Optimal. Leaf size=324 \[ \frac {692835 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{32768 b^{21/2}}-\frac {692835 a^8 \sqrt {a x+b x^{2/3}}}{32768 b^{10} x^{2/3}}+\frac {230945 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^9 x}-\frac {46189 a^6 \sqrt {a x+b x^{2/3}}}{4096 b^8 x^{4/3}}+\frac {138567 a^5 \sqrt {a x+b x^{2/3}}}{14336 b^7 x^{5/3}}-\frac {46189 a^4 \sqrt {a x+b x^{2/3}}}{5376 b^6 x^2}+\frac {20995 a^3 \sqrt {a x+b x^{2/3}}}{2688 b^5 x^{7/3}}-\frac {1615 a^2 \sqrt {a x+b x^{2/3}}}{224 b^4 x^{8/3}}+\frac {323 a \sqrt {a x+b x^{2/3}}}{48 b^3 x^3}-\frac {19 \sqrt {a x+b x^{2/3}}}{3 b^2 x^{10/3}}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}} \]
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Rubi [A] time = 0.60, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2023, 2025, 2029, 206} \[ -\frac {692835 a^8 \sqrt {a x+b x^{2/3}}}{32768 b^{10} x^{2/3}}+\frac {230945 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^9 x}-\frac {46189 a^6 \sqrt {a x+b x^{2/3}}}{4096 b^8 x^{4/3}}+\frac {138567 a^5 \sqrt {a x+b x^{2/3}}}{14336 b^7 x^{5/3}}-\frac {46189 a^4 \sqrt {a x+b x^{2/3}}}{5376 b^6 x^2}+\frac {20995 a^3 \sqrt {a x+b x^{2/3}}}{2688 b^5 x^{7/3}}-\frac {1615 a^2 \sqrt {a x+b x^{2/3}}}{224 b^4 x^{8/3}}+\frac {692835 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{32768 b^{21/2}}+\frac {323 a \sqrt {a x+b x^{2/3}}}{48 b^3 x^3}-\frac {19 \sqrt {a x+b x^{2/3}}}{3 b^2 x^{10/3}}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2023
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}+\frac {19 \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{b}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}-\frac {(323 a) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{18 b^2}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}+\frac {\left (1615 a^2\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{96 b^3}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}-\frac {\left (20995 a^3\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{1344 b^4}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}+\frac {\left (230945 a^4\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{16128 b^5}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}-\frac {\left (46189 a^5\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{3584 b^6}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}+\frac {\left (46189 a^6\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{4096 b^7}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac {46189 a^6 \sqrt {b x^{2/3}+a x}}{4096 b^8 x^{4/3}}-\frac {\left (230945 a^7\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{24576 b^8}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac {46189 a^6 \sqrt {b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac {230945 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^9 x}+\frac {\left (230945 a^8\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{32768 b^9}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac {46189 a^6 \sqrt {b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac {230945 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^9 x}-\frac {692835 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}-\frac {\left (230945 a^9\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{65536 b^{10}}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac {46189 a^6 \sqrt {b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac {230945 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^9 x}-\frac {692835 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}+\frac {\left (692835 a^9\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 )}{32768 b^{10}}\\ &=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac {46189 a^6 \sqrt {b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac {230945 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^9 x}-\frac {692835 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}+\frac {692835 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{21/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 48, normalized size = 0.15 \[ -\frac {6 a^9 \sqrt [3]{x} \, _2F_1\left (-\frac {1}{2},10;\frac {1}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^{10} \sqrt {a x+b x^{2/3}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 207, normalized size = 0.64 \[ -\frac {692835 \, a^{9} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{32768 \, \sqrt {-b} b^{10}} - \frac {6 \, a^{9}}{\sqrt {a x^{\frac {1}{3}} + b} b^{10}} - \frac {10420767 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{9} - 88937058 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{9} b + 334408914 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{9} b^{2} - 724860666 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{9} b^{3} + 993296384 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{9} b^{4} - 884769030 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{9} b^{5} + 503730990 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{9} b^{6} - 169799070 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{9} b^{7} + 26738145 \, \sqrt {a x^{\frac {1}{3}} + b} a^{9} b^{8}}{688128 \, a^{9} b^{10} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 159, normalized size = 0.49 \[ \frac {\left (a \,x^{\frac {1}{3}}+b \right ) \left (14549535 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{9} x^{3} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-14549535 a^{9} \sqrt {b}\, x^{3}-4849845 a^{8} b^{\frac {3}{2}} x^{\frac {8}{3}}+1939938 a^{7} b^{\frac {5}{2}} x^{\frac {7}{3}}-1108536 a^{6} b^{\frac {7}{2}} x^{2}+739024 a^{5} b^{\frac {9}{2}} x^{\frac {5}{3}}-537472 a^{4} b^{\frac {11}{2}} x^{\frac {4}{3}}+413440 a^{3} b^{\frac {13}{2}} x -330752 a^{2} b^{\frac {15}{2}} x^{\frac {2}{3}}+272384 a \,b^{\frac {17}{2}} x^{\frac {1}{3}}-229376 b^{\frac {19}{2}}\right )}{688128 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} b^{\frac {21}{2}} x^{2}} \]
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 \[ \int \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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