Optimal. Leaf size=295 \[ \frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)} \]
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Rubi [A] time = 0.42, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {961, 135, 133, 912} \[ \frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)}+\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 912
Rule 961
Rubi steps
\begin {align*} \int \frac {(g x)^m (d+e x)^n}{\left (a+c x^2\right )^2} \, dx &=\int \left (-\frac {c (g x)^m (d+e x)^n}{4 a \left (\sqrt {-a} \sqrt {c}-c x\right )^2}-\frac {c (g x)^m (d+e x)^n}{4 a \left (\sqrt {-a} \sqrt {c}+c x\right )^2}-\frac {c (g x)^m (d+e x)^n}{2 a \left (-a c-c^2 x^2\right )}\right ) \, dx\\ &=-\frac {c \int \frac {(g x)^m (d+e x)^n}{\left (\sqrt {-a} \sqrt {c}-c x\right )^2} \, dx}{4 a}-\frac {c \int \frac {(g x)^m (d+e x)^n}{\left (\sqrt {-a} \sqrt {c}+c x\right )^2} \, dx}{4 a}-\frac {c \int \frac {(g x)^m (d+e x)^n}{-a c-c^2 x^2} \, dx}{2 a}\\ &=-\frac {c \int \left (-\frac {\sqrt {-a} (g x)^m (d+e x)^n}{2 a c \left (\sqrt {-a}-\sqrt {c} x\right )}-\frac {\sqrt {-a} (g x)^m (d+e x)^n}{2 a c \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a}-\frac {\left (c (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}{\left (\sqrt {-a} \sqrt {c}-c x\right )^2} \, dx}{4 a}-\frac {\left (c (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}{\left (\sqrt {-a} \sqrt {c}+c x\right )^2} \, dx}{4 a}\\ &=\frac {(g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} F_1\left (1+m;-n,2;2+m;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 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,2;2+m;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (1+m)}+\frac {\int \frac {(g x)^m (d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 (-a)^{3/2}}+\frac {\int \frac {(g x)^m (d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 (-a)^{3/2}}\\ &=\frac {(g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} F_1\left (1+m;-n,2;2+m;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 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,2;2+m;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (1+m)}+\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}{4 (-a)^{3/2}}+\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}{4 (-a)^{3/2}}\\ &=\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 )}{4 a^2 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 )}{4 a^2 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,2;2+m;-\frac {e x}{d},-\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 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,2;2+m;-\frac {e x}{d},\frac {\sqrt {c} x}{\sqrt {-a}}\right )}{4 a^2 g (1+m)}\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {(g x)^m (d+e x)^n}{\left (a+c x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x \right )^{m} \left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} \left (g x\right )^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^n}{{\left (c\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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