Optimal. Leaf size=54 \[ \frac {(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {648} \[ \frac {(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \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 &=\frac {(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 d (1-m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.78 \[ -\frac {(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m}}{c d (m-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 57, normalized size = 1.06 \[ -\frac {{\left (c d x + a e\right )} {\left (e x + d\right )}^{m}}{{\left (c d m - c d\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 87, normalized size = 1.61 \[ -\frac {{\left (x e + d\right )}^{m} c d x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} + {\left (x e + d\right )}^{m} a e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )}}{c d m - c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 57, normalized size = 1.06 \[ -\frac {\left (c d x +a e \right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{\left (m -1\right ) c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 33, normalized size = 0.61 \[ -\frac {c d x + a e}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.25, size = 57, normalized size = 1.06 \[ -\frac {\left (a\,e+c\,d\,x\right )\,{\left (d+e\,x\right )}^m}{c\,d\,\left (m-1\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \]
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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