Optimal. Leaf size=354 \[ -\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{21/2}} \]
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Rubi [A]
time = 0.44, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2045, 2050,
2054, 212} \begin {gather*} -\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{262144 b^{21/2}}+\frac {12597 a^{10} \sqrt {a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac {4199 a^9 \sqrt {a x+b x^{2/3}}}{131072 b^9 x}+\frac {4199 a^8 \sqrt {a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac {12597 a^7 \sqrt {a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac {4199 a^6 \sqrt {a x+b x^{2/3}}}{215040 b^6 x^2}-\frac {4199 a^5 \sqrt {a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac {323 a^4 \sqrt {a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac {323 a^3 \sqrt {a x+b x^{2/3}}}{21120 b^3 x^3}+\frac {19 a^2 \sqrt {a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{220 b x^{11/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2045
Rule 2050
Rule 2054
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}+\frac {1}{22} a \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}-\frac {\left (19 a^2\right ) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{440 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}+\frac {\left (323 a^3\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{7920 b^2}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}-\frac {\left (323 a^4\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{8448 b^3}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}+\frac {\left (4199 a^5\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{118272 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}-\frac {\left (4199 a^6\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{129024 b^5}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}+\frac {\left (4199 a^7\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{143360 b^6}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}-\frac {\left (4199 a^8\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{163840 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}+\frac {\left (4199 a^9\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{196608 b^8}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}-\frac {\left (4199 a^{10}\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{262144 b^9}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}+\frac {\left (4199 a^{11}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{524288 b^{10}}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac {\left (12597 a^{11}\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 )}{262144 b^{10}}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{21/2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 186, normalized size = 0.53 \begin {gather*} \frac {\sqrt {b x^{2/3}+a x} \left (-82575360 b^{10}-4128768 a b^9 \sqrt [3]{x}+4358144 a^2 b^8 x^{2/3}-4630528 a^3 b^7 x+4961280 a^4 b^6 x^{4/3}-5374720 a^5 b^5 x^{5/3}+5912192 a^6 b^4 x^2-6651216 a^7 b^3 x^{7/3}+7759752 a^8 b^2 x^{8/3}-9699690 a^9 b x^3+14549535 a^{10} x^{10/3}\right )}{302776320 b^{10} x^4}-\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{21/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 209, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (14549535 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {21}{2}} b^{\frac {21}{2}}-155195040 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {19}{2}} b^{\frac {23}{2}}+749786037 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}} b^{\frac {25}{2}}-2163862272 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {27}{2}}+4139920070 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {29}{2}}-5503713280 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {31}{2}}+5174056250 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {33}{2}}-3424523520 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {35}{2}}+1551313995 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {37}{2}}-450357600 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {39}{2}}-14549535 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{10} a^{11} x^{\frac {11}{3}}-14549535 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}\right )}{302776320 x^{4} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}}\) | \(209\) |
default | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-14549535 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {21}{2}} b^{\frac {21}{2}}+155195040 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {19}{2}} b^{\frac {23}{2}}-749786037 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}} b^{\frac {25}{2}}+2163862272 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {27}{2}}-4139920070 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {29}{2}}+5503713280 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {31}{2}}-5174056250 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {33}{2}}+3424523520 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {35}{2}}-1551313995 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {37}{2}}+14549535 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{10} a^{11} x^{\frac {11}{3}}+450357600 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {39}{2}}+14549535 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}\right )}{302776320 x^{4} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}}\) | \(209\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.64, size = 228, normalized size = 0.64 \begin {gather*} \frac {\frac {14549535 \, a^{12} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{12} - 155195040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{12} b + 749786037 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{12} b^{2} - 2163862272 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{12} b^{3} + 4139920070 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{12} b^{4} - 5503713280 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{12} b^{5} + 5174056250 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{12} b^{6} - 3424523520 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{12} b^{7} + 1551313995 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{12} b^{8} - 450357600 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{12} b^{9} - 14549535 \, \sqrt {a x^{\frac {1}{3}} + b} a^{12} b^{10}}{a^{11} b^{10} x^{\frac {11}{3}}}}{302776320 \, a} \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 {\sqrt {a\,x+b\,x^{2/3}}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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