Optimal. Leaf size=327 \[ -\frac {2^{p-1} (2 c d-b e) \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {e \left (a+b x+c x^2\right )^{p+1} \left (-2 c (2 p+3) \left (c d^2 (2 p+5)-e (a e+b d (p+2))\right )-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right )}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \]
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Rubi [A] time = 0.42, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {742, 779, 624} \[ -\frac {2^{p-1} (2 c d-b e) \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {e \left (a+b x+c x^2\right )^{p+1} \left (-2 c (2 p+3) \left (c d^2 (2 p+5)-e (a e+b d (p+2))\right )-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right )}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 742
Rule 779
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}+\frac {\int (d+e x) \left (2 c d^2 (2+p)-e (2 a e+b d (1+p))+e (2 c d-b e) (3+p) x\right ) \left (a+b x+c x^2\right )^p \, dx}{2 c (2+p)}\\ &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac {e \left (b e (2 c d-b e) (2+p) (3+p)-2 c (3+2 p) \left (c d^2 (5+2 p)-e (a e+b d (2+p))\right )-2 c e (2 c d-b e) (1+p) (3+p) x\right ) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}+\frac {\left ((2 c d-b e) \left (b^2 e^2 (3+p)+2 c^2 d^2 (3+2 p)-2 c e (3 a e+b d (3+2 p))\right )\right ) \int \left (a+b x+c x^2\right )^p \, dx}{4 c^3 (3+2 p)}\\ &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac {e \left (b e (2 c d-b e) (2+p) (3+p)-2 c (3+2 p) \left (c d^2 (5+2 p)-e (a e+b d (2+p))\right )-2 c e (2 c d-b e) (1+p) (3+p) x\right ) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}-\frac {2^{-1+p} (2 c d-b e) \left (b^2 e^2 (3+p)+2 c^2 d^2 (3+2 p)-2 c e (3 a e+b d (3+2 p))\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} (1+p) (3+2 p)}\\ \end {align*}
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Mathematica [C] time = 1.47, size = 558, normalized size = 1.71 \[ \frac {1}{4} (a+x (b+c x))^p \left (6 d^2 e x^2 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )^{-p} F_1\left (2;-p,-p;3;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+4 d e^2 x^3 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )^{-p} F_1\left (3;-p,-p;4;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+e^3 x^4 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )^{-p} F_1\left (4;-p,-p;5;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+\frac {d^3 2^{p+1} \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac {-b-2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} {\left (c x^{2} + b x + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (e x + d\right )}^{3} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.47, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{p}\, dx \]
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 \[ \int {\left (e x + d\right )}^{3} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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