Optimal. Leaf size=165 \[ \frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{63 b^5 x}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{63 b^6 \sqrt {x}} \]
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Rubi [A]
time = 0.17, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {1024 a^4 \sqrt {a x+b \sqrt {x}}}{63 b^6 \sqrt {x}}+\frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{63 b^5 x}-\frac {128 a^2 \sqrt {a x+b \sqrt {x}}}{21 b^4 x^{3/2}}+\frac {320 a \sqrt {a x+b \sqrt {x}}}{63 b^3 x^2}-\frac {40 \sqrt {a x+b \sqrt {x}}}{9 b^2 x^{5/2}}+\frac {4}{b x^2 \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^{5/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}+\frac {10 \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}-\frac {(80 a) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{9 b^2}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}+\frac {\left (160 a^2\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{21 b^3}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}-\frac {\left (128 a^3\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{21 b^4}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{63 b^5 x}+\frac {\left (256 a^4\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{63 b^5}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{63 b^5 x}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{63 b^6 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 96, normalized size = 0.58 \begin {gather*} -\frac {4 \sqrt {b \sqrt {x}+a x} \left (7 b^5-10 a b^4 \sqrt {x}+16 a^2 b^3 x-32 a^3 b^2 x^{3/2}+128 a^4 b x^2+256 a^5 x^{5/2}\right )}{63 b^6 \left (b+a \sqrt {x}\right ) x^{5/2}} \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 = 0.40, size = 592, normalized size = 3.59
method | result | size |
derivativedivides | \(-\frac {4}{9 b \,x^{2} \sqrt {b \sqrt {x}+a x}}-\frac {20 a \left (-\frac {2}{7 b \,x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}-\frac {8 a \left (-\frac {2}{5 b x \sqrt {b \sqrt {x}+a x}}-\frac {6 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}+\frac {8 a \left (b +2 a \sqrt {x}\right )}{3 b^{3} \sqrt {b \sqrt {x}+a x}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\) | \(124\) |
default | \(\frac {4 \sqrt {b \sqrt {x}+a x}\, \left (126 x^{\frac {13}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {15}{2}}+126 x^{\frac {13}{2}} a^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}-315 x^{\frac {11}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {13}{2}}+63 x^{\frac {11}{2}} a^{\frac {13}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}}+126 x^{\frac {11}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {11}{2}} b^{2}+126 x^{\frac {11}{2}} a^{\frac {11}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{2}-128 x^{\frac {9}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2}+63 x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b -63 x^{\frac {13}{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^{7} b +252 x^{6} \sqrt {b \sqrt {x}+a x}\, a^{\frac {13}{2}} b +252 x^{6} a^{\frac {13}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b +63 x^{\frac {11}{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^{3}-63 x^{\frac {11}{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^{3}-508 x^{5} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {11}{2}} b -16 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4}+32 x^{4} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3}-7 x^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{6}+126 x^{6} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{6} b^{2}-126 x^{6} \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^{6} b^{2}+10 x^{3} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5}\right )}{63 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{7} x^{\frac {11}{2}} \sqrt {a}\, \left (a \sqrt {x}+b \right )^{2}}\) | \(592\) |
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 = 2.40, size = 101, normalized size = 0.61 \begin {gather*} \frac {4 \, {\left (128 \, a^{5} b x^{3} - 48 \, a^{3} b^{3} x^{2} - 17 \, a b^{5} x - {\left (256 \, a^{6} x^{3} - 160 \, a^{4} b^{2} x^{2} - 26 \, a^{2} b^{4} x - 7 \, b^{6}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{63 \, {\left (a^{2} b^{6} x^{4} - b^{8} x^{3}\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 {5}{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^{5/2}\,{\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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