Optimal. Leaf size=577 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{4 (2 p+1) (2 p+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (a e (2 p+3)+b d \left (2 p^2+8 p+9\right )\right )+b^2 e^2 \left (p^2+5 p+6\right )+2 c^2 d^2 \left (2 p^2+8 p+9\right )\right )}{4 (p+1) (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (p+3) (2 c d-b e) (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^2}-\frac{e (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.867737, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {744, 836, 806, 726} \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{4 (2 p+1) (2 p+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (a e (2 p+3)+b d \left (2 p^2+8 p+9\right )\right )+b^2 e^2 \left (p^2+5 p+6\right )+2 c^2 d^2 \left (2 p^2+8 p+9\right )\right )}{4 (p+1) (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (p+3) (2 c d-b e) (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^2}-\frac{e (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 744
Rule 836
Rule 806
Rule 726
Rubi steps
\begin{align*} \int (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^p \, dx &=-\frac{e (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+p)}-\frac{\int (d+e x)^{-4-2 p} (-2 c d (2+p)+b e (3+p)+2 c e x) \left (a+b x+c x^2\right )^p \, dx}{2 \left (c d^2-b d e+a e^2\right ) (2+p)}\\ &=-\frac{e (2 c d-b e) (3+p) (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 )^2 (2+p) (3+2 p)}-\frac{e (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+p)}+\frac{\int (d+e x)^{-3-2 p} \left (b^2 e^2 \left (6+5 p+p^2\right )+2 c^2 d^2 \left (6+7 p+2 p^2\right )-c e \left (2 a e (3+2 p)+b d \left (15+15 p+4 p^2\right )\right )-c e (2 c d-b e) (3+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) (3+2 p)}\\ &=-\frac{e (2 c d-b e) (3+p) (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 )^2 (2+p) (3+2 p)}-\frac{e \left (b^2 e^2 \left (6+5 p+p^2\right )+2 c^2 d^2 \left (9+8 p+2 p^2\right )-2 c e \left (a e (3+2 p)+b d \left (9+8 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 )^3 (1+p) (2+p) (3+2 p)}-\frac{e (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+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 (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 )^3 (3+2 p)}\\ &=-\frac{e (2 c d-b e) (3+p) (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 )^2 (2+p) (3+2 p)}-\frac{e \left (b^2 e^2 \left (6+5 p+p^2\right )+2 c^2 d^2 \left (9+8 p+2 p^2\right )-2 c e \left (a e (3+2 p)+b d \left (9+8 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 )^3 (1+p) (2+p) (3+2 p)}-\frac{e (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+p)}+\frac{(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 (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 )^3 (1+2 p) (3+2 p)}\\ \end{align*}
Mathematica [A] time = 5.97402, size = 731, normalized size = 1.27 \[ -\frac{(d+e x)^{-2 (p+2)} (a+x (b+c x))^p \left (-\frac{(p+3) (d+e x) (b e-2 c d) \left (\frac{(d+e x)^2 \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )}\right )^{-p-1} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}\right )}{(2 p+1) \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right ) \left (e (a e-b d)+c d^2\right )}+\frac{e (p+2) (d+e x) (a+x (b+c x)) (2 c d-b e)}{(p+1) \left (e (a e-b d)+c d^2\right )}+2 e (a+x (b+c x))\right )}{(2 p+3) \left (e (a e-b d)+c d^2\right )}+\frac{2 c (d+e x)^2 \left (\frac{(d+e x) \left (\sqrt{b^2-4 a c}+b+2 c x\right ) (b e-2 c d) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )}\right )^{-p-1} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}\right )}{(2 p+1) \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )}-\frac{e (a+x (b+c x))}{p+1}\right )}{e (a e-b d)+c d^2}+2 e (a+x (b+c x))\right )}{4 (p+2) \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.311, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-5-2\,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 - 5}\,{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 - 5}, 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 - 5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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