Optimal. Leaf size=45 \[ \frac {d^2 (b+2 c x)^5}{40 c^2}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^3}{24 c^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {683} \begin {gather*} \frac {d^2 (b+2 c x)^5}{40 c^2}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^3}{24 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 683
Rubi steps
\begin {align*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^2}{4 c}+\frac {(b d+2 c d x)^4}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3}{24 c^2}+\frac {d^2 (b+2 c x)^5}{40 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 64, normalized size = 1.42 \begin {gather*} d^2 \left (\frac {1}{3} c x^3 \left (4 a c+5 b^2\right )+\frac {1}{2} b x^2 \left (4 a c+b^2\right )+a b^2 x+2 b c^2 x^4+\frac {4 c^3 x^5}{5}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.36, size = 79, normalized size = 1.76 \begin {gather*} \frac {4}{5} x^{5} d^{2} c^{3} + 2 x^{4} d^{2} c^{2} b + \frac {5}{3} x^{3} d^{2} c b^{2} + \frac {4}{3} x^{3} d^{2} c^{2} a + \frac {1}{2} x^{2} d^{2} b^{3} + 2 x^{2} d^{2} c b a + x d^{2} b^{2} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 79, normalized size = 1.76 \begin {gather*} \frac {4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + \frac {5}{3} \, b^{2} c d^{2} x^{3} + \frac {4}{3} \, a c^{2} d^{2} x^{3} + \frac {1}{2} \, b^{3} d^{2} x^{2} + 2 \, a b c d^{2} x^{2} + a b^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 79, normalized size = 1.76 \begin {gather*} \frac {4 c^{3} d^{2} x^{5}}{5}+2 b \,c^{2} d^{2} x^{4}+a \,b^{2} d^{2} x +\frac {\left (4 c^{2} d^{2} a +5 b^{2} d^{2} c \right ) x^{3}}{3}+\frac {\left (4 b c \,d^{2} a +b^{3} d^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.33, size = 71, normalized size = 1.58 \begin {gather*} \frac {4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + a b^{2} d^{2} x + \frac {1}{3} \, {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (b^{3} + 4 \, a b c\right )} d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 69, normalized size = 1.53 \begin {gather*} \frac {4\,c^3\,d^2\,x^5}{5}+\frac {c\,d^2\,x^3\,\left (5\,b^2+4\,a\,c\right )}{3}+2\,b\,c^2\,d^2\,x^4+\frac {b\,d^2\,x^2\,\left (b^2+4\,a\,c\right )}{2}+a\,b^2\,d^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.09, size = 85, normalized size = 1.89 \begin {gather*} a b^{2} d^{2} x + 2 b c^{2} d^{2} x^{4} + \frac {4 c^{3} d^{2} x^{5}}{5} + x^{3} \left (\frac {4 a c^{2} d^{2}}{3} + \frac {5 b^{2} c d^{2}}{3}\right ) + x^{2} \left (2 a b c d^{2} + \frac {b^{3} d^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________