Optimal. Leaf size=107 \[ \frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{5 b^4 \sqrt {x}} \]
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Rubi [A]
time = 0.11, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2040, 2041,
2039} \begin {gather*} -\frac {64 a^2 \sqrt {a x+b \sqrt {x}}}{5 b^4 \sqrt {x}}+\frac {32 a \sqrt {a x+b \sqrt {x}}}{5 b^3 x}-\frac {24 \sqrt {a x+b \sqrt {x}}}{5 b^2 x^{3/2}}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2040
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}+\frac {6 \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}-\frac {(24 a) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{5 b^2}\\ &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}+\frac {\left (16 a^2\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{5 b^3}\\ &=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{5 b^4 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 70, normalized size = 0.65 \begin {gather*} -\frac {4 \sqrt {b \sqrt {x}+a x} \left (b^3-2 a b^2 \sqrt {x}+8 a^2 b x+16 a^3 x^{3/2}\right )}{5 b^4 \left (b+a \sqrt {x}\right ) x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.39, size = 548, normalized size = 5.12
method | result | size |
derivativedivides | \(-\frac {4}{5 b x \sqrt {b \sqrt {x}+a x}}-\frac {12 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}+\frac {8 a \left (b +2 a \sqrt {x}\right )}{3 b^{3} \sqrt {b \sqrt {x}+a x}}\right )}{5 b}\) | \(72\) |
default | \(\frac {2 \sqrt {b \sqrt {x}+a x}\, \left (10 x^{\frac {9}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {11}{2}}+10 x^{\frac {9}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {11}{2}}-30 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}}+10 x^{\frac {7}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} a^{\frac {9}{2}}+10 x^{\frac {7}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {7}{2}} b^{2}+10 x^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {7}{2}} b^{2}-5 x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{5} b +5 x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{5} b -16 x^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}+20 x^{4} \sqrt {b \sqrt {x}+a x}\, a^{\frac {9}{2}} b +20 x^{4} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {9}{2}} b -5 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}+5 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}-52 x^{3} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b -2 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{4}+4 x^{2} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3}-10 x^{4} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{4} b^{2}+10 x^{4} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{4} b^{2}\right )}{5 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{5} \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, x^{\frac {7}{2}}}\) | \(548\) |
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 [A]
time = 2.80, size = 79, normalized size = 0.74 \begin {gather*} \frac {4 \, {\left (8 \, a^{3} b x^{2} - 3 \, a b^{3} x - {\left (16 \, a^{4} x^{2} - 10 \, a^{2} b^{2} x - b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{5 \, {\left (a^{2} b^{4} x^{3} - b^{6} x^{2}\right )}} \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 {1}{x^{\frac {3}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {1}{x^{3/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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