right )^2} \, dx &=\int \left (\frac{(d+e x)^n}{a^2 x}-\frac{c x (d+e x)^n}{a \left (a+c x^2\right )^2}-\frac{c x (d+e x)^n}{a^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{(d+e x)^n}{x} \, dx}{a^2}-\frac{c \int \frac{x (d+e x)^n}{a+c x^2} \, dx}{a^2}-\frac{c \int \frac{x (d+e x)^n}{\left (a+c x^2\right )^2} \, dx}{a}\\ &=\frac{c (d-e x) (d+e x)^{1+n}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{e x}{d}\right )}{a^2 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^2}+\frac{\int \frac{(d+e x)^n \left (-a c d e n+a c e^2 n x\right )}{a+c x^2} \, dx}{2 a^2 \left (c d^2+a e^2\right )}\\ &=\frac{c (d-e x) (d+e x)^{1+n}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{e x}{d}\right )}{a^2 d (1+n)}+\frac{\sqrt{c} \int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{2 a^2}-\frac{\sqrt{c} \int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{2 a^2}+\frac{\int \left (\frac{\left (-\sqrt{-a} a c d e n-a^2 \sqrt{c} e^2 n\right ) (d+e x)^n}{2 a \left (\sqrt{-a}-\sqrt{c} x\right )}+\frac{\left (-\sqrt{-a} a c d e n+a^2 \sqrt{c} e^2 n\right ) (d+e x)^n}{2 a \left (\sqrt{-a}+\sqrt{c} x\right )}\right ) \, dx}{2 a^2 \left (c d^2+a e^2\right )}\\ &=\frac{c (d-e x) (d+e x)^{1+n}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\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^2 \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^2 \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^2 d (1+n)}-\frac{\left (\sqrt{c} e \left (\sqrt{-a} \sqrt{c} d-a e\right ) n\right ) \int \frac{(d+e x)^n}{\sqrt{-a}+\sqrt{c} x} \, dx}{4 a^2 \left (c d^2+a e^2\right )}-\frac{\left (\sqrt{c} e \left (\sqrt{-a} \sqrt{c} d+a e\right ) n\right ) \int \frac{(d+e x)^n}{\sqrt{-a}-\sqrt{c} x} \, dx}{4 a^2 \left (c d^2+a e^2\right )}\\ &=\frac{c (d-e x) (d+e x)^{1+n}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\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^2 \left (\sqrt{c} d-\sqrt{-a} e\right ) (1+n)}+\frac{\sqrt{c} e \left (\sqrt{-a} \sqrt{c} d-a e\right ) n (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 )}{4 a^2 \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\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^2 \left (\sqrt{c} d+\sqrt{-a} e\right ) (1+n)}-\frac{\sqrt{c} e \left (\sqrt{-a} \sqrt{c} d+a e\right ) n (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 )}{4 a^2 \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac{(d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{e x}{d}\right )}{a^2 d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.934426, size = 391, normalized size = 0.8 \[ \frac{(d+e x)^{n+1} \left (\frac{\sqrt{c} e n \left (\left (\sqrt{-a} c d^2-2 a \sqrt{c} d e+(-a)^{3/2} e^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )+\left (-\sqrt{-a} c d^2-2 a \sqrt{c} d e+\sqrt{-a} a e^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )\right )}{(n+1) \left (a e^2+c d^2\right )^2}+\frac{2 a c (d-e x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{2 \sqrt{c} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{(n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{2 \sqrt{c} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{(n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{4 \, _2F_1\left (1,n+1;n+2;\frac{d+e x}{d}\right )}{d n+d}\right )}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.73, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{n}}{x \left ( c{x}^{2}+a \right ) ^{2}}}\, 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 )}^{2} 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^{2} x^{5} + 2 \, a c x^{3} + a^{2} x}, 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 (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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