Optimal. Leaf size=41 \[ \frac {1}{8} a c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b c^2 x^8 \sqrt {c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac {1}{8} a c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b c^2 x^8 \sqrt {c x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin {align*} \int x^2 \left (c x^2\right )^{5/2} (a+b x) \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^7 (a+b x) \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^7+b x^8\right ) \, dx}{x}\\ &=\frac {1}{8} a c^2 x^7 \sqrt {c x^2}+\frac {1}{9} b c^2 x^8 \sqrt {c x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 24, normalized size = 0.59 \[ \frac {1}{72} x^3 \left (c x^2\right )^{5/2} (9 a+8 b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 28, normalized size = 0.68 \[ \frac {1}{72} \, {\left (8 \, b c^{2} x^{8} + 9 \, a c^{2} x^{7}\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.80, size = 28, normalized size = 0.68 \[ \frac {1}{72} \, {\left (8 \, b c^{2} x^{9} \mathrm {sgn}\relax (x) + 9 \, a c^{2} x^{8} \mathrm {sgn}\relax (x)\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 21, normalized size = 0.51 \[ \frac {\left (8 b x +9 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}} x^{3}}{72} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.30, size = 31, normalized size = 0.76 \[ \frac {\left (c x^{2}\right )^{\frac {7}{2}} b x^{2}}{9 \, c} + \frac {\left (c x^{2}\right )^{\frac {7}{2}} a x}{8 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,{\left (c\,x^2\right )}^{5/2}\,\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.10, size = 36, normalized size = 0.88 \[ \frac {a c^{\frac {5}{2}} x^{3} \left (x^{2}\right )^{\frac {5}{2}}}{8} + \frac {b c^{\frac {5}{2}} x^{4} \left (x^{2}\right )^{\frac {5}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________