Optimal. Leaf size=379 \[ -\frac {12597 a^{12} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{2097152 b^{21/2}}+\frac {12597 a^{11} \sqrt {a x+b x^{2/3}}}{2097152 b^{10} x^{2/3}}-\frac {4199 a^{10} \sqrt {a x+b x^{2/3}}}{1048576 b^9 x}+\frac {4199 a^9 \sqrt {a x+b x^{2/3}}}{1310720 b^8 x^{4/3}}-\frac {12597 a^8 \sqrt {a x+b x^{2/3}}}{4587520 b^7 x^{5/3}}+\frac {4199 a^7 \sqrt {a x+b x^{2/3}}}{1720320 b^6 x^2}-\frac {4199 a^6 \sqrt {a x+b x^{2/3}}}{1892352 b^5 x^{7/3}}+\frac {323 a^5 \sqrt {a x+b x^{2/3}}}{157696 b^4 x^{8/3}}-\frac {323 a^4 \sqrt {a x+b x^{2/3}}}{168960 b^3 x^3}+\frac {19 a^3 \sqrt {a x+b x^{2/3}}}{10560 b^2 x^{10/3}}-\frac {3 a^2 \sqrt {a x+b x^{2/3}}}{1760 b x^{11/3}}-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}-\frac {3 a \sqrt {a x+b x^{2/3}}}{88 x^4} \]
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Rubi [A] time = 0.72, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2029, 206} \[ \frac {12597 a^{11} \sqrt {a x+b x^{2/3}}}{2097152 b^{10} x^{2/3}}-\frac {4199 a^{10} \sqrt {a x+b x^{2/3}}}{1048576 b^9 x}+\frac {4199 a^9 \sqrt {a x+b x^{2/3}}}{1310720 b^8 x^{4/3}}-\frac {12597 a^8 \sqrt {a x+b x^{2/3}}}{4587520 b^7 x^{5/3}}+\frac {4199 a^7 \sqrt {a x+b x^{2/3}}}{1720320 b^6 x^2}-\frac {4199 a^6 \sqrt {a x+b x^{2/3}}}{1892352 b^5 x^{7/3}}+\frac {323 a^5 \sqrt {a x+b x^{2/3}}}{157696 b^4 x^{8/3}}-\frac {323 a^4 \sqrt {a x+b x^{2/3}}}{168960 b^3 x^3}+\frac {19 a^3 \sqrt {a x+b x^{2/3}}}{10560 b^2 x^{10/3}}-\frac {12597 a^{12} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{2097152 b^{21/2}}-\frac {3 a^2 \sqrt {a x+b x^{2/3}}}{1760 b x^{11/3}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{88 x^4}-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5} \]
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^6} \, dx &=-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {1}{8} a \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {1}{176} a^2 \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {\left (19 a^3\right ) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{3520 b}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {\left (323 a^4\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{63360 b^2}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {\left (323 a^5\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{67584 b^3}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {\left (4199 a^6\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{946176 b^4}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {\left (4199 a^7\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{1032192 b^5}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {\left (4199 a^8\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{1146880 b^6}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {\left (4199 a^9\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{1310720 b^7}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {\left (4199 a^{10}\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{1572864 b^8}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac {4199 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^9 x}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {\left (4199 a^{11}\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{2097152 b^9}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac {4199 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^9 x}+\frac {12597 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac {\left (4199 a^{12}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{4194304 b^{10}}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac {4199 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^9 x}+\frac {12597 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {\left (12597 a^{12}\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 )}{2097152 b^{10}}\\ &=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac {4199 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^9 x}+\frac {12597 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {12597 a^{12} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{21/2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 61, normalized size = 0.16 \[ -\frac {6 a^{12} \left (a \sqrt [3]{x}+b\right )^2 \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {5}{2},13;\frac {7}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{5 b^{13} \sqrt [3]{x}} \]
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 = 0.47, size = 245, normalized size = 0.65 \[ \frac {\frac {14549535 \, a^{13} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} a^{13} - 169744575 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{13} b + 904981077 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{13} b^{2} - 2913648309 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{13} b^{3} + 6303782342 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{13} b^{4} - 9643633350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{13} b^{5} + 10677769530 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{13} b^{6} - 8598579770 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{13} b^{7} + 4975837515 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{13} b^{8} - 2001671595 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{13} b^{9} - 169744575 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{13} b^{10} + 14549535 \, \sqrt {a x^{\frac {1}{3}} + b} a^{13} b^{11}}{a^{12} b^{10} x^{4}}}{2422210560 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 223, normalized size = 0.59 \[ \frac {\left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (-14549535 a^{12} b^{10} x^{4} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )+14549535 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {43}{2}}-169744575 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {41}{2}}-2001671595 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {5}{2}} b^{\frac {39}{2}}+4975837515 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {7}{2}} b^{\frac {37}{2}}-8598579770 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {9}{2}} b^{\frac {35}{2}}+10677769530 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {11}{2}} b^{\frac {33}{2}}-9643633350 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {13}{2}} b^{\frac {31}{2}}+6303782342 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {15}{2}} b^{\frac {29}{2}}-2913648309 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {17}{2}} b^{\frac {27}{2}}+904981077 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {19}{2}} b^{\frac {25}{2}}-169744575 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {21}{2}} b^{\frac {23}{2}}+14549535 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {23}{2}} b^{\frac {21}{2}}\right )}{2422210560 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {41}{2}} x^{5}} \]
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 \[ \int \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^6} \,d x \]
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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