Optimal. Leaf size=183 \[ \frac {165 a^9 \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac {55 a^9 \log (x)}{b^{12}}-\frac {30 a^9}{b^{11} \left (a \sqrt [3]{x}+b\right )}-\frac {3 a^9}{2 b^{10} \left (a \sqrt [3]{x}+b\right )^2}-\frac {135 a^8}{b^{11} \sqrt [3]{x}}+\frac {54 a^7}{b^{10} x^{2/3}}-\frac {28 a^6}{b^9 x}+\frac {63 a^5}{4 b^8 x^{4/3}}-\frac {9 a^4}{b^7 x^{5/3}}+\frac {5 a^3}{b^6 x^2}-\frac {18 a^2}{7 b^5 x^{7/3}}+\frac {9 a}{8 b^4 x^{8/3}}-\frac {1}{3 b^3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 44} \[ \frac {54 a^7}{b^{10} x^{2/3}}+\frac {63 a^5}{4 b^8 x^{4/3}}-\frac {9 a^4}{b^7 x^{5/3}}+\frac {5 a^3}{b^6 x^2}-\frac {18 a^2}{7 b^5 x^{7/3}}-\frac {30 a^9}{b^{11} \left (a \sqrt [3]{x}+b\right )}-\frac {3 a^9}{2 b^{10} \left (a \sqrt [3]{x}+b\right )^2}-\frac {135 a^8}{b^{11} \sqrt [3]{x}}-\frac {28 a^6}{b^9 x}+\frac {165 a^9 \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac {55 a^9 \log (x)}{b^{12}}+\frac {9 a}{8 b^4 x^{8/3}}-\frac {1}{3 b^3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right )^3 x^4} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{x^{10} (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{b^3 x^{10}}-\frac {3 a}{b^4 x^9}+\frac {6 a^2}{b^5 x^8}-\frac {10 a^3}{b^6 x^7}+\frac {15 a^4}{b^7 x^6}-\frac {21 a^5}{b^8 x^5}+\frac {28 a^6}{b^9 x^4}-\frac {36 a^7}{b^{10} x^3}+\frac {45 a^8}{b^{11} x^2}-\frac {55 a^9}{b^{12} x}+\frac {a^{10}}{b^{10} (b+a x)^3}+\frac {10 a^{10}}{b^{11} (b+a x)^2}+\frac {55 a^{10}}{b^{12} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^9}{2 b^{10} \left (b+a \sqrt [3]{x}\right )^2}-\frac {30 a^9}{b^{11} \left (b+a \sqrt [3]{x}\right )}-\frac {1}{3 b^3 x^3}+\frac {9 a}{8 b^4 x^{8/3}}-\frac {18 a^2}{7 b^5 x^{7/3}}+\frac {5 a^3}{b^6 x^2}-\frac {9 a^4}{b^7 x^{5/3}}+\frac {63 a^5}{4 b^8 x^{4/3}}-\frac {28 a^6}{b^9 x}+\frac {54 a^7}{b^{10} x^{2/3}}-\frac {135 a^8}{b^{11} \sqrt [3]{x}}+\frac {165 a^9 \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}-\frac {55 a^9 \log (x)}{b^{12}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 167, normalized size = 0.91 \[ -\frac {-27720 a^9 \log \left (a \sqrt [3]{x}+b\right )+9240 a^9 \log (x)+\frac {b \left (27720 a^{10} x^{10/3}+41580 a^9 b x^3+9240 a^8 b^2 x^{8/3}-2310 a^7 b^3 x^{7/3}+924 a^6 b^4 x^2-462 a^5 b^5 x^{5/3}+264 a^4 b^6 x^{4/3}-165 a^3 b^7 x+110 a^2 b^8 x^{2/3}-77 a b^9 \sqrt [3]{x}+56 b^{10}\right )}{x^3 \left (a \sqrt [3]{x}+b\right )^2}}{168 b^{12}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 263, normalized size = 1.44 \[ -\frac {9240 \, a^{12} b^{3} x^{4} + 13860 \, a^{9} b^{6} x^{3} + 3080 \, a^{6} b^{9} x^{2} - 728 \, a^{3} b^{12} x + 56 \, b^{15} - 27720 \, {\left (a^{15} x^{5} + 2 \, a^{12} b^{3} x^{4} + a^{9} b^{6} x^{3}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 27720 \, {\left (a^{15} x^{5} + 2 \, a^{12} b^{3} x^{4} + a^{9} b^{6} x^{3}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, {\left (1540 \, a^{14} b x^{4} + 2695 \, a^{11} b^{4} x^{3} + 990 \, a^{8} b^{7} x^{2} - 99 \, a^{5} b^{10} x + 24 \, a^{2} b^{13}\right )} x^{\frac {2}{3}} - 63 \, {\left (220 \, a^{13} b^{2} x^{4} + 352 \, a^{10} b^{5} x^{3} + 99 \, a^{7} b^{8} x^{2} - 18 \, a^{4} b^{11} x + 3 \, a b^{14}\right )} x^{\frac {1}{3}}}{168 \, {\left (a^{6} b^{12} x^{5} + 2 \, a^{3} b^{15} x^{4} + b^{18} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 156, normalized size = 0.85 \[ \frac {165 \, a^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{12}} - \frac {55 \, a^{9} \log \left ({\left | x \right |}\right )}{b^{12}} - \frac {27720 \, a^{10} b x^{\frac {10}{3}} + 41580 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac {8}{3}} - 2310 \, a^{7} b^{4} x^{\frac {7}{3}} + 924 \, a^{6} b^{5} x^{2} - 462 \, a^{5} b^{6} x^{\frac {5}{3}} + 264 \, a^{4} b^{7} x^{\frac {4}{3}} - 165 \, a^{3} b^{8} x + 110 \, a^{2} b^{9} x^{\frac {2}{3}} - 77 \, a b^{10} x^{\frac {1}{3}} + 56 \, b^{11}}{168 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{12} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 156, normalized size = 0.85 \[ -\frac {3 a^{9}}{2 \left (a \,x^{\frac {1}{3}}+b \right )^{2} b^{10}}-\frac {30 a^{9}}{\left (a \,x^{\frac {1}{3}}+b \right ) b^{11}}-\frac {55 a^{9} \ln \relax (x )}{b^{12}}+\frac {165 a^{9} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{b^{12}}-\frac {135 a^{8}}{b^{11} x^{\frac {1}{3}}}+\frac {54 a^{7}}{b^{10} x^{\frac {2}{3}}}-\frac {28 a^{6}}{b^{9} x}+\frac {63 a^{5}}{4 b^{8} x^{\frac {4}{3}}}-\frac {9 a^{4}}{b^{7} x^{\frac {5}{3}}}+\frac {5 a^{3}}{b^{6} x^{2}}-\frac {18 a^{2}}{7 b^{5} x^{\frac {7}{3}}}+\frac {9 a}{8 b^{4} x^{\frac {8}{3}}}-\frac {1}{3 b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 197, normalized size = 1.08 \[ \frac {165 \, a^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{12}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{9}}{3 \, b^{12}} + \frac {33 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8} a}{8 \, b^{12}} - \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a^{2}}{7 \, b^{12}} + \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{3}}{2 \, b^{12}} - \frac {198 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{4}}{b^{12}} + \frac {693 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{5}}{2 \, b^{12}} - \frac {462 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{6}}{b^{12}} + \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{7}}{b^{12}} - \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{8}}{b^{12}} + \frac {33 \, a^{10}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{12}} - \frac {3 \, a^{11}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.31, size = 159, normalized size = 0.87 \[ \frac {330\,a^9\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^{12}}-\frac {\frac {1}{3\,b}-\frac {11\,a\,x^{1/3}}{24\,b^2}-\frac {55\,a^3\,x}{56\,b^4}+\frac {55\,a^2\,x^{2/3}}{84\,b^3}+\frac {11\,a^6\,x^2}{2\,b^7}+\frac {11\,a^4\,x^{4/3}}{7\,b^5}-\frac {11\,a^5\,x^{5/3}}{4\,b^6}+\frac {495\,a^9\,x^3}{2\,b^{10}}-\frac {55\,a^7\,x^{7/3}}{4\,b^8}+\frac {55\,a^8\,x^{8/3}}{b^9}+\frac {165\,a^{10}\,x^{10/3}}{b^{11}}}{a^2\,x^{11/3}+b^2\,x^3+2\,a\,b\,x^{10/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 40.81, size = 848, normalized size = 4.63 \[ \begin {cases} \frac {\tilde {\infty }}{x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 b^{3} x^{3}} & \text {for}\: a = 0 \\- \frac {1}{4 a^{3} x^{4}} & \text {for}\: b = 0 \\- \frac {9240 a^{11} x^{\frac {16}{3}} \log {\relax (x )}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {27720 a^{11} x^{\frac {16}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {18480 a^{10} b x^{5} \log {\relax (x )}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {55440 a^{10} b x^{5} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {27720 a^{10} b x^{5}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {9240 a^{9} b^{2} x^{\frac {14}{3}} \log {\relax (x )}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {27720 a^{9} b^{2} x^{\frac {14}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {41580 a^{9} b^{2} x^{\frac {14}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {9240 a^{8} b^{3} x^{\frac {13}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {2310 a^{7} b^{4} x^{4}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {924 a^{6} b^{5} x^{\frac {11}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {462 a^{5} b^{6} x^{\frac {10}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {264 a^{4} b^{7} x^{3}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {165 a^{3} b^{8} x^{\frac {8}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {110 a^{2} b^{9} x^{\frac {7}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} + \frac {77 a b^{10} x^{2}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} - \frac {56 b^{11} x^{\frac {5}{3}}}{168 a^{2} b^{12} x^{\frac {16}{3}} + 336 a b^{13} x^{5} + 168 b^{14} x^{\frac {14}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________