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 = 0.18, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 65
Rule 68
Rule 831
Rule 961
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*}
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Mathematica [A] time = 0.12, 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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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n}}{c x^{3} + a x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{n}}{{\left (c x^{2} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^n}{x\,\left (c\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{n}}{x \left (a + c x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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