Optimal. Leaf size=84 \[ -\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}+\frac {16 a \sqrt {b \sqrt {x}+a x}}{15 b^2 x}-\frac {32 a^2 \sqrt {b \sqrt {x}+a x}}{15 b^3 \sqrt {x}} \]
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Rubi [A]
time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2041, 2039}
\begin {gather*} -\frac {32 a^2 \sqrt {a x+b \sqrt {x}}}{15 b^3 \sqrt {x}}+\frac {16 a \sqrt {a x+b \sqrt {x}}}{15 b^2 x}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}-\frac {(4 a) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{5 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}+\frac {16 a \sqrt {b \sqrt {x}+a x}}{15 b^2 x}+\frac {\left (8 a^2\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{15 b^2}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b x^{3/2}}+\frac {16 a \sqrt {b \sqrt {x}+a x}}{15 b^2 x}-\frac {32 a^2 \sqrt {b \sqrt {x}+a x}}{15 b^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 48, normalized size = 0.57 \begin {gather*} -\frac {4 \sqrt {b \sqrt {x}+a x} \left (3 b^2-4 a b \sqrt {x}+8 a^2 x\right )}{15 b^3 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 = 0.38, size = 218, normalized size = 2.60
method | result | size |
derivativedivides | \(-\frac {4 \sqrt {b \sqrt {x}+a x}}{5 b \,x^{\frac {3}{2}}}-\frac {8 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {4 a \sqrt {b \sqrt {x}+a x}}{3 b^{2} \sqrt {x}}\right )}{5 b}\) | \(67\) |
default | \(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (60 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {5}{2}} a^{\frac {5}{2}}-30 \sqrt {b \sqrt {x}+a x}\, x^{\frac {7}{2}} a^{\frac {7}{2}}-30 x^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {7}{2}}-15 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b +15 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b +12 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {3}{2}} \sqrt {a}\, b^{2}-28 a^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b \,x^{2}\right )}{15 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{4} x^{\frac {7}{2}} \sqrt {a}}\) | \(218\) |
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 = 4.04, size = 42, normalized size = 0.50 \begin {gather*} \frac {4 \, {\left (4 \, a b x - {\left (8 \, a^{2} x + 3 \, b^{2}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{15 \, b^{3} x^{2}} \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^{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 = 2.37, size = 84, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (20 \, a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 15 \, \sqrt {a} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 3 \, b^{2}\right )}}{15 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{5}} \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^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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