Optimal. Leaf size=136 \[ -\frac {2^{1+q} \left (-\frac {e-\sqrt {-16 a c+e^2}+4 c x}{\sqrt {-16 a c+e^2}}\right )^{-1-p-q} \left (2 a+e x+2 c x^2\right )^{1+p+q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac {e+\sqrt {-16 a c+e^2}+4 c x}{2 \sqrt {-16 a c+e^2}}\right )}{\sqrt {-16 a c+e^2} (1+p+q)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {981, 638}
\begin {gather*} -\frac {2^{q+1} \left (-\frac {-\sqrt {e^2-16 a c}+4 c x+e}{\sqrt {e^2-16 a c}}\right )^{-p-q-1} \left (2 a+2 c x^2+e x\right )^{p+q+1} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac {e+4 c x+\sqrt {e^2-16 a c}}{2 \sqrt {e^2-16 a c}}\right )}{(p+q+1) \sqrt {e^2-16 a c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 638
Rule 981
Rubi steps
\begin {align*} \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx &=2^{-p} \int \left (2 a+e x+2 c x^2\right )^{p+q} \, dx\\ &=-\frac {2^{1+q} \left (-\frac {e-\sqrt {-16 a c+e^2}+4 c x}{\sqrt {-16 a c+e^2}}\right )^{-1-p-q} \left (2 a+e x+2 c x^2\right )^{1+p+q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac {e+\sqrt {-16 a c+e^2}+4 c x}{2 \sqrt {-16 a c+e^2}}\right )}{\sqrt {-16 a c+e^2} (1+p+q)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 142, normalized size = 1.04 \begin {gather*} \frac {2^{-2+q} \left (e-\sqrt {-16 a c+e^2}+4 c x\right ) \left (\frac {e+\sqrt {-16 a c+e^2}+4 c x}{\sqrt {-16 a c+e^2}}\right )^{-p-q} (2 a+x (e+2 c x))^{p+q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac {-e+\sqrt {-16 a c+e^2}-4 c x}{2 \sqrt {-16 a c+e^2}}\right )}{c (1+p+q)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (a +\frac {1}{2} e x +c \,x^{2}\right )^{p} \left (2 c \,x^{2}+e x +2 a \right )^{q}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\left (c\,x^2+\frac {e\,x}{2}+a\right )}^p\,{\left (2\,c\,x^2+e\,x+2\,a\right )}^q \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________