Optimal. Leaf size=158 \[ \frac {3 a f^2 \tanh ^{-1}\left (\frac {\sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {d}}\right )}{2 d^{5/2} e}-\frac {a f^2 \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}{2 d^2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{e \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}} \]
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Rubi [A] time = 0.16, 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 = {2117, 897, 1259, 453, 206} \begin {gather*} -\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}}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 453
Rule 897
Rule 1259
Rule 2117
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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \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 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 = 0.43, size = 167, normalized size = 1.06 \begin {gather*} \frac {\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 )}{d^{5/2}}+\frac {-2 d^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )-a f^2 \left (3 f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+3 e x\right )}{d^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right ) \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.31, size = 194, normalized size = 1.23 \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 {\left (-3 a f^3-2 d^2 f\right ) \sqrt {a+\frac {e^2 x^2}{f^2}}-a d f^2-3 a e f^2 x-2 d^2 e x}{2 d^2 e^2 x \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}+2 d^2 e f \sqrt {a+\frac {e^2 x^2}{f^2}} \sqrt {f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 487, normalized size = 3.08 \begin {gather*} \left [\frac {3 \, {\left (a^{2} f^{4} - 2 \, a d e f^{2} x - a d^{2} f^{2}\right )} \sqrt {d} \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 (2 \, d^{2} e^{2} x^{2} - 2 \, a d^{2} f^{2} - 2 \, d^{4} - {\left (3 \, a d e f^{2} + d^{3} e\right )} x + {\left (3 \, a d f^{3} - 2 \, d^{2} e f x + d^{3} f\right )} \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}}{4 \, {\left (a d^{3} e f^{2} - 2 \, d^{4} e^{2} x - d^{5} e\right )}}, -\frac {3 \, {\left (a^{2} f^{4} - 2 \, a d e f^{2} x - a d^{2} f^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt {-d}}{d}\right ) + {\left (2 \, d^{2} e^{2} x^{2} - 2 \, a d^{2} f^{2} - 2 \, d^{4} - {\left (3 \, a d e f^{2} + d^{3} e\right )} x + {\left (3 \, a d f^{3} - 2 \, d^{2} e f x + d^{3} f\right )} \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}}{2 \, {\left (a d^{3} e f^{2} - 2 \, d^{4} e^{2} x - d^{5} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, f \right )^{\frac {3}{2}}}\, dx \end {gather*}
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*} \int \frac {1}{{\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac {3}{2}}}\,{d x} \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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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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