Optimal. Leaf size=233 \[ -\frac{f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{4 \sqrt{2} e^{5/2} \sqrt{2 d e-b f^2}}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e} \]
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Rubi [A] time = 0.304588, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2116, 897, 1257, 1153, 208} \[ -\frac{f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{4 \sqrt{2} e^{5/2} \sqrt{2 d e-b f^2}}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e} \]
Antiderivative was successfully verified.
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Rule 2116
Rule 897
Rule 1257
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2\right )}{\left (-2 d e+b f^2+2 e x\right )^2} \, dx,x,d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^2 \left (d^2 e-(b d-a e) f^2+\left (-2 d e+b f^2\right ) x^2+e x^4\right )}{\left (-2 d e+b f^2+2 e x^2\right )^2} \, dx,x,\sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}\right )\\ &=-\frac{f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right ) \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac{\operatorname{Subst}\left (\int \frac{-e f^2 \left (4 a e^2-b^2 f^2\right )+4 e^2 \left (2 d e-b f^2\right ) x^2-8 e^3 x^4}{-2 d e+b f^2+2 e x^2} \, dx,x,\sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}\right )}{4 e^3}\\ &=-\frac{f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right ) \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac{\operatorname{Subst}\left (\int \left (-4 e^2 x^2-\frac{e f^2 \left (4 a e^2-b^2 f^2\right )}{-2 d e+b f^2+2 e x^2}\right ) \, dx,x,\sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}\right )}{4 e^3}\\ &=\frac{\left (d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac{f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right ) \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac{1}{4} \left (f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 d e+b f^2+2 e x^2} \, dx,x,\sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}\right )\\ &=\frac{\left (d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac{f^2 \left (4 a-\frac{b^2 f^2}{e^2}\right ) \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}}{\sqrt{2 d e-b f^2}}\right )}{4 \sqrt{2} e^{5/2} \sqrt{2 d e-b f^2}}\\ \end{align*}
Mathematica [A] time = 0.401768, size = 223, normalized size = 0.96 \[ \frac{\frac{\left (b^2 e f^4-4 a e^3 f^2\right ) \sqrt{f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x}}{2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2}-\frac{\sqrt{e} f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{\sqrt{4 d e-2 b f^2}}+\frac{4}{3} e^2 \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x\right )^{3/2}}{4 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int \sqrt{d+ex+f\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + \sqrt{b x + \frac{e^{2} x^{2}}{f^{2}} + a} f + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61705, size = 1472, normalized size = 6.32 \begin{align*} \left [-\frac{3 \,{\left (b^{2} f^{4} - 4 \, a e^{2} f^{2}\right )} \sqrt{-2 \, b e f^{2} + 4 \, d e^{2}} \log \left (-b^{2} f^{4} + 4 \,{\left (b d e - a e^{2}\right )} f^{2} - 4 \,{\left (b e^{2} f^{2} - 2 \, d e^{3}\right )} x - 2 \,{\left (2 \, \sqrt{-2 \, b e f^{2} + 4 \, d e^{2}} e f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} - \sqrt{-2 \, b e f^{2} + 4 \, d e^{2}}{\left (b f^{2} + 2 \, e^{2} x\right )}\right )} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d} + 4 \,{\left (b e f^{3} - 2 \, d e^{2} f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) - 4 \,{\left (3 \, b^{2} e f^{4} - 2 \, b d e^{2} f^{2} - 8 \, d^{2} e^{3} + 10 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )} x - 2 \,{\left (b e^{2} f^{3} - 2 \, d e^{3} f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{48 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )}}, \frac{3 \,{\left (b^{2} f^{4} - 4 \, a e^{2} f^{2}\right )} \sqrt{2 \, b e f^{2} - 4 \, d e^{2}} \arctan \left (\frac{\sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}{\left (\sqrt{2 \, b e f^{2} - 4 \, d e^{2}} f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} - \sqrt{2 \, b e f^{2} - 4 \, d e^{2}}{\left (e x + d\right )}\right )}}{2 \,{\left (a e f^{2} - d^{2} e +{\left (b e f^{2} - 2 \, d e^{2}\right )} x\right )}}\right ) + 2 \,{\left (3 \, b^{2} e f^{4} - 2 \, b d e^{2} f^{2} - 8 \, d^{2} e^{3} + 10 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )} x - 2 \,{\left (b e^{2} f^{3} - 2 \, d e^{3} f\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{24 \,{\left (b e^{3} f^{2} - 2 \, d e^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d + e x + f \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + \sqrt{b x + \frac{e^{2} x^{2}}{f^{2}} + a} f + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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