Integrand size = 19, antiderivative size = 313 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {262144 b^9 \sqrt {b x^{2/3}+a x}}{323323 a^{10}}+\frac {524288 b^{10} \sqrt {b x^{2/3}+a x}}{323323 a^{11} \sqrt [3]{x}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a} \]
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Time = 0.34 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027, 2039} \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {524288 b^{10} \sqrt {a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac {262144 b^9 \sqrt {a x+b x^{2/3}}}{323323 a^{10}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {a x+b x^{2/3}}}{323323 a^8}+\frac {20480 b^6 x \sqrt {a x+b x^{2/3}}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {a x+b x^{2/3}}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {a x+b x^{2/3}}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {a x+b x^{2/3}}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {a x+b x^{2/3}}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {a x+b x^{2/3}}}{133 a^2}+\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a} \]
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Rule 2027
Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {(20 b) \int \frac {x^{8/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{21 a} \\ & = -\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (120 b^2\right ) \int \frac {x^{7/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{133 a^2} \\ & = \frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (1920 b^3\right ) \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx}{2261 a^3} \\ & = -\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (256 b^4\right ) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{323 a^4} \\ & = \frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (3072 b^5\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{4199 a^5} \\ & = -\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (30720 b^6\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{46189 a^6} \\ & = \frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (81920 b^7\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{138567 a^7} \\ & = -\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (163840 b^8\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{323323 a^8} \\ & = \frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (131072 b^9\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{323323 a^9} \\ & = -\frac {262144 b^9 \sqrt {b x^{2/3}+a x}}{323323 a^{10}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (262144 b^{10}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{969969 a^{10}} \\ & = -\frac {262144 b^9 \sqrt {b x^{2/3}+a x}}{323323 a^{10}}+\frac {524288 b^{10} \sqrt {b x^{2/3}+a x}}{323323 a^{11} \sqrt [3]{x}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.47 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (262144 b^{10}-131072 a b^9 \sqrt [3]{x}+98304 a^2 b^8 x^{2/3}-81920 a^3 b^7 x+71680 a^4 b^6 x^{4/3}-64512 a^5 b^5 x^{5/3}+59136 a^6 b^4 x^2-54912 a^7 b^3 x^{7/3}+51480 a^8 b^2 x^{8/3}-48620 a^9 b x^3+46189 a^{10} x^{10/3}\right )}{323323 a^{11} \sqrt [3]{x}} \]
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Time = 1.76 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.43
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 x^{2} a^{6} b^{4}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} x^{\frac {4}{3}} b^{6}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right )}{323323 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{11}}\) | \(134\) |
default | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 x^{2} a^{6} b^{4}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} x^{\frac {4}{3}} b^{6}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right )}{323323 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{11}}\) | \(134\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (233) = 466\).
Time = 176.92 (sec) , antiderivative size = 1031, normalized size of antiderivative = 3.29 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^{3}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
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\[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {x^{3}}{\sqrt {a x + b x^{\frac {2}{3}}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.52 \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {524288 \, b^{\frac {21}{2}}}{323323 \, a^{11}} + \frac {2 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{323323 \, a^{11}} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^3}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]
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