Optimal. Leaf size=90 \[ -\frac {2 d^2 (b+2 c x)^3 \left (\frac {1}{4} \left (4 a-\frac {b^2}{c}\right )+\frac {(b+2 c x)^2}{4 c}\right )^{1+p} \, _2F_1\left (1,\frac {5}{2}+p;\frac {5}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 85, normalized size of antiderivative = 0.94, 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 {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^p \left (1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{6 c} \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)^2 \left (a+b x+c x^2\right )^p \, dx &=\frac {\text {Subst}\left (\int x^2 \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^2 \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^2 (b+2 c x)^3 \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 {3}{2},-p;\frac {5}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.42, size = 92, normalized size = 1.02 \begin {gather*} \frac {2^{-1-2 p} d^2 (b+2 c x)^3 (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 {3}{2},-p;\frac {5}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \left (2 c d x +b d \right )^{2} \left (c \,x^{2}+b x +a \right )^{p}\, dx\]
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*} d^{2} \left (\int b^{2} \left (a + b x + c x^{2}\right )^{p}\, dx + \int 4 c^{2} x^{2} \left (a + b x + c x^{2}\right )^{p}\, dx + \int 4 b c x \left (a + b x + c x^{2}\right )^{p}\, dx\right ) \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 )}^2\,{\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]
________________________________________________________________________________________