Optimal. Leaf size=291 \[ \frac {429 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac {429 a^8 \sqrt {a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac {143 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^6 x}-\frac {143 a^6 \sqrt {a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac {143 a^4 \sqrt {a x+b x^{2/3}}}{26880 b^3 x^2}+\frac {13 a^3 \sqrt {a x+b x^{2/3}}}{2688 b^2 x^{7/3}}-\frac {a^2 \sqrt {a x+b x^{2/3}}}{224 b x^{8/3}}-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}-\frac {a \sqrt {a x+b x^{2/3}}}{16 x^3} \]
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Rubi [A] time = 0.52, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2029, 206} \[ -\frac {429 a^8 \sqrt {a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac {143 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^6 x}-\frac {143 a^6 \sqrt {a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac {143 a^4 \sqrt {a x+b x^{2/3}}}{26880 b^3 x^2}+\frac {13 a^3 \sqrt {a x+b x^{2/3}}}{2688 b^2 x^{7/3}}+\frac {429 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac {a^2 \sqrt {a x+b x^{2/3}}}{224 b x^{8/3}}-\frac {a \sqrt {a x+b x^{2/3}}}{16 x^3}-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2020
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx &=-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {1}{6} a \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {1}{96} a^2 \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (13 a^3\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{1344 b}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (143 a^4\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{16128 b^2}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (143 a^5\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{17920 b^3}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (143 a^6\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{20480 b^4}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (143 a^7\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{24576 b^5}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (143 a^8\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{32768 b^6}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (143 a^9\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{65536 b^7}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (429 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^7}\\ &=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {429 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{15/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 61, normalized size = 0.21 \[ \frac {6 a^9 \left (a \sqrt [3]{x}+b\right )^2 \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {5}{2},10;\frac {7}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{5 b^{10} \sqrt [3]{x}} \]
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 = 194, normalized size = 0.67 \[ -\frac {\frac {45045 \, a^{10} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{10} - 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{10} b + 1495494 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{10} b^{2} - 3317886 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{10} b^{3} + 4685824 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{10} b^{4} - 4349826 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{10} b^{5} + 2633274 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{10} b^{6} + 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{10} b^{7} - 45045 \, \sqrt {a x^{\frac {1}{3}} + b} a^{10} b^{8}}{a^{9} b^{7} x^{3}}}{3440640 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 181, normalized size = 0.62 \[ -\frac {\left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (-45045 a^{9} b^{7} x^{3} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-45045 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {31}{2}}+390390 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {29}{2}}+2633274 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {5}{2}} b^{\frac {27}{2}}-4349826 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {7}{2}} b^{\frac {25}{2}}+4685824 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {9}{2}} b^{\frac {23}{2}}-3317886 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {11}{2}} b^{\frac {21}{2}}+1495494 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {13}{2}} b^{\frac {19}{2}}-390390 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {15}{2}} b^{\frac {17}{2}}+45045 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {17}{2}} b^{\frac {15}{2}}\right )}{3440640 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {29}{2}} x^{4}} \]
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 \[ \int \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{5}}\,{d x} \]
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 \[ \int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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