Optimal. Leaf size=113 \[ \frac {(a e+c d x) (d+e x)^{-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (p,1+p;2+p;-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)} \]
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Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {693, 691, 72,
71} \begin {gather*} \frac {(d+e x)^{-2 p} (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (p,p+1;p+2;-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 691
Rule 693
Rubi steps
\begin {align*} \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\left ((d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^{2 p}\right ) \int \left (1+\frac {e x}{d}\right )^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\\ &=\left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (1+\frac {e x}{d}\right )^{-p} \, dx\\ &=\left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^{-2 p} \left (\frac {c d^2 \left (1+\frac {e x}{d}\right )}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (\frac {c d^2}{c d^2-a e^2}+\frac {c d e x}{c d^2-a e^2}\right )^{-p} \, dx\\ &=\frac {(a e+c d x) (d+e x)^{-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (p,1+p;2+p;-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 97, normalized size = 0.86 \begin {gather*} \frac {(d+e x)^{-1-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p ((a e+c d x) (d+e x))^{1+p} \, _2F_1\left (p,1+p;2+p;\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{p} \left (e x +d \right )^{-2 p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p}{{\left (d+e\,x\right )}^{2\,p}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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