Optimal. Leaf size=162 \[ -\frac {3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {1}{3 a^2 x^3}+\frac {3 b}{4 a^3 x^{8/3}}-\frac {9 b^2}{7 a^4 x^{7/3}}+\frac {2 b^3}{a^5 x^2}-\frac {3 b^4}{a^6 x^{5/3}}+\frac {9 b^5}{2 a^7 x^{4/3}}-\frac {7 b^6}{a^8 x}+\frac {12 b^7}{a^9 x^{2/3}}-\frac {27 b^8}{a^{10} \sqrt [3]{x}}+\frac {30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac {10 b^9 \log (x)}{a^{11}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46}
\begin {gather*} \frac {30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac {10 b^9 \log (x)}{a^{11}}-\frac {3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {27 b^8}{a^{10} \sqrt [3]{x}}+\frac {12 b^7}{a^9 x^{2/3}}-\frac {7 b^6}{a^8 x}+\frac {9 b^5}{2 a^7 x^{4/3}}-\frac {3 b^4}{a^6 x^{5/3}}+\frac {2 b^3}{a^5 x^2}-\frac {9 b^2}{7 a^4 x^{7/3}}+\frac {3 b}{4 a^3 x^{8/3}}-\frac {1}{3 a^2 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx &=3 \text {Subst}\left (\int \frac {1}{x^{10} (a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {1}{a^2 x^{10}}-\frac {2 b}{a^3 x^9}+\frac {3 b^2}{a^4 x^8}-\frac {4 b^3}{a^5 x^7}+\frac {5 b^4}{a^6 x^6}-\frac {6 b^5}{a^7 x^5}+\frac {7 b^6}{a^8 x^4}-\frac {8 b^7}{a^9 x^3}+\frac {9 b^8}{a^{10} x^2}-\frac {10 b^9}{a^{11} x}+\frac {b^{10}}{a^{10} (a+b x)^2}+\frac {10 b^{10}}{a^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {1}{3 a^2 x^3}+\frac {3 b}{4 a^3 x^{8/3}}-\frac {9 b^2}{7 a^4 x^{7/3}}+\frac {2 b^3}{a^5 x^2}-\frac {3 b^4}{a^6 x^{5/3}}+\frac {9 b^5}{2 a^7 x^{4/3}}-\frac {7 b^6}{a^8 x}+\frac {12 b^7}{a^9 x^{2/3}}-\frac {27 b^8}{a^{10} \sqrt [3]{x}}+\frac {30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac {10 b^9 \log (x)}{a^{11}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 154, normalized size = 0.95 \begin {gather*} -\frac {\frac {a \left (28 a^9-35 a^8 b \sqrt [3]{x}+45 a^7 b^2 x^{2/3}-60 a^6 b^3 x+84 a^5 b^4 x^{4/3}-126 a^4 b^5 x^{5/3}+210 a^3 b^6 x^2-420 a^2 b^7 x^{7/3}+1260 a b^8 x^{8/3}+2520 b^9 x^3\right )}{\left (a+b \sqrt [3]{x}\right ) x^3}-2520 b^9 \log \left (a+b \sqrt [3]{x}\right )+840 b^9 \log (x)}{84 a^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 139, normalized size = 0.86
method | result | size |
derivativedivides | \(-\frac {3 b^{9}}{a^{10} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{3 a^{2} x^{3}}+\frac {3 b}{4 a^{3} x^{\frac {8}{3}}}-\frac {9 b^{2}}{7 a^{4} x^{\frac {7}{3}}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{4}}{a^{6} x^{\frac {5}{3}}}+\frac {9 b^{5}}{2 a^{7} x^{\frac {4}{3}}}-\frac {7 b^{6}}{a^{8} x}+\frac {12 b^{7}}{a^{9} x^{\frac {2}{3}}}-\frac {27 b^{8}}{a^{10} x^{\frac {1}{3}}}+\frac {30 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{11}}-\frac {10 b^{9} \ln \left (x \right )}{a^{11}}\) | \(139\) |
default | \(-\frac {3 b^{9}}{a^{10} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{3 a^{2} x^{3}}+\frac {3 b}{4 a^{3} x^{\frac {8}{3}}}-\frac {9 b^{2}}{7 a^{4} x^{\frac {7}{3}}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{4}}{a^{6} x^{\frac {5}{3}}}+\frac {9 b^{5}}{2 a^{7} x^{\frac {4}{3}}}-\frac {7 b^{6}}{a^{8} x}+\frac {12 b^{7}}{a^{9} x^{\frac {2}{3}}}-\frac {27 b^{8}}{a^{10} x^{\frac {1}{3}}}+\frac {30 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{11}}-\frac {10 b^{9} \ln \left (x \right )}{a^{11}}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 143, normalized size = 0.88 \begin {gather*} -\frac {2520 \, b^{9} x^{3} + 1260 \, a b^{8} x^{\frac {8}{3}} - 420 \, a^{2} b^{7} x^{\frac {7}{3}} + 210 \, a^{3} b^{6} x^{2} - 126 \, a^{4} b^{5} x^{\frac {5}{3}} + 84 \, a^{5} b^{4} x^{\frac {4}{3}} - 60 \, a^{6} b^{3} x + 45 \, a^{7} b^{2} x^{\frac {2}{3}} - 35 \, a^{8} b x^{\frac {1}{3}} + 28 \, a^{9}}{84 \, {\left (a^{10} b x^{\frac {10}{3}} + a^{11} x^{3}\right )}} + \frac {30 \, b^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{11}} - \frac {10 \, b^{9} \log \left (x\right )}{a^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 196, normalized size = 1.21 \begin {gather*} -\frac {840 \, a^{3} b^{9} x^{3} + 420 \, a^{6} b^{6} x^{2} - 140 \, a^{9} b^{3} x + 28 \, a^{12} - 2520 \, {\left (b^{12} x^{4} + a^{3} b^{9} x^{3}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 2520 \, {\left (b^{12} x^{4} + a^{3} b^{9} x^{3}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, {\left (140 \, a b^{11} x^{3} + 105 \, a^{4} b^{8} x^{2} - 15 \, a^{7} b^{5} x + 6 \, a^{10} b^{2}\right )} x^{\frac {2}{3}} - 63 \, {\left (20 \, a^{2} b^{10} x^{3} + 12 \, a^{5} b^{7} x^{2} - 3 \, a^{8} b^{4} x + a^{11} b\right )} x^{\frac {1}{3}}}{84 \, {\left (a^{11} b^{3} x^{4} + a^{14} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 505 vs.
\(2 (163) = 326\).
time = 5.70, size = 505, normalized size = 3.12 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 a^{2} x^{3}} & \text {for}\: b = 0 \\- \frac {3}{11 b^{2} x^{\frac {11}{3}}} & \text {for}\: a = 0 \\- \frac {28 a^{10} x^{\frac {2}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {35 a^{9} b x}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {45 a^{8} b^{2} x^{\frac {4}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {60 a^{7} b^{3} x^{\frac {5}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {84 a^{6} b^{4} x^{2}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {126 a^{5} b^{5} x^{\frac {7}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {210 a^{4} b^{6} x^{\frac {8}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {420 a^{3} b^{7} x^{3}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {1260 a^{2} b^{8} x^{\frac {10}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {840 a b^{9} x^{\frac {11}{3}} \log {\left (x \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {2520 a b^{9} x^{\frac {11}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {2520 a b^{9} x^{\frac {11}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {840 b^{10} x^{4} \log {\left (x \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {2520 b^{10} x^{4} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.57, size = 145, normalized size = 0.90 \begin {gather*} \frac {30 \, b^{9} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{11}} - \frac {10 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{11}} - \frac {2520 \, a b^{9} x^{3} + 1260 \, a^{2} b^{8} x^{\frac {8}{3}} - 420 \, a^{3} b^{7} x^{\frac {7}{3}} + 210 \, a^{4} b^{6} x^{2} - 126 \, a^{5} b^{5} x^{\frac {5}{3}} + 84 \, a^{6} b^{4} x^{\frac {4}{3}} - 60 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} x^{\frac {2}{3}} - 35 \, a^{9} b x^{\frac {1}{3}} + 28 \, a^{10}}{84 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{11} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 136, normalized size = 0.84 \begin {gather*} \frac {60\,b^9\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^{11}}-\frac {\frac {1}{3\,a}-\frac {5\,b\,x^{1/3}}{12\,a^2}-\frac {5\,b^3\,x}{7\,a^4}+\frac {15\,b^2\,x^{2/3}}{28\,a^3}+\frac {5\,b^6\,x^2}{2\,a^7}+\frac {b^4\,x^{4/3}}{a^5}-\frac {3\,b^5\,x^{5/3}}{2\,a^6}+\frac {30\,b^9\,x^3}{a^{10}}-\frac {5\,b^7\,x^{7/3}}{a^8}+\frac {15\,b^8\,x^{8/3}}{a^9}}{a\,x^3+b\,x^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________