Optimal. Leaf size=195 \[ \frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{231 b^7 \sqrt {x}}-\frac {2048 a^4 \sqrt {a x+b \sqrt {x}}}{231 b^6 x}+\frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{77 b^5 x^{3/2}}-\frac {1280 a^2 \sqrt {a x+b \sqrt {x}}}{231 b^4 x^2}+\frac {160 a \sqrt {a x+b \sqrt {x}}}{33 b^3 x^{5/2}}-\frac {48 \sqrt {a x+b \sqrt {x}}}{11 b^2 x^3}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.30, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ \frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{77 b^5 x^{3/2}}-\frac {1280 a^2 \sqrt {a x+b \sqrt {x}}}{231 b^4 x^2}+\frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{231 b^7 \sqrt {x}}-\frac {2048 a^4 \sqrt {a x+b \sqrt {x}}}{231 b^6 x}+\frac {160 a \sqrt {a x+b \sqrt {x}}}{33 b^3 x^{5/2}}-\frac {48 \sqrt {a x+b \sqrt {x}}}{11 b^2 x^3}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}+\frac {12 \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {48 \sqrt {b \sqrt {x}+a x}}{11 b^2 x^3}-\frac {(120 a) \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{11 b^2}\\ &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {48 \sqrt {b \sqrt {x}+a x}}{11 b^2 x^3}+\frac {160 a \sqrt {b \sqrt {x}+a x}}{33 b^3 x^{5/2}}+\frac {\left (320 a^2\right ) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{33 b^3}\\ &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {48 \sqrt {b \sqrt {x}+a x}}{11 b^2 x^3}+\frac {160 a \sqrt {b \sqrt {x}+a x}}{33 b^3 x^{5/2}}-\frac {1280 a^2 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^2}-\frac {\left (640 a^3\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{77 b^4}\\ &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {48 \sqrt {b \sqrt {x}+a x}}{11 b^2 x^3}+\frac {160 a \sqrt {b \sqrt {x}+a x}}{33 b^3 x^{5/2}}-\frac {1280 a^2 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^2}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{77 b^5 x^{3/2}}+\frac {\left (512 a^4\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{77 b^5}\\ &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {48 \sqrt {b \sqrt {x}+a x}}{11 b^2 x^3}+\frac {160 a \sqrt {b \sqrt {x}+a x}}{33 b^3 x^{5/2}}-\frac {1280 a^2 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^2}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{77 b^5 x^{3/2}}-\frac {2048 a^4 \sqrt {b \sqrt {x}+a x}}{231 b^6 x}-\frac {\left (1024 a^5\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{231 b^6}\\ &=\frac {4}{b x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {48 \sqrt {b \sqrt {x}+a x}}{11 b^2 x^3}+\frac {160 a \sqrt {b \sqrt {x}+a x}}{33 b^3 x^{5/2}}-\frac {1280 a^2 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^2}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{77 b^5 x^{3/2}}-\frac {2048 a^4 \sqrt {b \sqrt {x}+a x}}{231 b^6 x}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{231 b^7 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 96, normalized size = 0.49 \[ \frac {4 \left (1024 a^6 x^3+512 a^5 b x^{5/2}-128 a^4 b^2 x^2+64 a^3 b^3 x^{3/2}-40 a^2 b^4 x+28 a b^5 \sqrt {x}-21 b^6\right )}{231 b^7 x^{5/2} \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 109, normalized size = 0.56 \[ -\frac {4 \, {\left (512 \, a^{6} b x^{3} - 192 \, a^{4} b^{3} x^{2} - 68 \, a^{2} b^{5} x - 21 \, b^{7} - {\left (1024 \, a^{7} x^{3} - 640 \, a^{5} b^{2} x^{2} - 104 \, a^{3} b^{4} x - 49 \, a b^{6}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{231 \, {\left (a^{2} b^{7} x^{4} - b^{9} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 614, normalized size = 3.15 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (1155 a^{8} b \,x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-1155 a^{8} b \,x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2310 a^{7} b^{2} x^{7} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2310 a^{7} b^{2} x^{7} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+1155 a^{6} b^{3} x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-1155 a^{6} b^{3} x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2310 \sqrt {a x +b \sqrt {x}}\, a^{\frac {17}{2}} x^{\frac {15}{2}}-2310 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {17}{2}} x^{\frac {15}{2}}-4620 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}} b \,x^{7}-4620 \sqrt {a x +b \sqrt {x}}\, a^{\frac {15}{2}} b \,x^{7}-2310 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {13}{2}} b^{2} x^{\frac {13}{2}}-2310 \sqrt {a x +b \sqrt {x}}\, a^{\frac {13}{2}} b^{2} x^{\frac {13}{2}}+5544 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {15}{2}} x^{\frac {13}{2}}-924 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {15}{2}} x^{\frac {13}{2}}+8716 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} b \,x^{6}+2048 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b^{2} x^{\frac {11}{2}}-512 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{3} x^{5}+256 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{4} x^{\frac {9}{2}}-160 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{5} x^{4}+112 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{6} x^{\frac {7}{2}}-84 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{7} x^{3}\right )}{231 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{8} x^{\frac {13}{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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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