Optimal. Leaf size=287 \[ \frac {B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac {(2 a B c (3+2 p)+b (2+p) (2 A c (2+p)-b B (3+p))-2 c (1+p) (2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}-\frac {2^{-1+p} \left (6 a b B c-4 a A c^2+2 A b^2 c (2+p)-b^3 B (3+p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} (1+p) (3+2 p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.19, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {846, 793, 638}
\begin {gather*} -\frac {2^{p-1} \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 (-4 a A c^2+6 a b B c+2 A b^2 c (p+2)+b^3 (-B) (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 {\left (a+b x+c x^2\right )^{p+1} (2 a B c (2 p+3)-2 c (p+1) x (2 A c (p+2)-b B (p+3))+b (p+2) (2 A c (p+2)-b B (p+3)))}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac {B x^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 638
Rule 793
Rule 846
Rubi steps
\begin {align*} \int x^2 (A+B x) \left (a+b x+c x^2\right )^p \, dx &=\frac {B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}+\frac {\int x (-2 a B+(2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^p \, dx}{2 c (2+p)}\\ &=\frac {B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac {(2 a B c (3+2 p)+b (2+p) (2 A c (2+p)-b B (3+p))-2 c (1+p) (2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}+\frac {\left (6 a b B c-4 a A c^2+2 A b^2 c (2+p)-b^3 B (3+p)\right ) \int \left (a+b x+c x^2\right )^p \, dx}{4 c^3 (3+2 p)}\\ &=\frac {B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac {(2 a B c (3+2 p)+b (2+p) (2 A c (2+p)-b B (3+p))-2 c (1+p) (2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}-\frac {2^{-1+p} \left (6 a b B c-4 a A c^2+2 A b^2 c (2+p)-b^3 B (3+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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.42, size = 210, normalized size = 0.73 \begin {gather*} \frac {1}{12} x^3 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} (a+x (b+c x))^p \left (4 A F_1\left (3;-p,-p;4;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+3 B x F_1\left (4;-p,-p;5;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int x^{2} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________