Optimal. Leaf size=98 \[ -\frac {b^5 (a x+b)^9}{9 a^6}+\frac {b^4 (a x+b)^{10}}{2 a^6}-\frac {10 b^3 (a x+b)^{11}}{11 a^6}+\frac {5 b^2 (a x+b)^{12}}{6 a^6}+\frac {(a x+b)^{14}}{14 a^6}-\frac {5 b (a x+b)^{13}}{13 a^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 98, 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 = {263, 43} \[ \frac {5 b^2 (a x+b)^{12}}{6 a^6}-\frac {10 b^3 (a x+b)^{11}}{11 a^6}+\frac {b^4 (a x+b)^{10}}{2 a^6}-\frac {b^5 (a x+b)^9}{9 a^6}+\frac {(a x+b)^{14}}{14 a^6}-\frac {5 b (a x+b)^{13}}{13 a^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 263
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^8 x^{13} \, dx &=\int x^5 (b+a x)^8 \, dx\\ &=\int \left (-\frac {b^5 (b+a x)^8}{a^5}+\frac {5 b^4 (b+a x)^9}{a^5}-\frac {10 b^3 (b+a x)^{10}}{a^5}+\frac {10 b^2 (b+a x)^{11}}{a^5}-\frac {5 b (b+a x)^{12}}{a^5}+\frac {(b+a x)^{13}}{a^5}\right ) \, dx\\ &=-\frac {b^5 (b+a x)^9}{9 a^6}+\frac {b^4 (b+a x)^{10}}{2 a^6}-\frac {10 b^3 (b+a x)^{11}}{11 a^6}+\frac {5 b^2 (b+a x)^{12}}{6 a^6}-\frac {5 b (b+a x)^{13}}{13 a^6}+\frac {(b+a x)^{14}}{14 a^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 106, normalized size = 1.08 \[ \frac {a^8 x^{14}}{14}+\frac {8}{13} a^7 b x^{13}+\frac {7}{3} a^6 b^2 x^{12}+\frac {56}{11} a^5 b^3 x^{11}+7 a^4 b^4 x^{10}+\frac {56}{9} a^3 b^5 x^9+\frac {7}{2} a^2 b^6 x^8+\frac {8}{7} a b^7 x^7+\frac {b^8 x^6}{6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.03, size = 90, normalized size = 0.92 \[ \frac {1}{14} \, a^{8} x^{14} + \frac {8}{13} \, a^{7} b x^{13} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {8}{7} \, a b^{7} x^{7} + \frac {1}{6} \, b^{8} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 90, normalized size = 0.92 \[ \frac {1}{14} \, a^{8} x^{14} + \frac {8}{13} \, a^{7} b x^{13} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {8}{7} \, a b^{7} x^{7} + \frac {1}{6} \, b^{8} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 91, normalized size = 0.93 \[ \frac {1}{14} a^{8} x^{14}+\frac {8}{13} a^{7} b \,x^{13}+\frac {7}{3} a^{6} b^{2} x^{12}+\frac {56}{11} a^{5} b^{3} x^{11}+7 a^{4} b^{4} x^{10}+\frac {56}{9} a^{3} b^{5} x^{9}+\frac {7}{2} a^{2} b^{6} x^{8}+\frac {8}{7} a \,b^{7} x^{7}+\frac {1}{6} b^{8} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.03, size = 90, normalized size = 0.92 \[ \frac {1}{14} \, a^{8} x^{14} + \frac {8}{13} \, a^{7} b x^{13} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {8}{7} \, a b^{7} x^{7} + \frac {1}{6} \, b^{8} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 90, normalized size = 0.92 \[ \frac {a^8\,x^{14}}{14}+\frac {8\,a^7\,b\,x^{13}}{13}+\frac {7\,a^6\,b^2\,x^{12}}{3}+\frac {56\,a^5\,b^3\,x^{11}}{11}+7\,a^4\,b^4\,x^{10}+\frac {56\,a^3\,b^5\,x^9}{9}+\frac {7\,a^2\,b^6\,x^8}{2}+\frac {8\,a\,b^7\,x^7}{7}+\frac {b^8\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 105, normalized size = 1.07 \[ \frac {a^{8} x^{14}}{14} + \frac {8 a^{7} b x^{13}}{13} + \frac {7 a^{6} b^{2} x^{12}}{3} + \frac {56 a^{5} b^{3} x^{11}}{11} + 7 a^{4} b^{4} x^{10} + \frac {56 a^{3} b^{5} x^{9}}{9} + \frac {7 a^{2} b^{6} x^{8}}{2} + \frac {8 a b^{7} x^{7}}{7} + \frac {b^{8} x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________