Optimal. Leaf size=19 \[ \frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {629} \begin {gather*} \frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 629
Rubi steps
\begin {align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {2}{7} d \left (a+b x+c x^2\right )^{7/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 0.95 \begin {gather*} \frac {2}{7} d (a+x (b+c x))^{7/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 94, normalized size = 4.95 \begin {gather*} \frac {2}{7} \, {\left (c^{3} d x^{6} + 3 \, b c^{2} d x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} d x^{4} + 3 \, a^{2} b d x + {\left (b^{3} + 6 \, a b c\right )} d x^{3} + a^{3} d + 3 \, {\left (a b^{2} + a^{2} c\right )} d x^{2}\right )} \sqrt {c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 16, normalized size = 0.84 \begin {gather*} \frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} d}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,d\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.84, size = 260, normalized size = 13.68 \begin {gather*} \frac {2 a^{3} d \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a^{2} b d x \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a^{2} c d x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a b^{2} d x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {12 a b c d x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a c^{2} d x^{4} \sqrt {a + b x + c x^{2}}}{7} + \frac {2 b^{3} d x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 b^{2} c d x^{4} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 b c^{2} d x^{5} \sqrt {a + b x + c x^{2}}}{7} + \frac {2 c^{3} d x^{6} \sqrt {a + b x + c x^{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________