Optimal. Leaf size=809 \[ -\frac{e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac{e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]
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Rubi [A] time = 1.71996, antiderivative size = 809, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {744, 836, 806, 726} \[ -\frac{e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac{e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]
Antiderivative was successfully verified.
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Rule 744
Rule 836
Rule 806
Rule 726
Rubi steps
\begin{align*} \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx &=-\frac{e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac{\int (d+e x)^{-5-2 p} (b e (4+p)-c d (5+2 p)+3 c e x) \left (a+b x+c x^2\right )^p \, dx}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac{e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac{\int (d+e x)^{-4-2 p} \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (10+9 p+2 p^2\right )-2 c e \left (3 a e (2+p)+2 b d \left (7+5 p+p^2\right )\right )-2 c e (2 c d-b e) (4+p) x\right ) \left (a+b x+c x^2\right )^p \, dx}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac{e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac{e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}-\frac{\int (d+e x)^{-3-2 p} \left (b^3 e^3 \left (24+26 p+9 p^2+p^3\right )-2 c^3 d^3 \left (30+47 p+24 p^2+4 p^3\right )-b c e^2 (3+p) \left (2 a e (8+5 p)+b d \left (28+25 p+6 p^2\right )\right )+2 c^2 d e \left (a e \left (42+43 p+10 p^2\right )+b d \left (54+76 p+37 p^2+6 p^3\right )\right )+c e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^p \, dx}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac{e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac{e (2 c d-b e) (3+p) \left (b^2 e^2 \left (8+6 p+p^2\right )+2 c^2 d^2 \left (8+7 p+2 p^2\right )-2 c e \left (a e (8+5 p)+b d \left (8+7 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{4 \left (c d^2-b d e+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac{e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac{\left (b^4 e^4 \left (12+7 p+p^2\right )+4 c^4 d^4 \left (15+16 p+4 p^2\right )-8 c^3 d^2 e (5+2 p) (3 a e+b d (3+2 p))-4 b^2 c e^3 (3+p) (3 a e+b d (5+2 p))+12 c^2 e^2 \left (a^2 e^2+2 a b d e (5+2 p)+b^2 d^2 \left (10+9 p+2 p^2\right )\right )\right ) \int (d+e x)^{-2-2 p} \left (a+b x+c x^2\right )^p \, dx}{4 \left (c d^2-b d e+a e^2\right )^4 (3+2 p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac{e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac{e (2 c d-b e) (3+p) \left (b^2 e^2 \left (8+6 p+p^2\right )+2 c^2 d^2 \left (8+7 p+2 p^2\right )-2 c e \left (a e (8+5 p)+b d \left (8+7 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{4 \left (c d^2-b d e+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac{e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac{\left (b^4 e^4 \left (12+7 p+p^2\right )+4 c^4 d^4 \left (15+16 p+4 p^2\right )-8 c^3 d^2 e (5+2 p) (3 a e+b d (3+2 p))-4 b^2 c e^3 (3+p) (3 a e+b d (5+2 p))+12 c^2 e^2 \left (a^2 e^2+2 a b d e (5+2 p)+b^2 d^2 \left (10+9 p+2 p^2\right )\right )\right ) \left (b-\sqrt{b^2-4 a c}+2 c x\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b-\sqrt{b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b-\sqrt{b^2-4 a c}+2 c x\right )}\right )}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)}\\ \end{align*}
Mathematica [A] time = 6.27372, size = 1577, normalized size = 1.95 \[ \frac{e (d+e x)^{1-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{\left (c d^2-b e d+a e^2\right ) (1-2 (p+3))}+\frac{3 c \left (\frac{e (d+e x)^{3-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{\left (c d^2-b e d+a e^2\right ) (3-2 (p+3))}+\frac{\frac{\left (b (c d e+(b e (p+2)-c d (2 p+3)) e)-2 \left (a c e^2+c d (b e (p+2)-c d (2 p+3))\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-b e-\sqrt{b^2-4 a c} e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} (d+e x)^{5-2 (p+3)} \left (c x^2+b x+a\right )^p \, _2F_1\left (-p,5-2 (p+3);6-2 (p+3);-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt{b^2-4 a c} e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 \left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \left (c d^2-b e d+a e^2\right ) (5-2 (p+3))}-\frac{(e (b e (p+2)-c d (2 p+3))-c d e) (d+e x)^{4-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{2 \left (c d^2-b e d+a e^2\right ) (p+1)}}{\left (c d^2-b e d+a e^2\right ) (3-2 (p+3))}\right )+\frac{(e (b e (p+4)-c d (2 p+5))-3 c d e) \left (\frac{e (d+e x)^{2-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{\left (c d^2-b e d+a e^2\right ) (2-2 (p+3))}+\frac{2 c \left (\frac{e (d+e x)^{4-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{\left (c d^2-b e d+a e^2\right ) (4-2 (p+3))}-\frac{(2 c d-b e) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-b e-\sqrt{b^2-4 a c} e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} (d+e x)^{5-2 (p+3)} \left (c x^2+b x+a\right )^p \, _2F_1\left (-p,5-2 (p+3);6-2 (p+3);-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt{b^2-4 a c} e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 \left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \left (c d^2-b e d+a e^2\right ) (5-2 (p+3))}\right )+\frac{(e (b e (p+3)-2 c d (p+2))-2 c d e) \left (\frac{e (d+e x)^{3-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{\left (c d^2-b e d+a e^2\right ) (3-2 (p+3))}+\frac{\frac{\left (b (c d e+(b e (p+2)-c d (2 p+3)) e)-2 \left (a c e^2+c d (b e (p+2)-c d (2 p+3))\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-b e-\sqrt{b^2-4 a c} e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} (d+e x)^{5-2 (p+3)} \left (c x^2+b x+a\right )^p \, _2F_1\left (-p,5-2 (p+3);6-2 (p+3);-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt{b^2-4 a c} e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 \left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \left (c d^2-b e d+a e^2\right ) (5-2 (p+3))}-\frac{(e (b e (p+2)-c d (2 p+3))-c d e) (d+e x)^{4-2 (p+3)} \left (c x^2+b x+a\right )^{p+1}}{2 \left (c d^2-b e d+a e^2\right ) (p+1)}}{\left (c d^2-b e d+a e^2\right ) (3-2 (p+3))}\right )}{e}}{\left (c d^2-b e d+a e^2\right ) (2-2 (p+3))}\right )}{e}}{\left (c d^2-b e d+a e^2\right ) (1-2 (p+3))} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.268, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-6-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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