Optimal. Leaf size=147 \[ -\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e}-\frac {a f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 \sqrt {d} e} \]
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Rubi [A] time = 0.12, 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 = {2117, 897, 1257, 1153, 206} \[ -\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e}-\frac {a f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 \sqrt {d} e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 897
Rule 1153
Rule 1257
Rule 2117
Rubi steps
\begin {align*} \int \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}} \, dx &=\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 = 0.33, size = 139, normalized size = 0.95 \[ -\frac {\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x}-\frac {2}{3} \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}+\frac {a f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 301, normalized size = 2.05 \[ \left [\frac {3 \, a \sqrt {d} f^{2} \log \left (a f^{2} - 2 \, d e x + 2 \, d f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, {\left (\sqrt {d} e x - \sqrt {d} f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d}\right ) + 2 \, {\left (5 \, d e x - d f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{12 \, d e}, \frac {3 \, a \sqrt {-d} f^{2} \arctan \left (\frac {\sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt {-d}}{d}\right ) + {\left (5 \, d e x - d f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d^{2}\right )} \sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{6 \, d e}\right ] \]
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 \[ \int \sqrt {e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \sqrt {e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, f}\, dx \]
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 \[ \int \sqrt {e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d}\,{d x} \]
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 \[ \int \sqrt {d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}} \,d x \]
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 \[ \int \sqrt {d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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