Optimal. Leaf size=291 \[ -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {429 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{15/2}} \]
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Rubi [A]
time = 0.35, 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 = {2045, 2050,
2054, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2045
Rule 2050
Rule 2054
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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.06, size = 61, normalized size = 0.21 \begin {gather*} \frac {6 a^9 \left (b+a \sqrt [3]{x}\right )^2 \sqrt {b x^{2/3}+a x} \, _2F_1\left (\frac {5}{2},10;\frac {7}{2};1+\frac {a \sqrt [3]{x}}{b}\right )}{5 b^{10} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 181, normalized size = 0.62
method | result | size |
derivativedivides | \(-\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}} b^{\frac {15}{2}}-390390 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {17}{2}}+1495494 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {19}{2}}-3317886 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {21}{2}}+4685824 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {23}{2}}-4349826 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {25}{2}}+2633274 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {27}{2}}-45045 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}+390390 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}-45045 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {31}{2}}\right )}{3440640 x^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}}\) | \(181\) |
default | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (-45045 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}} b^{\frac {15}{2}}+390390 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {17}{2}}-1495494 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {19}{2}}+3317886 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {21}{2}}-4685824 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {23}{2}}+4349826 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {25}{2}}-2633274 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {27}{2}}-390390 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}+45045 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {31}{2}}+45045 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}\right )}{3440640 x^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}}\) | \(181\) |
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 {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 194, normalized size = 0.67 \begin {gather*} -\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} \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 {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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