Optimal. Leaf size=147 \[ -\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac {a f^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{\sqrt {d}}\right )}{2 \sqrt {d} e} \]
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Rubi [A]
time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2142, 911,
1271, 1167, 212} \begin {gather*} -\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e}-\frac {a f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 \sqrt {d} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 911
Rule 1167
Rule 1271
Rule 2142
Rubi steps
\begin {align*} \int \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {x} \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \frac {x^2 \left (d^2+a f^2-2 d x^2+x^4\right )}{\left (d-x^2\right )^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{e}\\ &=-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\text {Subst}\left (\int \frac {-a f^2+2 d x^2-2 x^4}{d-x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ &=-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\text {Subst}\left (\int \left (2 x^2-\frac {a f^2}{d-x^2}\right ) \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ &=-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac {\left (a f^2\right ) \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ &=-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac {a f^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{\sqrt {d}}\right )}{2 \sqrt {d} e}\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 142, normalized size = 0.97 \begin {gather*} \frac {\frac {\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}} \left (-a f^2+2 (d+2 e x) \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )}{e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}-\frac {3 a f^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{\sqrt {d}}\right )}{\sqrt {d}}}{6 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {d +e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}\, dx\]
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 = 0.40, size = 299, normalized size = 2.03 \begin {gather*} \left [\frac {{\left (3 \, a \sqrt {d} f^{2} \log \left (a f^{2} - 2 \, d x e + 2 \, d f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + 2 \, {\left (\sqrt {d} x e - \sqrt {d} f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + d}\right ) + 2 \, {\left (5 \, d x e - d f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt {x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + d}\right )} e^{\left (-1\right )}}{12 \, d}, \frac {{\left (3 \, a \sqrt {-d} f^{2} \arctan \left (\frac {\sqrt {x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + d} \sqrt {-d}}{d}\right ) + {\left (5 \, d x e - d f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt {x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + d}\right )} e^{\left (-1\right )}}{6 \, d}\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 \sqrt {d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{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 \sqrt {d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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