Integrand size = 19, antiderivative size = 412 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {50702925 a^{12} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{27/2}} \]
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Time = 0.53 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2048, 2050, 2054, 212} \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {50702925 a^{12} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{2097152 b^{27/2}}+\frac {50702925 a^{11} \sqrt {a x+b x^{2/3}}}{2097152 b^{13} x^{2/3}}-\frac {16900975 a^{10} \sqrt {a x+b x^{2/3}}}{1048576 b^{12} x}+\frac {3380195 a^9 \sqrt {a x+b x^{2/3}}}{262144 b^{11} x^{4/3}}-\frac {1448655 a^8 \sqrt {a x+b x^{2/3}}}{131072 b^{10} x^{5/3}}+\frac {482885 a^7 \sqrt {a x+b x^{2/3}}}{49152 b^9 x^2}-\frac {2414425 a^6 \sqrt {a x+b x^{2/3}}}{270336 b^8 x^{7/3}}+\frac {185725 a^5 \sqrt {a x+b x^{2/3}}}{22528 b^7 x^{8/3}}-\frac {260015 a^4 \sqrt {a x+b x^{2/3}}}{33792 b^6 x^3}+\frac {15295 a^3 \sqrt {a x+b x^{2/3}}}{2112 b^5 x^{10/3}}-\frac {2415 a^2 \sqrt {a x+b x^{2/3}}}{352 b^4 x^{11/3}}+\frac {575 a \sqrt {a x+b x^{2/3}}}{88 b^3 x^4}-\frac {25 \sqrt {a x+b x^{2/3}}}{4 b^2 x^{13/3}}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}} \]
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Rule 212
Rule 2048
Rule 2050
Rule 2054
Rubi steps \begin{align*} \text {integral}& = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}+\frac {25 \int \frac {1}{x^{14/3} \sqrt {b x^{2/3}+a x}} \, dx}{b} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}-\frac {(575 a) \int \frac {1}{x^{13/3} \sqrt {b x^{2/3}+a x}} \, dx}{24 b^2} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}+\frac {\left (4025 a^2\right ) \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx}{176 b^3} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}-\frac {\left (15295 a^3\right ) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{704 b^4} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}+\frac {\left (260015 a^4\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{12672 b^5} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}-\frac {\left (1300075 a^5\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{67584 b^6} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}+\frac {\left (2414425 a^6\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{135168 b^7} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}-\frac {\left (2414425 a^7\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{147456 b^8} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}+\frac {\left (482885 a^8\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{32768 b^9} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}-\frac {\left (3380195 a^9\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{262144 b^{10}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}+\frac {\left (16900975 a^{10}\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{1572864 b^{11}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}-\frac {\left (16900975 a^{11}\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{2097152 b^{12}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}+\frac {\left (16900975 a^{12}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{4194304 b^{13}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {\left (50702925 a^{12}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{13}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {50702925 a^{12} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{27/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6 a^{12} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},13,\frac {1}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{b^{13} \sqrt {b x^{2/3}+a x}} \]
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Time = 1.99 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.47
method | result | size |
derivativedivides | \(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (17301504 b^{\frac {25}{2}}+1673196525 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{12} x^{4}-1673196525 a^{12} x^{4} \sqrt {b}-19660800 b^{\frac {23}{2}} a \,x^{\frac {1}{3}}+22609920 b^{\frac {21}{2}} a^{2} x^{\frac {2}{3}}-26378240 b^{\frac {19}{2}} a^{3} x +31324160 b^{\frac {17}{2}} a^{4} x^{\frac {4}{3}}-38036480 b^{\frac {15}{2}} a^{5} x^{\frac {5}{3}}+47545600 b^{\frac {13}{2}} a^{6} x^{2}-61809280 b^{\frac {11}{2}} a^{7} x^{\frac {7}{3}}+84987760 b^{\frac {9}{2}} a^{8} x^{\frac {8}{3}}-127481640 b^{\frac {7}{2}} a^{9} x^{3}+223092870 b^{\frac {5}{2}} a^{10} x^{\frac {10}{3}}-557732175 b^{\frac {3}{2}} a^{11} x^{\frac {11}{3}}\right )}{69206016 x^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {27}{2}}}\) | \(192\) |
default | \(\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (1673196525 a^{12} x^{4} \sqrt {b}-17301504 b^{\frac {25}{2}}-1673196525 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{12} x^{4}+19660800 b^{\frac {23}{2}} a \,x^{\frac {1}{3}}-22609920 b^{\frac {21}{2}} a^{2} x^{\frac {2}{3}}+26378240 b^{\frac {19}{2}} a^{3} x -31324160 b^{\frac {17}{2}} a^{4} x^{\frac {4}{3}}+38036480 b^{\frac {15}{2}} a^{5} x^{\frac {5}{3}}-47545600 b^{\frac {13}{2}} a^{6} x^{2}+61809280 b^{\frac {11}{2}} a^{7} x^{\frac {7}{3}}-84987760 b^{\frac {9}{2}} a^{8} x^{\frac {8}{3}}+127481640 b^{\frac {7}{2}} a^{9} x^{3}-223092870 b^{\frac {5}{2}} a^{10} x^{\frac {10}{3}}+557732175 b^{\frac {3}{2}} a^{11} x^{\frac {11}{3}}\right )}{69206016 x^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {27}{2}}}\) | \(192\) |
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Timed out. \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {50702925 \, a^{12} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{2097152 \, \sqrt {-b} b^{13}} + \frac {6 \, a^{12}}{\sqrt {a x^{\frac {1}{3}} + b} b^{13}} + \frac {1257960429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} a^{12} - 14537792973 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{12} b + 76667241519 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{12} b^{2} - 243717614415 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{12} b^{3} + 519393101810 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{12} b^{4} - 780150847218 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{12} b^{5} + 844265343246 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{12} b^{6} - 659969685518 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{12} b^{7} + 366679446705 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{12} b^{8} - 138840292305 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{12} b^{9} + 32660709939 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{12} b^{10} - 3724872723 \, \sqrt {a x^{\frac {1}{3}} + b} a^{12} b^{11}}{69206016 \, a^{12} b^{13} x^{4}} \]
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Timed out. \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^4\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \]
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