Optimal. Leaf size=107 \[ -\frac {2 (b d+2 c d x)^{1+m} \left (\frac {1}{4} \left (4 a-\frac {b^2}{c}\right )+\frac {(b+2 c x)^2}{4 c}\right )^{1+p} \, _2F_1\left (1,\frac {1}{2} (3+m+2 p);\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 0.95, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {708, 372, 371}
\begin {gather*} \frac {\left (a+b x+c x^2\right )^p \left (1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )^{-p} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 372
Rule 708
Rubi steps
\begin {align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^p \, dx &=\frac {\text {Subst}\left (\int x^m \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^p \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac {\left (2^{-1+2 p} \left (a+b x+c x^2\right )^p \left (4+\frac {(b d+2 c d x)^2}{\left (a-\frac {b^2}{4 c}\right ) c d^2}\right )^{-p}\right ) \text {Subst}\left (\int x^m \left (1+\frac {x^2}{4 \left (a-\frac {b^2}{4 c}\right ) c d^2}\right )^p \, dx,x,b d+2 c d x\right )}{c d}\\ &=\frac {2^{-1+2 p} (d (b+2 c x))^{1+m} \left (a+b x+c x^2\right )^p \left (4-\frac {4 (b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c d (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.21, size = 113, normalized size = 1.06 \begin {gather*} \frac {2^{-1-2 p} (b+2 c x) (d (b+2 c x))^m (a+x (b+c x))^p \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{-p} \, _2F_1\left (\frac {1}{2}+\frac {m}{2},-p;\frac {3}{2}+\frac {m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.47, size = 0, normalized size = 0.00 \[\int \left (2 c d x +b d \right )^{m} \left (c \,x^{2}+b x +a \right )^{p}\, dx\]
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(-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 (b\,d+2\,c\,d\,x\right )}^m\,{\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]
________________________________________________________________________________________