Optimal. Leaf size=92 \[ \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,2 p;p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(1-p) (d+e x) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.30, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {677, 70, 69} \[ \frac {c d (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (2-p,p+1;p+2;-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 677
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx &=\frac {\left ((a e+c d x)^{-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 (a e+c d x)^p \left (1+\frac {e x}{d}\right )^{-2+p} \, dx}{d^2}\\ &=\frac {\left (c^2 (a e+c d x)^{-p} \left (\frac {c d \left (1+\frac {e x}{d}\right )}{c d-\frac {a e^2}{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 (a e+c d x)^p \left (\frac {c d^2}{c d^2-a e^2}+\frac {c d e x}{c d^2-a e^2}\right )^{-2+p} \, dx}{\left (c d-\frac {a e^2}{d}\right )^2}\\ &=\frac {c d (a e+c d x) \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 (2-p,1+p;2+p;-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 108, normalized size = 1.17 \[ \frac {c d (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (2-p,p+1;p+2;\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{(p+1) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{p}}{\left (e x +d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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