Optimal. Leaf size=158 \[ -\frac {1+\frac {a f^2}{d^2}}{e \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 d^2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {3 a f^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{\sqrt {d}}\right )}{2 d^{5/2} e} \]
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Rubi [A]
time = 0.11, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2142, 911,
1273, 464, 212} \begin {gather*} \frac {3 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 d^{5/2} e}-\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 d^2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{e \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 464
Rule 911
Rule 1273
Rule 2142
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {d^2+a f^2-2 d x+x^2}{(d-x)^2 x^{3/2}} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \frac {d^2+a f^2-2 d x^2+x^4}{x^2 \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 d^2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {\text {Subst}\left (\int \frac {-2 d \left (d^2+a f^2\right )+\left (2 d^2-a f^2\right ) x^2}{x^2 \left (d-x^2\right )} \, dx,x,\sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 d^2 e}\\ &=-\frac {d^2+a f^2}{d^2 e \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 d^2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {\left (3 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 d^2 e}\\ &=-\frac {d^2+a f^2}{d^2 e \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}-\frac {a f^2 \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}{2 d^2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\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 )}{2 d^{5/2} e}\\ \end {align*}
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Mathematica [A]
time = 1.41, size = 169, normalized size = 1.07 \begin {gather*} \frac {-\frac {\sqrt {d} \left (2 d^2 \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )+a f^2 \left (d+3 e x+3 f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )}{\left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right ) \sqrt {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}}+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 )}{2 d^{5/2} e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d +e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {3}{2}}}\, dx\]
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 = 0.43, size = 490, normalized size = 3.10 \begin {gather*} \left [-\frac {3 \, {\left (a^{2} f^{4} - 2 \, a d f^{2} x e - a d^{2} f^{2}\right )} \sqrt {d} \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 (2 \, a d^{2} f^{2} - 2 \, d^{2} x^{2} e^{2} + 2 \, d^{4} + {\left (3 \, a d f^{2} + d^{3}\right )} x e - {\left (3 \, a d f^{3} - 2 \, d^{2} f x e + d^{3} f\right )} \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}}{4 \, {\left (2 \, d^{4} x e^{2} - {\left (a d^{3} f^{2} - d^{5}\right )} e\right )}}, \frac {3 \, {\left (a^{2} f^{4} - 2 \, a d f^{2} x e - a d^{2} f^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} + d} \sqrt {-d}}{d}\right ) - {\left (2 \, a d^{2} f^{2} - 2 \, d^{2} x^{2} e^{2} + 2 \, d^{4} + {\left (3 \, a d f^{2} + d^{3}\right )} x e - {\left (3 \, a d f^{3} - 2 \, d^{2} f x e + d^{3} f\right )} \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}}{2 \, {\left (2 \, d^{4} x e^{2} - {\left (a d^{3} f^{2} - d^{5}\right )} e\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}{\left (d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}\right )^{\frac {3}{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 \frac {1}{{\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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