Optimal. Leaf size=150 \[ \frac {g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac {(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]
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Rubi [A] time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {794, 648} \[ \frac {g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac {(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\frac {g (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac {\left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c d e (2-m)}\\ &=-\frac {\left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) (d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 e (1-m) (2-m)}+\frac {g (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d e (2-m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 67, normalized size = 0.45 \[ -\frac {(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} (a e g+c d (f (m-2)+g (m-1) x))}{c^2 d^2 (m-2) (m-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 145, normalized size = 0.97 \[ -\frac {{\left (a c d e f m - 2 \, a c d e f + a^{2} e^{2} g + {\left (c^{2} d^{2} g m - c^{2} d^{2} g\right )} x^{2} - {\left (2 \, c^{2} d^{2} f - {\left (c^{2} d^{2} f + a c d e g\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{{\left (c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 369, normalized size = 2.46 \[ -\frac {{\left (x e + d\right )}^{m} c^{2} d^{2} g m x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} + {\left (x e + d\right )}^{m} c^{2} d^{2} f m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} - {\left (x e + d\right )}^{m} c^{2} d^{2} g x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} + {\left (x e + d\right )}^{m} a c d g m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} - 2 \, {\left (x e + d\right )}^{m} c^{2} d^{2} f x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} + {\left (x e + d\right )}^{m} a c d f m e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} - 2 \, {\left (x e + d\right )}^{m} a c d f e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} + {\left (x e + d\right )}^{m} a^{2} g e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 2\right )}}{c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 89, normalized size = 0.59 \[ -\frac {\left (c d g m x +c d f m -c d g x +a e g -2 c d f \right ) \left (c d x +a e \right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{\left (m^{2}-3 m +2\right ) c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 94, normalized size = 0.63 \[ -\frac {{\left (c d x + a e\right )} f}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} - \frac {{\left (c^{2} d^{2} {\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} g}{{\left (m^{2} - 3 \, m + 2\right )} {\left (c d x + a e\right )}^{m} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 139, normalized size = 0.93 \[ -\frac {\frac {g\,x^2\,\left (m-1\right )\,{\left (d+e\,x\right )}^m}{m^2-3\,m+2}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a\,e\,g\,m-2\,c\,d\,f+c\,d\,f\,m\right )}{c\,d\,\left (m^2-3\,m+2\right )}+\frac {a\,e\,{\left (d+e\,x\right )}^m\,\left (a\,e\,g-2\,c\,d\,f+c\,d\,f\,m\right )}{c^2\,d^2\,\left (m^2-3\,m+2\right )}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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