Optimal. Leaf size=248 \[ -\frac{2^p \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{e (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.209372, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {742, 640, 624} \[ -\frac{2^p \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{e (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 742
Rule 640
Rule 624
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^p \, dx &=\frac{e (d+e x) \left (a+b x+c x^2\right )^{1+p}}{c (3+2 p)}+\frac{\int \left (c d^2 (3+2 p)-e (a e+b d (1+p))+e (2 c d-b e) (2+p) x\right ) \left (a+b x+c x^2\right )^p \, dx}{c (3+2 p)}\\ &=\frac{e (2 c d-b e) (2+p) \left (a+b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{1+p}}{c (3+2 p)}+\frac{\left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 c e (a e+b d (3+2 p))\right ) \int \left (a+b x+c x^2\right )^p \, dx}{2 c^2 (3+2 p)}\\ &=\frac{e (2 c d-b e) (2+p) \left (a+b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{1+p}}{c (3+2 p)}-\frac{2^p \left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 c e (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^2 \sqrt{b^2-4 a c} (1+p) (3+2 p)}\\ \end{align*}
Mathematica [C] time = 0.811754, size = 414, normalized size = 1.67 \[ \frac{1}{6} (a+x (b+c x))^p \left (6 d e x^2 \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )^{-p} F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+2 e^2 x^3 \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )^{-p} F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+\frac{3 d^2 2^p \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.245, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (c x^{2} + b x + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c x^{2} + b x + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (c x^{2} + b x + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]