Optimal. Leaf size=74 \[ \frac {a^2 \left (a+b x^3\right )^{p+1}}{3 b^3 (p+1)}-\frac {2 a \left (a+b x^3\right )^{p+2}}{3 b^3 (p+2)}+\frac {\left (a+b x^3\right )^{p+3}}{3 b^3 (p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {a^2 \left (a+b x^3\right )^{p+1}}{3 b^3 (p+1)}-\frac {2 a \left (a+b x^3\right )^{p+2}}{3 b^3 (p+2)}+\frac {\left (a+b x^3\right )^{p+3}}{3 b^3 (p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^8 \left (a+b x^3\right )^p \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^2 (a+b x)^p \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^p}{b^2}-\frac {2 a (a+b x)^{1+p}}{b^2}+\frac {(a+b x)^{2+p}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac {a^2 \left (a+b x^3\right )^{1+p}}{3 b^3 (1+p)}-\frac {2 a \left (a+b x^3\right )^{2+p}}{3 b^3 (2+p)}+\frac {\left (a+b x^3\right )^{3+p}}{3 b^3 (3+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 64, normalized size = 0.86 \[ \frac {\left (a+b x^3\right )^{p+1} \left (2 a^2-2 a b (p+1) x^3+b^2 \left (p^2+3 p+2\right ) x^6\right )}{3 b^3 (p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 98, normalized size = 1.32 \[ \frac {{\left ({\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} x^{9} - 2 \, a^{2} b p x^{3} + {\left (a b^{2} p^{2} + a b^{2} p\right )} x^{6} + 2 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{p}}{3 \, {\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 231, normalized size = 3.12 \[ \frac {{\left (b x^{3} + a\right )}^{3} {\left (b x^{3} + a\right )}^{p} p^{2} - 2 \, {\left (b x^{3} + a\right )}^{2} {\left (b x^{3} + a\right )}^{p} a p^{2} + {\left (b x^{3} + a\right )} {\left (b x^{3} + a\right )}^{p} a^{2} p^{2} + 3 \, {\left (b x^{3} + a\right )}^{3} {\left (b x^{3} + a\right )}^{p} p - 8 \, {\left (b x^{3} + a\right )}^{2} {\left (b x^{3} + a\right )}^{p} a p + 5 \, {\left (b x^{3} + a\right )} {\left (b x^{3} + a\right )}^{p} a^{2} p + 2 \, {\left (b x^{3} + a\right )}^{3} {\left (b x^{3} + a\right )}^{p} - 6 \, {\left (b x^{3} + a\right )}^{2} {\left (b x^{3} + a\right )}^{p} a + 6 \, {\left (b x^{3} + a\right )} {\left (b x^{3} + a\right )}^{p} a^{2}}{3 \, {\left (b^{2} p^{3} + 6 \, b^{2} p^{2} + 11 \, b^{2} p + 6 \, b^{2}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 80, normalized size = 1.08 \[ \frac {\left (b^{2} p^{2} x^{6}+3 b^{2} p \,x^{6}+2 b^{2} x^{6}-2 a b p \,x^{3}-2 a b \,x^{3}+2 a^{2}\right ) \left (b \,x^{3}+a \right )^{p +1}}{3 \left (p^{3}+6 p^{2}+11 p +6\right ) b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.44, size = 73, normalized size = 0.99 \[ \frac {{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{9} + {\left (p^{2} + p\right )} a b^{2} x^{6} - 2 \, a^{2} b p x^{3} + 2 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{p}}{3 \, {\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.09, size = 118, normalized size = 1.59 \[ {\left (b\,x^3+a\right )}^p\,\left (\frac {2\,a^3}{3\,b^3\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {x^9\,\left (p^2+3\,p+2\right )}{3\,\left (p^3+6\,p^2+11\,p+6\right )}-\frac {2\,a^2\,p\,x^3}{3\,b^2\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {a\,p\,x^6\,\left (p+1\right )}{3\,b\,\left (p^3+6\,p^2+11\,p+6\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 32.72, size = 1370, normalized size = 18.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________