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 = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, 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 68
Rule 831
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*}
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Mathematica [A] time = 0.06, 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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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n} x}{c x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x}{c x^{2} + a}\,{d x} \]
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 \[ \int \frac {x \left (e x +d \right )^{n}}{c \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \]
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 \[ \int \frac {x \left (d + e x\right )^{n}}{a + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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