Optimal. Leaf size=329 \[ -\frac {138567 a^{10} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac {138567 a^9 \sqrt {a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac {46189 a^8 \sqrt {a x+b x^{2/3}}}{65536 b^9 x}+\frac {46189 a^7 \sqrt {a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac {138567 a^6 \sqrt {a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac {46189 a^5 \sqrt {a x+b x^{2/3}}}{107520 b^6 x^2}-\frac {4199 a^4 \sqrt {a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac {323 a^3 \sqrt {a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac {323 a^2 \sqrt {a x+b x^{2/3}}}{960 b^3 x^3}+\frac {19 a \sqrt {a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}} \]
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Rubi [A] time = 0.58, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \begin {gather*} \frac {138567 a^9 \sqrt {a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac {46189 a^8 \sqrt {a x+b x^{2/3}}}{65536 b^9 x}+\frac {46189 a^7 \sqrt {a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac {138567 a^6 \sqrt {a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac {46189 a^5 \sqrt {a x+b x^{2/3}}}{107520 b^6 x^2}-\frac {4199 a^4 \sqrt {a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac {323 a^3 \sqrt {a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac {323 a^2 \sqrt {a x+b x^{2/3}}}{960 b^3 x^3}-\frac {138567 a^{10} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac {19 a \sqrt {a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}-\frac {(19 a) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{20 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}+\frac {\left (323 a^2\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{360 b^2}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}-\frac {\left (323 a^3\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{384 b^3}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}+\frac {\left (4199 a^4\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{5376 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}-\frac {\left (46189 a^5\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{64512 b^5}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}+\frac {\left (46189 a^6\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{71680 b^6}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}-\frac {\left (46189 a^7\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{81920 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}+\frac {\left (46189 a^8\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{98304 b^8}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}-\frac {\left (46189 a^9\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{131072 b^9}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}+\frac {\left (46189 a^{10}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{262144 b^{10}}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac {\left (138567 a^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{10}}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac {138567 a^{10} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{21/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 48, normalized size = 0.15 \begin {gather*} -\frac {6 a^{10} \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {1}{2},11;\frac {3}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^{11} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 175, normalized size = 0.53 \begin {gather*} \frac {\sqrt {a x+b x^{2/3}} \left (14549535 a^9 x^3-9699690 a^8 b x^{8/3}+7759752 a^7 b^2 x^{7/3}-6651216 a^6 b^3 x^2+5912192 a^5 b^4 x^{5/3}-5374720 a^4 b^5 x^{4/3}+4961280 a^3 b^6 x-4630528 a^2 b^7 x^{2/3}+4358144 a b^8 \sqrt [3]{x}-4128768 b^9\right )}{13762560 b^{10} x^{11/3}}-\frac {138567 a^{10} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{131072 b^{21/2}} \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 [A] time = 0.36, size = 211, normalized size = 0.64 \begin {gather*} \frac {\frac {14549535 \, a^{11} \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 {19}{2}} a^{11} - 140645505 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{11} b + 609140532 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{11} b^{2} - 1554721740 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{11} b^{3} + 2585198330 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{11} b^{4} - 2918514950 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{11} b^{5} + 2255541300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{11} b^{6} - 1168982220 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{11} b^{7} + 382331775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{11} b^{8} - 68025825 \, \sqrt {a x^{\frac {1}{3}} + b} a^{11} b^{9}}{a^{10} b^{10} x^{\frac {10}{3}}}}{13762560 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 248, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {a \,x^{\frac {1}{3}}+b}\, \left (14549535 a^{10} b \,x^{\frac {19}{3}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-14549535 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{9} b^{\frac {3}{2}} x^{6}+9699690 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{8} b^{\frac {5}{2}} x^{\frac {17}{3}}-7759752 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{7} b^{\frac {7}{2}} x^{\frac {16}{3}}+6651216 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{6} b^{\frac {9}{2}} x^{5}-5912192 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{5} b^{\frac {11}{2}} x^{\frac {14}{3}}+5374720 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{4} b^{\frac {13}{2}} x^{\frac {13}{3}}-4961280 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{3} b^{\frac {15}{2}} x^{4}+4630528 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{2} b^{\frac {17}{2}} x^{\frac {11}{3}}-4358144 \sqrt {a \,x^{\frac {1}{3}}+b}\, a \,b^{\frac {19}{2}} x^{\frac {10}{3}}+4128768 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {21}{2}} x^{3}\right )}{13762560 \sqrt {a x +b \,x^{\frac {2}{3}}}\, b^{\frac {23}{2}} x^{6}} \end {gather*}
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}{\sqrt {a x + b x^{\frac {2}{3}}} x^{4}}\,{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^4\,\sqrt {a\,x+b\,x^{2/3}}} \,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^{4} \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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