Optimal. Leaf size=266 \[ \frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^3}-\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^3}-\frac {2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )} \]
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Rubi [A] time = 0.23, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2116, 893} \begin {gather*} \frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^3}-\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \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 {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \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 )}{\left (2 d e-b f^2\right )^3}-\frac {2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 893
Rule 2116
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx &=2 \operatorname {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}{x^2 \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 )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {d^2 e-b d f^2+a e f^2}{\left (2 d e-b f^2\right )^2 x^2}+\frac {4 a e^2 f^2-b^2 f^4}{\left (2 d e-b f^2\right )^3 x}+\frac {4 a e^3 f^2-b^2 e f^4}{\left (2 d e-b f^2\right )^2 \left (2 d e-b f^2-2 e x\right )^2}+\frac {2 \left (4 a e^3 f^2-b^2 e f^4\right )}{\left (2 d e-b f^2\right )^3 \left (2 d e-b f^2-2 e x\right )}\right ) \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )\\ &=-\frac {2 \left (d^2 e-b d f^2+a e f^2\right )}{\left (2 d e-b f^2\right )^2 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \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 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{\left (2 d e-b f^2\right )^3}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \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 )}{\left (2 d e-b f^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 237, normalized size = 0.89 \begin {gather*} -\frac {\frac {f^2 \left (b^2 f^2-4 a e^2\right ) \left (b f^2-2 d e\right )}{2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2}+2 f^2 \left (b^2 f^2-4 a e^2\right ) \log \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )-2 f^2 \left (b^2 f^2-4 a e^2\right ) \log \left (-2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )-b f^2\right )+\frac {2 \left (2 d e-b f^2\right ) \left (a e f^2-b d f^2+d^2 e\right )}{f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}}{\left (2 d e-b f^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 5.25, size = 614, normalized size = 2.31 \begin {gather*} \frac {2 \left (2 a^2 e^2 f^4 x-2 a b d e f^4 x+a b e^2 f^4 x^2-2 a d e^3 f^2 x^2+b^2 d^2 f^4 x-2 b d^3 e f^2 x-b d^2 e^2 f^2 x^2+2 d^4 e^2 x+2 d^3 e^3 x^2\right )}{\left (d^2-a f^2\right ) \left (2 d e-b f^2\right )^2 \left (-a f^2-b f^2 x+d^2+2 d e x\right )}+\frac {\left (-4 a e^3 f^2+4 a e^2 f^3 \sqrt {\frac {e^2}{f^2}}-b^2 f^5 \sqrt {\frac {e^2}{f^2}}+b^2 e f^4\right ) \log \left (2 e \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+b f-2 e x \sqrt {\frac {e^2}{f^2}}\right )}{e \left (2 d e-b f^2\right )^3}+\frac {\left (4 a e^3 f^2-4 a e^2 f^3 \sqrt {\frac {e^2}{f^2}}+b^2 f^5 \sqrt {\frac {e^2}{f^2}}-b^2 e f^4\right ) \log \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d-f x \sqrt {\frac {e^2}{f^2}}\right )}{e \left (2 d e-b f^2\right )^3}+\frac {\left (4 a e^3 f^2+4 a e^2 f^3 \sqrt {\frac {e^2}{f^2}}-b^2 f^5 \sqrt {\frac {e^2}{f^2}}-b^2 e f^4\right ) \log \left (\left (b f^2-2 d e\right ) \sqrt {a+b x+\frac {e^2 x^2}{f^2}}-2 a e f+\sqrt {\frac {e^2}{f^2}} \left (2 d e x-b f^2 x\right )+b d f\right )}{e \left (2 d e-b f^2\right )^3}+\frac {2 \sqrt {a+b x+\frac {e^2 x^2}{f^2}} \left (-2 a e f^3+b d f^3-b e f^3 x+2 d e^2 f x\right )}{\left (b f^2-2 d e\right )^2 \left (a f^2+b f^2 x-d^2-2 d e x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 3.69, size = 826, normalized size = 3.11 \begin {gather*} -\frac {a b^{2} f^{6} + {\left (3 \, b^{2} d^{2} - 14 \, a b d e + 8 \, a^{2} e^{2}\right )} f^{4} - 2 \, {\left (b d^{3} e - 4 \, a d^{2} e^{2}\right )} f^{2} - 4 \, {\left (b^{2} e^{2} f^{4} - 4 \, b d e^{3} f^{2} + 4 \, d^{2} e^{4}\right )} x^{2} + {\left (b^{3} f^{6} - 8 \, b^{2} d e f^{4} + 20 \, b d^{2} e^{2} f^{2} - 16 \, d^{3} e^{3}\right )} x - 2 \, {\left (a b^{2} f^{6} + 4 \, a d^{2} e^{2} f^{2} - {\left (b^{2} d^{2} + 4 \, a^{2} e^{2}\right )} f^{4} + {\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} x\right )} \log \left (-4 \, a d e^{2} f^{2} - {\left (b^{2} d - 4 \, a b e\right )} f^{4} + 4 \, {\left (b e^{3} f^{2} - 2 \, d e^{4}\right )} x^{2} + {\left (3 \, b^{2} e f^{4} - 4 \, {\left (2 \, b d e^{2} - a e^{3}\right )} f^{2}\right )} x - {\left (b^{2} f^{5} - 4 \, {\left (b d e - a e^{2}\right )} f^{3} + 4 \, {\left (b e^{2} f^{3} - 2 \, d e^{3} f\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) - 2 \, {\left (a b^{2} f^{6} + 4 \, a d^{2} e^{2} f^{2} - {\left (b^{2} d^{2} + 4 \, a^{2} e^{2}\right )} f^{4} + {\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} x\right )} \log \left (a f^{2} - d^{2} + {\left (b f^{2} - 2 \, d e\right )} x\right ) + 2 \, {\left (a b^{2} f^{6} + 4 \, a d^{2} e^{2} f^{2} - {\left (b^{2} d^{2} + 4 \, a^{2} e^{2}\right )} f^{4} + {\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} x\right )} \log \left (-e x + f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) - 4 \, {\left ({\left (b^{2} d - 2 \, a b e\right )} f^{5} - 2 \, {\left (b d^{2} e - 2 \, a d e^{2}\right )} f^{3} - {\left (b^{2} e f^{5} - 4 \, b d e^{2} f^{3} + 4 \, d^{2} e^{3} f\right )} x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \, {\left (a b^{3} f^{8} + 8 \, d^{5} e^{3} - {\left (b^{3} d^{2} + 6 \, a b^{2} d e\right )} f^{6} + 6 \, {\left (b^{2} d^{3} e + 2 \, a b d^{2} e^{2}\right )} f^{4} - 4 \, {\left (3 \, b d^{4} e^{2} + 2 \, a d^{3} e^{3}\right )} f^{2} + {\left (b^{4} f^{8} - 8 \, b^{3} d e f^{6} + 24 \, b^{2} d^{2} e^{2} f^{4} - 32 \, b d^{3} e^{3} f^{2} + 16 \, d^{4} e^{4}\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 = 58067, normalized size = 218.30 \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 {b x + \frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{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 = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^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 + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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