Optimal. Leaf size=54 \[ -\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^2 \sqrt {x}} \]
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Rubi [A]
time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2041, 2039}
\begin {gather*} \frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}-\frac {(2 a) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{3 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 35, normalized size = 0.65 \begin {gather*} -\frac {4 \left (b-2 a \sqrt {x}\right ) \sqrt {b \sqrt {x}+a x}}{3 b^2 x} \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 = 0.52, size = 194, normalized size = 3.59
method | result | size |
derivativedivides | \(-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^{2} \sqrt {x}}\) | \(41\) |
default | \(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (6 x^{\frac {5}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {5}{2}}+6 x^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {5}{2}}-3 x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b +3 x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b -12 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}}+4 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b \sqrt {a}\, x \right )}{3 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{3} \sqrt {a}\, x^{\frac {5}{2}}}\) | \(194\) |
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 = 3.74, size = 29, normalized size = 0.54 \begin {gather*} \frac {4 \, \sqrt {a x + b \sqrt {x}} {\left (2 \, a \sqrt {x} - b\right )}}{3 \, b^{2} x} \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}} \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 53, normalized size = 0.98 \begin {gather*} \frac {4 \, {\left (3 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )}}{3 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3}} \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 = -0.02 \begin {gather*} \int \frac {1}{x^{3/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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