Optimal. Leaf size=330 \[ -\frac {2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}+\frac {6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^4}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac {6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^4}-\frac {a e f^2-b d f^2+d^2 e}{\left (2 d e-b f^2\right )^2 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^2} \]
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Rubi [A] time = 0.29, antiderivative size = 330, 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} \[ -\frac {2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}+\frac {6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^4}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac {6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^4}-\frac {a e f^2-b d f^2+d^2 e}{\left (2 d e-b f^2\right )^2 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^2} \]
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 )^3} \, 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^3 \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^3}+\frac {4 a e^2 f^2-b^2 f^4}{\left (2 d e-b f^2\right )^3 x^2}+\frac {3 \left (4 a e^3 f^2-b^2 e f^4\right )}{\left (2 d e-b f^2\right )^4 x}+\frac {2 \left (4 a e^4 f^2-b^2 e^2 f^4\right )}{\left (2 d e-b f^2\right )^3 \left (2 d e-b f^2-2 e x\right )^2}+\frac {6 \left (4 a e^4 f^2-b^2 e^2 f^4\right )}{\left (2 d e-b f^2\right )^4 \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 {d^2 e-b d f^2+a e f^2}{\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 )^2}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \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 {6 e 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 )^4}-\frac {6 e 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 )^4}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 300, normalized size = 0.91 \[ -\frac {\frac {2 f^2 \left (b^2 f^2-4 a e^2\right ) \left (b f^2-2 d e\right )}{f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x}+\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )}{2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2}-6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )+6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (-2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )-b f^2\right )+\frac {\left (b f^2-2 d e\right )^2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+d+e x\right )^2}}{\left (b f^2-2 d e\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 19.38, size = 1954, normalized size = 5.92 \[ \text {result too large to display} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 295147, normalized size = 894.38 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (e x + \sqrt {b x + \frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^3} \,d x \]
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 \[ \int \frac {1}{\left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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