Optimal. Leaf size=148 \[ \frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{2 a g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{2 a g (m+1)} \]
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Rubi [A] time = 0.18, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {912, 135, 133} \[ \frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{2 a g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{2 a g (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 912
Rubi steps
\begin {align*} \int \frac {(g x)^m (d+e x)^n}{a+c x^2} \, dx &=\int \left (\frac {\sqrt {-a} (g x)^m (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\sqrt {-a} (g x)^m (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {(g x)^m (d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {(g x)^m (d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 \sqrt {-a}}\\ &=-\frac {\left ((d+e x)^n \left (1+\frac {e x}{d}\right )^{-n}\right ) \int \frac {(g x)^m \left (1+\frac {e x}{d}\right )^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 \sqrt {-a}}-\frac {\left ((d+e x)^n \left (1+\frac {e x}{d}\right )^{-n}\right ) \int \frac {(g x)^m \left (1+\frac {e x}{d}\right )^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 \sqrt {-a}}\\ &=\frac {(g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{2 a g (1+m)}+\frac {(g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{2 a g (1+m)}\\ \end {align*}
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Mathematica [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {(g x)^m (d+e x)^n}{a+c x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{c x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x \right )^{m} \left (e x +d \right )^{n}}{c \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{c x^{2} + a}\,{d x} \]
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 \[ \int \frac {{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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