Optimal. Leaf size=193 \[ -\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 d^4 e}+\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{2 d^4 e}-\frac {a f^2}{2 d^3 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {a f^2}{d^3 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{4 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2} \]
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Rubi [A] time = 0.13, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2117, 893} \begin {gather*} -\frac {a f^2}{2 d^3 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {a f^2}{d^3 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{4 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}-\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 d^4 e}+\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{2 d^4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 893
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} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {d^2+a f^2-2 d x+x^2}{(d-x)^2 x^3} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a f^2}{d^3 (d-x)^2}+\frac {3 a f^2}{d^4 (d-x)}+\frac {d^2+a f^2}{d^2 x^3}+\frac {2 a f^2}{d^3 x^2}+\frac {3 a f^2}{d^4 x}\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac {a f^2}{2 d^3 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {1+\frac {a f^2}{d^2}}{4 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}-\frac {a f^2}{d^3 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {3 a f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^4 e}+\frac {3 a f^2 \log \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^4 e}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 180, normalized size = 0.93 \begin {gather*} \frac {-\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{d^4}+\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{d^4}+\frac {a f^2}{d^3 \left (f \left (-\sqrt {a+\frac {e^2 x^2}{f^2}}\right )-e x\right )}-\frac {2 a f^2}{d^3 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.54, size = 528, normalized size = 2.74 \begin {gather*} \frac {\sqrt {a+\frac {e^2 x^2}{f^2}} \left (-3 a^2 f^5+5 a d^2 f^3+9 a d e f^3 x-3 d^3 e f x-4 d^2 e^2 f x^2\right )}{2 d^3 e \left (-a f^2+d^2+2 d e x\right )^2}+\frac {3 a^4 f^8 x-9 a^3 d^2 f^6 x-9 a^3 d e f^6 x^2+3 a^2 d^4 f^4 x+15 a^2 d^3 e f^4 x^2+4 a^2 d^2 e^2 f^4 x^3+a d^6 f^2 x-3 a d^5 e f^2 x^2-8 a d^4 e^2 f^2 x^3+2 d^8 x+5 d^7 e x^2+4 d^6 e^2 x^3}{2 d^3 \left (d^2-a f^2\right )^2 \left (-a f^2+d^2+2 d e x\right )^2}+\frac {3 a f^3 \sqrt {\frac {e^2}{f^2}} \log \left (d \sqrt {a+\frac {e^2 x^2}{f^2}}+a f+d x \left (-\sqrt {\frac {e^2}{f^2}}\right )\right )}{4 d^4 e^2}-\frac {3 a \left (e f^2-f^3 \sqrt {\frac {e^2}{f^2}}\right ) \log \left (\sqrt {a+\frac {e^2 x^2}{f^2}}-x \sqrt {\frac {e^2}{f^2}}\right )}{4 d^4 e^2}+\frac {3 a \left (e f^2-f^3 \sqrt {\frac {e^2}{f^2}}\right ) \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d-f x \sqrt {\frac {e^2}{f^2}}\right )}{4 d^4 e^2}+\frac {3 a f^2 \log \left (d^4 e \sqrt {a+\frac {e^2 x^2}{f^2}}+a d^3 e f+d^4 (-e) x \sqrt {\frac {e^2}{f^2}}\right )}{4 d^4 e} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 536, normalized size = 2.78 \begin {gather*} \frac {5 \, a^{3} f^{6} + 8 \, d^{3} e^{3} x^{3} - 6 \, a^{2} d^{2} f^{4} - 3 \, a d^{4} f^{2} + 2 \, {\left (a d^{2} e^{2} f^{2} + 5 \, d^{4} e^{2}\right )} x^{2} - 2 \, {\left (7 \, a^{2} d e f^{4} + a d^{3} e f^{2} - 2 \, d^{5} e\right )} x + 3 \, {\left (a^{3} f^{6} + 4 \, a d^{2} e^{2} f^{2} x^{2} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} - 4 \, {\left (a^{2} d e f^{4} - a d^{3} e f^{2}\right )} x\right )} \log \left (-a e f^{2} x + 2 \, d e^{2} x^{2} + a d f^{2} + {\left (a f^{3} - 2 \, d e f x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \, {\left (a^{3} f^{6} + 4 \, a d^{2} e^{2} f^{2} x^{2} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} - 4 \, {\left (a^{2} d e f^{4} - a d^{3} e f^{2}\right )} x\right )} \log \left (-a f^{2} + 2 \, d e x + d^{2}\right ) - 3 \, {\left (a^{3} f^{6} + 4 \, a d^{2} e^{2} f^{2} x^{2} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} - 4 \, {\left (a^{2} d e f^{4} - a d^{3} e f^{2}\right )} x\right )} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) - 2 \, {\left (3 \, a^{2} d f^{5} + 4 \, d^{3} e^{2} f x^{2} - 5 \, a d^{3} f^{3} - 3 \, {\left (3 \, a d^{2} e f^{3} - d^{4} e f\right )} x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}}{4 \, {\left (a^{2} d^{4} e f^{4} + 4 \, d^{6} e^{3} x^{2} - 2 \, a d^{6} e f^{2} + d^{8} e - 4 \, {\left (a d^{5} e^{2} f^{2} - d^{7} e^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 9721, normalized size = 50.37 \begin {gather*} \text {output too large to display} \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 )}^{3}}\,{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} \,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 )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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