Optimal. Leaf size=244 \[ \frac {\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{e}-\frac {f^2 \left (4 a e-\frac {b^2 f^2}{e}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{2 \left (2 d e-b f^2\right ) \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{\sqrt {2 d e-b f^2}}\right )}{2 \sqrt {2} e^{3/2} \left (2 d e-b f^2\right )^{3/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2141, 911,
1171, 396, 214} \begin {gather*} -\frac {f^2 \left (4 a e-\frac {b^2 f^2}{e}\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{2 \left (2 d e-b f^2\right ) \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 )}{2 \sqrt {2} e^{3/2} \left (2 d e-b f^2\right )^{3/2}}+\frac {\sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 396
Rule 911
Rule 1171
Rule 2141
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}} \, dx &=2 \text {Subst}\left (\int \frac {d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2}{\sqrt {x} \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 \text {Subst}\left (\int \frac {d^2 e-(b d-a e) f^2+\left (-2 d e+b f^2\right ) x^2+e x^4}{\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 e-\frac {b^2 f^2}{e}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{2 \left (2 d e-b f^2\right ) \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 {2 \text {Subst}\left (\int \frac {\frac {1}{4} \left (-8 d^2 e+8 b d f^2-4 a e f^2-\frac {b^2 f^4}{e}\right )+\left (2 d e-b f^2\right ) x^2}{-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 )}{2 d e-b f^2}\\ &=\frac {\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{e}-\frac {f^2 \left (4 a e-\frac {b^2 f^2}{e}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{2 \left (2 d e-b f^2\right ) \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 {\left (f^2 \left (4 a e^2-b^2 f^2\right )\right ) \text {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 )}{2 e \left (2 d e-b f^2\right )}\\ &=\frac {\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{e}-\frac {f^2 \left (4 a e-\frac {b^2 f^2}{e}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{2 \left (2 d e-b f^2\right ) \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 )}{2 \sqrt {2} e^{3/2} \left (2 d e-b f^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.58, size = 341, normalized size = 1.40 \begin {gather*} \frac {\frac {2 \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}} \left (-b^2 f^4-4 b e f^2 \left (-d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 e^2 \left (-a f^2+2 d e x+2 d f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{\left (2 d e-b f^2\right ) \left (b f^2+2 e \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}+\frac {4 \sqrt {2} a e^2 f^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}}{\sqrt {-2 d e+b f^2}}\right )}{\left (-2 d e+b f^2\right )^{3/2}}-\frac {\sqrt {2} b^2 f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}}{\sqrt {-2 d e+b f^2}}\right )}{\left (-2 d e+b f^2\right )^{3/2}}}{4 e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}}\, dx\]
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 = 0.52, size = 620, normalized size = 2.54 \begin {gather*} \left [-\frac {{\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \sqrt {-2 \, b f^{2} e + 4 \, d e^{2}} \log \left (b^{2} f^{4} - 4 \, b d f^{2} e - 8 \, d x e^{3} + 4 \, {\left (b f^{2} x + a f^{2}\right )} e^{2} + 2 \, {\left (2 \, \sqrt {-2 \, b f^{2} e + 4 \, d e^{2}} f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} e - \sqrt {-2 \, b f^{2} e + 4 \, d e^{2}} {\left (b f^{2} + 2 \, x e^{2}\right )}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d} - 4 \, {\left (b f^{3} e - 2 \, d f e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) - 4 \, {\left (b^{2} f^{4} e - 6 \, b d f^{2} e^{2} + 4 \, d x e^{4} - 2 \, {\left (b f^{2} x - 4 \, d^{2}\right )} e^{3} + 2 \, {\left (b f^{3} e^{2} - 2 \, d f e^{3}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}}{8 \, {\left (b^{2} f^{4} e^{2} - 4 \, b d f^{2} e^{3} + 4 \, d^{2} e^{4}\right )}}, \frac {{\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \sqrt {2 \, b f^{2} e - 4 \, d e^{2}} \arctan \left (-\frac {\sqrt {2 \, b f^{2} e - 4 \, d e^{2}} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}}{b f^{2} - 2 \, d e}\right ) + 2 \, {\left (b^{2} f^{4} e - 6 \, b d f^{2} e^{2} + 4 \, d x e^{4} - 2 \, {\left (b f^{2} x - 4 \, d^{2}\right )} e^{3} + 2 \, {\left (b f^{3} e^{2} - 2 \, d f e^{3}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}}{4 \, {\left (b^{2} f^{4} e^{2} - 4 \, b d f^{2} e^{3} + 4 \, d^{2} e^{4}\right )}}\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 \frac {1}{\sqrt {d + e x + f \sqrt {a + b x + \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 = -0.00 \begin {gather*} \int \frac {1}{\sqrt {d+e\,x+f\,\sqrt {a+b\,x+\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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