Optimal. Leaf size=84 \[ \frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {\left (a+b x^3\right )^9}{36 a x^{36}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {\left (a+b x^3\right )^9}{36 a x^{36}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{13}} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^3\right )}{12 a}\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^3\right )}{66 a^2}\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}-\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{660 a^3}\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 108, normalized size = 1.29 \[ -\frac {a^8}{36 x^{36}}-\frac {8 a^7 b}{33 x^{33}}-\frac {14 a^6 b^2}{15 x^{30}}-\frac {56 a^5 b^3}{27 x^{27}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {8 a^3 b^5}{3 x^{21}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {8 a b^7}{15 x^{15}}-\frac {b^8}{12 x^{12}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 92, normalized size = 1.10 \[ -\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 92, normalized size = 1.10 \[ -\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 91, normalized size = 1.08 \[ -\frac {b^{8}}{12 x^{12}}-\frac {8 a \,b^{7}}{15 x^{15}}-\frac {14 a^{2} b^{6}}{9 x^{18}}-\frac {8 a^{3} b^{5}}{3 x^{21}}-\frac {35 a^{4} b^{4}}{12 x^{24}}-\frac {56 a^{5} b^{3}}{27 x^{27}}-\frac {14 a^{6} b^{2}}{15 x^{30}}-\frac {8 a^{7} b}{33 x^{33}}-\frac {a^{8}}{36 x^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 92, normalized size = 1.10 \[ -\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 92, normalized size = 1.10 \[ -\frac {\frac {a^8}{36}+\frac {8\,a^7\,b\,x^3}{33}+\frac {14\,a^6\,b^2\,x^6}{15}+\frac {56\,a^5\,b^3\,x^9}{27}+\frac {35\,a^4\,b^4\,x^{12}}{12}+\frac {8\,a^3\,b^5\,x^{15}}{3}+\frac {14\,a^2\,b^6\,x^{18}}{9}+\frac {8\,a\,b^7\,x^{21}}{15}+\frac {b^8\,x^{24}}{12}}{x^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.49, size = 99, normalized size = 1.18 \[ \frac {- 165 a^{8} - 1440 a^{7} b x^{3} - 5544 a^{6} b^{2} x^{6} - 12320 a^{5} b^{3} x^{9} - 17325 a^{4} b^{4} x^{12} - 15840 a^{3} b^{5} x^{15} - 9240 a^{2} b^{6} x^{18} - 3168 a b^{7} x^{21} - 495 b^{8} x^{24}}{5940 x^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________