Optimal. Leaf size=87 \[ -\frac {3 b \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\sqrt {x} \sqrt {b \sqrt {x}+a x}}{a}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2043, 684, 654,
634, 212} \begin {gather*} \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{2 a^{5/2}}-\frac {3 b \sqrt {a x+b \sqrt {x}}}{2 a^2}+\frac {\sqrt {x} \sqrt {a x+b \sqrt {x}}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rule 684
Rule 2043
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x} \sqrt {b \sqrt {x}+a x}}{a}-\frac {(3 b) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{2 a}\\ &=-\frac {3 b \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\sqrt {x} \sqrt {b \sqrt {x}+a x}}{a}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {3 b \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\sqrt {x} \sqrt {b \sqrt {x}+a x}}{a}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^2}\\ &=-\frac {3 b \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\sqrt {x} \sqrt {b \sqrt {x}+a x}}{a}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 86, normalized size = 0.99 \begin {gather*} \frac {\left (-3 b+2 a \sqrt {x}\right ) \sqrt {b \sqrt {x}+a x}}{2 a^2}-\frac {3 b^2 \log \left (a^2 b+2 a^3 \sqrt {x}-2 a^{5/2} \sqrt {b \sqrt {x}+a x}\right )}{4 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(159\) vs.
\(2(63)=126\).
time = 0.38, size = 160, normalized size = 1.84
method | result | size |
derivativedivides | \(\frac {\sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{a}-\frac {3 b \left (\frac {\sqrt {b \sqrt {x}+a x}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{2 a^{\frac {3}{2}}}\right )}{2 a}\) | \(74\) |
default | \(\frac {\sqrt {b \sqrt {x}+a x}\, \left (4 \sqrt {x}\, \sqrt {b \sqrt {x}+a x}\, a^{\frac {5}{2}}+2 \sqrt {b \sqrt {x}+a x}\, a^{\frac {3}{2}} b -8 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {3}{2}} b +4 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{2}-b^{2} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \right )}{4 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {7}{2}}}\) | \(160\) |
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\sqrt {x}}{\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 = 0.59, size = 69, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {a x + b \sqrt {x}} {\left (\frac {2 \, \sqrt {x}}{a} - \frac {3 \, b}{a^{2}}\right )} - \frac {3 \, b^{2} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{4 \, a^{\frac {5}{2}}} \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 {\sqrt {x}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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