Optimal. Leaf size=95 \[ -\frac {(d+e x)^2 (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,2 p+3;p+3;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(p+2) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 152, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 624} \[ \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{2 c d (p+1)}-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{2 c d (p+1)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 640
Rubi steps
\begin {align*} \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{2 c d (1+p)}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{2 d}\\ &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{2 c d (1+p)}-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-1-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{2 c d (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 94, normalized size = 0.99 \[ \frac {\left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p-1} ((d+e x) (a e+c d x))^{p+1} \, _2F_1\left (-p-1,p+1;p+2;\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{c d (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}, x\right ) \]
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 \[ \int {\left (e x + d\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.49, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{p}\, dx \]
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 \[ \int {\left (e x + d\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}\,{d x} \]
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 \[ \int \left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p \,d x \]
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 \[ \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p} \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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