Optimal. Leaf size=62 \[ \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+3 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^8}{b^2}-\frac {2 a (a+b x)^9}{b^2}+\frac {(a+b x)^{10}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 40, normalized size = 0.65 \[ \frac {\left (a+b x^n\right )^9 \left (a^2-9 a b x^n+45 b^2 x^{2 n}\right )}{495 b^3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.84, size = 113, normalized size = 1.82 \[ \frac {45 \, b^{8} x^{11 \, n} + 396 \, a b^{7} x^{10 \, n} + 1540 \, a^{2} b^{6} x^{9 \, n} + 3465 \, a^{3} b^{5} x^{8 \, n} + 4950 \, a^{4} b^{4} x^{7 \, n} + 4620 \, a^{5} b^{3} x^{6 \, n} + 2772 \, a^{6} b^{2} x^{5 \, n} + 990 \, a^{7} b x^{4 \, n} + 165 \, a^{8} x^{3 \, n}}{495 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{8} x^{3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 136, normalized size = 2.19 \[ \frac {a^{8} x^{3 n}}{3 n}+\frac {2 a^{7} b \,x^{4 n}}{n}+\frac {28 a^{6} b^{2} x^{5 n}}{5 n}+\frac {28 a^{5} b^{3} x^{6 n}}{3 n}+\frac {10 a^{4} b^{4} x^{7 n}}{n}+\frac {7 a^{3} b^{5} x^{8 n}}{n}+\frac {28 a^{2} b^{6} x^{9 n}}{9 n}+\frac {4 a \,b^{7} x^{10 n}}{5 n}+\frac {b^{8} x^{11 n}}{11 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.46, size = 135, normalized size = 2.18 \[ \frac {b^{8} x^{11 \, n}}{11 \, n} + \frac {4 \, a b^{7} x^{10 \, n}}{5 \, n} + \frac {28 \, a^{2} b^{6} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{3} b^{5} x^{8 \, n}}{n} + \frac {10 \, a^{4} b^{4} x^{7 \, n}}{n} + \frac {28 \, a^{5} b^{3} x^{6 \, n}}{3 \, n} + \frac {28 \, a^{6} b^{2} x^{5 \, n}}{5 \, n} + \frac {2 \, a^{7} b x^{4 \, n}}{n} + \frac {a^{8} x^{3 \, n}}{3 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.45, size = 135, normalized size = 2.18 \[ \frac {a^8\,x^{3\,n}}{3\,n}+\frac {b^8\,x^{11\,n}}{11\,n}+\frac {28\,a^6\,b^2\,x^{5\,n}}{5\,n}+\frac {28\,a^5\,b^3\,x^{6\,n}}{3\,n}+\frac {10\,a^4\,b^4\,x^{7\,n}}{n}+\frac {7\,a^3\,b^5\,x^{8\,n}}{n}+\frac {28\,a^2\,b^6\,x^{9\,n}}{9\,n}+\frac {2\,a^7\,b\,x^{4\,n}}{n}+\frac {4\,a\,b^7\,x^{10\,n}}{5\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________