Optimal. Leaf size=106 \[ -\frac {\left (a+b x^3\right )^9}{39 a x^{39}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac {b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac {b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac {b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 47, 37}
\begin {gather*} -\frac {b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}+\frac {b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac {b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac {\left (a+b x^3\right )^9}{39 a x^{39}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^{14}} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^9}{39 a x^{39}}-\frac {(4 b) \text {Subst}\left (\int \frac {(a+b x)^8}{x^{13}} \, dx,x,x^3\right )}{39 a}\\ &=-\frac {\left (a+b x^3\right )^9}{39 a x^{39}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^3\right )}{39 a^2}\\ &=-\frac {\left (a+b x^3\right )^9}{39 a x^{39}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac {b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^3\right )}{429 a^3}\\ &=-\frac {\left (a+b x^3\right )^9}{39 a x^{39}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac {b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac {b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}+\frac {b^4 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{2145 a^4}\\ &=-\frac {\left (a+b x^3\right )^9}{39 a x^{39}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac {b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac {b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac {b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 108, normalized size = 1.02 \begin {gather*} -\frac {a^8}{39 x^{39}}-\frac {2 a^7 b}{9 x^{36}}-\frac {28 a^6 b^2}{33 x^{33}}-\frac {28 a^5 b^3}{15 x^{30}}-\frac {70 a^4 b^4}{27 x^{27}}-\frac {7 a^3 b^5}{3 x^{24}}-\frac {4 a^2 b^6}{3 x^{21}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{15 x^{15}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 91, normalized size = 0.86
method | result | size |
default | \(-\frac {b^{8}}{15 x^{15}}-\frac {a^{8}}{39 x^{39}}-\frac {4 a^{2} b^{6}}{3 x^{21}}-\frac {2 b \,a^{7}}{9 x^{36}}-\frac {28 a^{6} b^{2}}{33 x^{33}}-\frac {7 a^{3} b^{5}}{3 x^{24}}-\frac {4 a \,b^{7}}{9 x^{18}}-\frac {70 a^{4} b^{4}}{27 x^{27}}-\frac {28 a^{5} b^{3}}{15 x^{30}}\) | \(91\) |
risch | \(\frac {-\frac {4}{9} a \,b^{7} x^{21}-\frac {1}{15} b^{8} x^{24}-\frac {28}{33} a^{6} b^{2} x^{6}-\frac {28}{15} a^{5} b^{3} x^{9}-\frac {70}{27} a^{4} b^{4} x^{12}-\frac {7}{3} a^{3} b^{5} x^{15}-\frac {4}{3} a^{2} b^{6} x^{18}-\frac {2}{9} a^{7} b \,x^{3}-\frac {1}{39} a^{8}}{x^{39}}\) | \(92\) |
gosper | \(-\frac {1287 b^{8} x^{24}+8580 a \,b^{7} x^{21}+25740 a^{2} b^{6} x^{18}+45045 a^{3} b^{5} x^{15}+50050 a^{4} b^{4} x^{12}+36036 a^{5} b^{3} x^{9}+16380 a^{6} b^{2} x^{6}+4290 a^{7} b \,x^{3}+495 a^{8}}{19305 x^{39}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 92, normalized size = 0.87 \begin {gather*} -\frac {1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 92, normalized size = 0.87 \begin {gather*} -\frac {1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.63, size = 99, normalized size = 0.93 \begin {gather*} \frac {- 495 a^{8} - 4290 a^{7} b x^{3} - 16380 a^{6} b^{2} x^{6} - 36036 a^{5} b^{3} x^{9} - 50050 a^{4} b^{4} x^{12} - 45045 a^{3} b^{5} x^{15} - 25740 a^{2} b^{6} x^{18} - 8580 a b^{7} x^{21} - 1287 b^{8} x^{24}}{19305 x^{39}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.33, size = 92, normalized size = 0.87 \begin {gather*} -\frac {1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.00, size = 92, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^8}{39}+\frac {2\,a^7\,b\,x^3}{9}+\frac {28\,a^6\,b^2\,x^6}{33}+\frac {28\,a^5\,b^3\,x^9}{15}+\frac {70\,a^4\,b^4\,x^{12}}{27}+\frac {7\,a^3\,b^5\,x^{15}}{3}+\frac {4\,a^2\,b^6\,x^{18}}{3}+\frac {4\,a\,b^7\,x^{21}}{9}+\frac {b^8\,x^{24}}{15}}{x^{39}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________