Optimal. Leaf size=200 \[ -\frac {a^2 \text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9-a b^5\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 b^3}-\frac {3 \left (a^3 x^3+b^2 x^2\right )^{2/3} \left (-7290 a^{15} x^5+4860 a^{12} b^2 x^4-4050 a^9 b^4 x^3-11781 a^7 b^5 x^5+3600 a^6 b^6 x^2+7854 a^4 b^7 x^4-3300 a^3 b^8 x-6545 a b^9 x^3+3080 b^{10}\right )}{52360 b^{13} x^7} \]
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Rubi [B] time = 3.97, antiderivative size = 2325, normalized size of antiderivative = 11.62, number of steps used = 27, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2056, 6725, 129, 155, 12, 91}
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Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 91
Rule 129
Rule 155
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{b^2+a^3 x} \left (b+a x^3\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {1}{3 b^{2/3} x^{20/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{20/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{20/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{20/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{20/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{20/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \left (15 a^{8/3}+17 b^{5/3}\right )-5 a^{10/3} x}{x^{17/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{17 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right )+5 \sqrt [3]{-1} a^{10/3} x}{x^{17/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{17 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {1}{3} \sqrt [3]{a} \sqrt [3]{b} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right )-5 (-1)^{2/3} a^{10/3} x}{x^{17/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{17 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right )+\frac {4}{3} a^{11/3} \sqrt [3]{b} \left (15 a^{8/3}+17 b^{5/3}\right ) x}{x^{14/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{238 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right )-\frac {4}{3} \sqrt [3]{-1} a^{11/3} \sqrt [3]{b} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) x}{x^{14/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{238 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right )+\frac {4}{3} (-1)^{2/3} a^{11/3} \sqrt [3]{b} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) x}{x^{14/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{238 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {2}{27} a b \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right )+\frac {2}{3} a^4 b^{2/3} \left (90 \sqrt [3]{-1} a^{16/3}-102 (-1)^{2/3} a^{8/3} b^{5/3}-119 b^{10/3}\right ) x}{x^{11/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2618 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {2}{27} a b \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right )-\frac {2}{3} a^4 b^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) x}{x^{11/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2618 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {2}{27} a b \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right )-\frac {2}{3} a^4 b^{2/3} \left (90 (-1)^{2/3} a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 b^{10/3}\right ) x}{x^{11/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2618 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{81} \left (2430 a^{12} b^{4/3}+2754 a^{28/3} b^3+3213 a^{20/3} b^{14/3}+3927 a^4 b^{19/3}+5236 a^{4/3} b^8\right )+\frac {4}{27} a^{13/3} b \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right ) x}{x^{8/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{20944 b^{10} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{81} \left (2430 a^{12} b^{4/3}+2754 (-1)^{2/3} a^{28/3} b^3-3213 \sqrt [3]{-1} a^{20/3} b^{14/3}+3927 a^4 b^{19/3}+5236 (-1)^{2/3} a^{4/3} b^8\right )+\frac {4}{27} (-1)^{2/3} a^{13/3} b \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right ) x}{x^{8/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{20944 b^{10} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {4}{81} \left (2430 a^{12} b^{4/3}-2754 \sqrt [3]{-1} a^{28/3} b^3+3213 (-1)^{2/3} a^{20/3} b^{14/3}+3927 a^4 b^{19/3}-5236 \sqrt [3]{-1} a^{4/3} b^8\right )-\frac {4}{27} \sqrt [3]{-1} a^{13/3} b \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right ) x}{x^{8/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{20944 b^{10} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}+3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}-2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}+3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}+3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (81 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {4}{243} \left (7290 a^{15} b^{5/3}+8262 a^{37/3} b^{10/3}+9639 a^{29/3} b^5+11781 a^7 b^{20/3}+15708 a^{13/3} b^{25/3}+26180 a^{5/3} b^{10}\right )-\frac {4}{81} a^{14/3} b^{4/3} \left (2430 a^{32/3}+2754 a^8 b^{5/3}+3213 a^{16/3} b^{10/3}+3927 a^{8/3} b^5+5236 b^{20/3}\right ) x}{x^{5/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{104720 b^{37/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (81 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {4}{243} \left (7290 a^{15} b^{5/3}-8262 \sqrt [3]{-1} a^{37/3} b^{10/3}+9639 (-1)^{2/3} a^{29/3} b^5+11781 a^7 b^{20/3}-15708 \sqrt [3]{-1} a^{13/3} b^{25/3}+26180 (-1)^{2/3} a^{5/3} b^{10}\right )+\frac {4}{81} \sqrt [3]{-1} a^{14/3} b^{4/3} \left (2430 a^{32/3}-2754 \sqrt [3]{-1} a^8 b^{5/3}+3213 (-1)^{2/3} a^{16/3} b^{10/3}+3927 a^{8/3} b^5-5236 \sqrt [3]{-1} b^{20/3}\right ) x}{x^{5/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{104720 b^{37/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (81 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-\frac {4}{243} \left (7290 a^{15} b^{5/3}+8262 (-1)^{2/3} a^{37/3} b^{10/3}-9639 \sqrt [3]{-1} a^{29/3} b^5+11781 a^7 b^{20/3}+15708 (-1)^{2/3} a^{13/3} b^{25/3}-26180 \sqrt [3]{-1} a^{5/3} b^{10}\right )-\frac {4}{81} (-1)^{2/3} a^{14/3} b^{4/3} \left (2430 a^{32/3}+2754 (-1)^{2/3} a^8 b^{5/3}-3213 \sqrt [3]{-1} a^{16/3} b^{10/3}+3927 a^{8/3} b^5+5236 (-1)^{2/3} b^{20/3}\right ) x}{x^{5/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{104720 b^{37/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (7290 a^{15}+8262 a^{37/3} b^{5/3}+9639 a^{29/3} b^{10/3}+11781 a^7 b^5+15708 a^{13/3} b^{20/3}+26180 a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (7290 a^{15}+8262 (-1)^{2/3} a^{37/3} b^{5/3}-9639 \sqrt [3]{-1} a^{29/3} b^{10/3}+11781 a^7 b^5+15708 (-1)^{2/3} a^{13/3} b^{20/3}-26180 \sqrt [3]{-1} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (7290 a^{15}-8262 \sqrt [3]{-1} a^{37/3} b^{5/3}+9639 (-1)^{2/3} a^{29/3} b^{10/3}+11781 a^7 b^5-15708 \sqrt [3]{-1} a^{13/3} b^{20/3}+26180 (-1)^{2/3} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}+3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}-2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}+3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}+3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {209440 a^2 b^{12}}{729 x^{2/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{209440 b^{44/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {209440 a^2 b^{12}}{729 x^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{209440 b^{44/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {209440 a^2 b^{12}}{729 x^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{209440 b^{44/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (7290 a^{15}+8262 a^{37/3} b^{5/3}+9639 a^{29/3} b^{10/3}+11781 a^7 b^5+15708 a^{13/3} b^{20/3}+26180 a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (7290 a^{15}+8262 (-1)^{2/3} a^{37/3} b^{5/3}-9639 \sqrt [3]{-1} a^{29/3} b^{10/3}+11781 a^7 b^5+15708 (-1)^{2/3} a^{13/3} b^{20/3}-26180 \sqrt [3]{-1} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (7290 a^{15}-8262 \sqrt [3]{-1} a^{37/3} b^{5/3}+9639 (-1)^{2/3} a^{29/3} b^{10/3}+11781 a^7 b^5-15708 \sqrt [3]{-1} a^{13/3} b^{20/3}+26180 (-1)^{2/3} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}+3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}-2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}+3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}+3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{8/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{8/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{8/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (7290 a^{15}+8262 a^{37/3} b^{5/3}+9639 a^{29/3} b^{10/3}+11781 a^7 b^5+15708 a^{13/3} b^{20/3}+26180 a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (7290 a^{15}+8262 (-1)^{2/3} a^{37/3} b^{5/3}-9639 \sqrt [3]{-1} a^{29/3} b^{10/3}+11781 a^7 b^5+15708 (-1)^{2/3} a^{13/3} b^{20/3}-26180 \sqrt [3]{-1} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (7290 a^{15}-8262 \sqrt [3]{-1} a^{37/3} b^{5/3}+9639 (-1)^{2/3} a^{29/3} b^{10/3}+11781 a^7 b^5-15708 \sqrt [3]{-1} a^{13/3} b^{20/3}+26180 (-1)^{2/3} a^{5/3} b^{25/3}\right ) \left (b^2+a^3 x\right )}{52360 b^{13} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 \left (b^2+a^3 x\right )}{17 b^3 x^5 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}-17 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{a} \left (15 a^{8/3}+17 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{238 b^5 x^4 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 a^{8/3} b^{5/3}+119 b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}+102 (-1)^{2/3} a^{8/3} b^{5/3}-119 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{2/3} \left (90 a^{16/3}-102 \sqrt [3]{-1} a^{8/3} b^{5/3}+119 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{1309 b^7 x^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 a^{16/3} b^{5/3}+1071 a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8+918 (-1)^{2/3} a^{16/3} b^{5/3}-1071 \sqrt [3]{-1} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a \left (810 a^8-918 \sqrt [3]{-1} a^{16/3} b^{5/3}+1071 (-1)^{2/3} a^{8/3} b^{10/3}+1309 b^5\right ) \left (b^2+a^3 x\right )}{10472 b^9 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 a^{28/3} b^{5/3}+3213 a^{20/3} b^{10/3}+3927 a^4 b^5+5236 a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}-2754 \sqrt [3]{-1} a^{28/3} b^{5/3}+3213 (-1)^{2/3} a^{20/3} b^{10/3}+3927 a^4 b^5-5236 \sqrt [3]{-1} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (2430 a^{12}+2754 (-1)^{2/3} a^{28/3} b^{5/3}-3213 \sqrt [3]{-1} a^{20/3} b^{10/3}+3927 a^4 b^5+5236 (-1)^{2/3} a^{4/3} b^{20/3}\right ) \left (b^2+a^3 x\right )}{26180 b^{11} x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^3 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^3 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^3 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )}{6 b^3 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}{6 b^3 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{6 b^3 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-b^{5/3}}}\right )}{2 b^3 \sqrt [3]{a^{8/3}-b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}}}\right )}{2 b^3 \sqrt [3]{a^{8/3}+\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{17/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}}}\right )}{2 b^3 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.73, size = 314, normalized size = 1.57 \begin {gather*} \frac {21870 a^{18} x^6+7290 a^{15} b^2 x^5-2430 a^{12} b^4 x^4+35343 a^{10} b^5 x^6+1350 a^9 b^6 x^3+11781 a^7 b^7 x^5-900 a^6 b^8 x^2-3927 a^4 b^9 x^4+660 a^3 b^{10} x+52360 a^2 b^{10} x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^3 x-\sqrt [3]{a} b^{5/3} x}{x a^3+b^2}\right )+52360 a^2 b^{10} x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x a^3+\left (1-i \sqrt {3}\right ) b^{5/3} x \sqrt [3]{a}}{2 \left (x a^3+b^2\right )}\right )+52360 a^2 b^{10} x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x a^3+\left (1+i \sqrt {3}\right ) b^{5/3} x \sqrt [3]{a}}{2 \left (x a^3+b^2\right )}\right )+19635 a b^{11} x^3-9240 b^{12}}{52360 b^{13} x^5 \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
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IntegrateAlgebraic [A] time = 1.23, size = 200, normalized size = 1.00 \begin {gather*} -\frac {3 \left (b^2 x^2+a^3 x^3\right )^{2/3} \left (3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3-6545 a b^9 x^3+4860 a^{12} b^2 x^4+7854 a^4 b^7 x^4-7290 a^{15} x^5-11781 a^7 b^5 x^5\right )}{52360 b^{13} x^7}-\frac {a^2 \text {RootSum}\left [a^9-a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{6} \left (a \,x^{3}+b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} + b\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^6\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (a\,x^3+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{3} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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