Optimal. Leaf size=266 \[ -\frac {2 \left (d^2 e-b d f^2+a e f^2\right )}{\left (2 d e-b f^2\right )^2 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{\left (2 d e-b f^2\right )^3}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{\left (2 d e-b f^2\right )^3} \]
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Rubi [A]
time = 0.16, 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 = {2141, 907}
\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 907
Rule 2141
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 \text {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 \text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(551\) vs. \(2(266)=532\).
time = 3.77, size = 551, normalized size = 2.07 \begin {gather*} \frac {2 \left (b f^3 (-d+e x)+2 e f \left (a f^2-d e x\right )\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}{\left (-2 d e+b f^2\right )^2 \left (d^2+2 d e x-f^2 (a+b x)\right )}-\frac {2 \left (4 a^2 e^3 f^2 x+b d x \left (2 d^2 e^2-2 b d e f^2+b^2 f^4+2 d e^3 x-b e^2 f^2 x\right )+a \left (4 d^3 e^2+2 b e^3 f^2 x^2+d^2 \left (-4 b e f^2+4 e^3 x\right )+d \left (b^2 f^4-6 b e^2 f^2 x-4 e^4 x^2\right )\right )\right )}{(b d-2 a e) \left (-2 d e+b f^2\right )^2 \left (-d^2-2 d e x+f^2 (a+b x)\right )}+\frac {f^2 \left (e+\sqrt {\frac {e^2}{f^2}} f\right ) \left (4 a e^2-b^2 f^2\right ) \log \left (b d f-2 a e f+\sqrt {\frac {e^2}{f^2}} \left (2 d e-b f^2\right ) x+\left (-2 d e+b f^2\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}{e \left (2 d e-b f^2\right )^3}-\frac {f^2 \left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \left (4 a e^2-b^2 f^2\right ) \log \left (b f+2 e \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{e \left (2 d e-b f^2\right )^3}+\frac {f^2 \left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \left (4 a e^2-b^2 f^2\right ) \log \left (d+f \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{e \left (2 d e-b f^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58068\) vs.
\(2(258)=516\).
time = 0.10, size = 58069, normalized size = 218.30
method | result | size |
default | \(\text {Expression too large to display}\) | \(58069\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 803 vs.
\(2 (256) = 512\).
time = 3.04, size = 803, normalized size = 3.02 \begin {gather*} -\frac {b^{3} f^{6} x + a b^{2} f^{6} + 3 \, b^{2} d^{2} f^{4} - 16 \, d^{2} x^{2} e^{4} + 16 \, {\left (b d f^{2} x^{2} - d^{3} x\right )} e^{3} - 4 \, {\left (b^{2} f^{4} x^{2} - 2 \, a^{2} f^{4} - 5 \, b d^{2} f^{2} x - 2 \, a d^{2} f^{2}\right )} e^{2} - 2 \, {\left (4 \, b^{2} d f^{4} x + 7 \, a b d f^{4} + b d^{3} f^{2}\right )} e - 2 \, {\left (b^{3} f^{6} x + a b^{2} f^{6} - 2 \, b^{2} d f^{4} x e - b^{2} d^{2} f^{4} + 8 \, a d f^{2} x e^{3} - 4 \, {\left (a b f^{4} x + a^{2} f^{4} - a d^{2} f^{2}\right )} e^{2}\right )} \log \left (b^{2} d f^{4} + 8 \, d x^{2} e^{4} - 4 \, {\left (b f^{2} x^{2} + a f^{2} x\right )} e^{3} + 4 \, {\left (2 \, b d f^{2} x + a d f^{2}\right )} e^{2} - {\left (3 \, b^{2} f^{4} x + 4 \, a b f^{4}\right )} e + {\left (b^{2} f^{5} - 4 \, b d f^{3} e - 8 \, d f x e^{3} + 4 \, {\left (b f^{3} x + a f^{3}\right )} e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) - 2 \, {\left (b^{3} f^{6} x + a b^{2} f^{6} - 2 \, b^{2} d f^{4} x e - b^{2} d^{2} f^{4} + 8 \, a d f^{2} x e^{3} - 4 \, {\left (a b f^{4} x + a^{2} f^{4} - a d^{2} f^{2}\right )} e^{2}\right )} \log \left (-b f^{2} x - a f^{2} + 2 \, d x e + d^{2}\right ) + 2 \, {\left (b^{3} f^{6} x + a b^{2} f^{6} - 2 \, b^{2} d f^{4} x e - b^{2} d^{2} f^{4} + 8 \, a d f^{2} x e^{3} - 4 \, {\left (a b f^{4} x + a^{2} f^{4} - a d^{2} f^{2}\right )} e^{2}\right )} \log \left (-x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} - d\right ) - 4 \, {\left (b^{2} d f^{5} - 4 \, d^{2} f x e^{3} + 4 \, {\left (b d f^{3} x + a d f^{3}\right )} e^{2} - {\left (b^{2} f^{5} x + 2 \, a b f^{5} + 2 \, b d^{2} f^{3}\right )} e\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}}{2 \, {\left (b^{4} f^{8} x + a b^{3} f^{8} - b^{3} d^{2} f^{6} + 16 \, d^{4} x e^{4} - 8 \, {\left (4 \, b d^{3} f^{2} x + a d^{3} f^{2} - d^{5}\right )} e^{3} + 12 \, {\left (2 \, b^{2} d^{2} f^{4} x + a b d^{2} f^{4} - b d^{4} f^{2}\right )} e^{2} - 2 \, {\left (4 \, b^{3} d f^{6} x + 3 \, a b^{2} d f^{6} - 3 \, b^{2} d^{3} f^{4}\right )} e\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 + 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1243 vs.
\(2 (256) = 512\).
time = 4.27, size = 1243, normalized size = 4.67 \begin {gather*} \frac {2 \, x e^{2}}{b^{2} f^{4} - 4 \, b d f^{2} e + 4 \, d^{2} e^{2}} + \frac {{\left (b^{3} f^{5} {\left | f \right |} - 2 \, b^{2} d f^{3} {\left | f \right |} e - 4 \, a b f^{3} {\left | f \right |} e^{2} + 8 \, a d f {\left | f \right |} e^{3}\right )} \log \left ({\left | {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b f^{2} - b d f^{2} + 2 \, a f^{2} e - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d e \right |}\right )}{b^{4} f^{8} - 8 \, b^{3} d f^{6} e + 24 \, b^{2} d^{2} f^{4} e^{2} - 32 \, b d^{3} f^{2} e^{3} + 16 \, d^{4} e^{4}} + \frac {{\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \log \left ({\left | b f^{2} x + a f^{2} - 2 \, d x e - d^{2} \right |}\right )}{b^{3} f^{6} - 6 \, b^{2} d f^{4} e + 12 \, b d^{2} f^{2} e^{2} - 8 \, d^{3} e^{3}} + \frac {{\left (b^{2} f^{3} {\left | f \right |} e - 4 \, a f {\left | f \right |} e^{3}\right )} \log \left ({\left | b f^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right )}{b^{3} f^{6} e - 6 \, b^{2} d f^{4} e^{2} + 12 \, b d^{2} f^{2} e^{3} - 8 \, d^{3} e^{4}} - \frac {{\left (b^{2} f^{3} {\left | f \right |} - 4 \, a f {\left | f \right |} e^{2}\right )} \log \left ({\left | -x e - d + \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} \right |}\right )}{b^{3} f^{6} - 6 \, b^{2} d f^{4} e + 12 \, b d^{2} f^{2} e^{2} - 8 \, d^{3} e^{3}} - \frac {2 \, {\left (b^{3} f^{6} {\left | f \right |} e - 6 \, b^{2} d f^{4} {\left | f \right |} e^{2} + 12 \, b d^{2} f^{2} {\left | f \right |} e^{3} - 8 \, d^{3} {\left | f \right |} e^{4}\right )} \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}}{b^{5} f^{11} - 10 \, b^{4} d f^{9} e + 40 \, b^{3} d^{2} f^{7} e^{2} - 80 \, b^{2} d^{3} f^{5} e^{3} + 80 \, b d^{4} f^{3} e^{4} - 32 \, d^{5} f e^{5}} - \frac {2 \, {\left ({\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b^{3} d f^{6} {\left | f \right |} - {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} a b^{2} f^{6} {\left | f \right |} e + a b^{2} d f^{6} {\left | f \right |} e - a^{2} b f^{6} {\left | f \right |} e^{2} - 3 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b^{2} d^{2} f^{4} {\left | f \right |} e + b^{2} d^{3} f^{4} {\left | f \right |} e - 6 \, a b d^{2} f^{4} {\left | f \right |} e^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} a^{2} f^{4} {\left | f \right |} e^{3} + 4 \, a^{2} d f^{4} {\left | f \right |} e^{3} + 4 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b d^{3} f^{2} {\left | f \right |} e^{2} - b d^{4} f^{2} {\left | f \right |} e^{2} + 4 \, a d^{3} f^{2} {\left | f \right |} e^{3} - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d^{4} {\left | f \right |} e^{3}\right )}}{{\left (b^{3} f^{7} - 6 \, b^{2} d f^{5} e + 12 \, b d^{2} f^{3} e^{2} - 8 \, d^{3} f e^{3}\right )} {\left ({\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )}^{2} b f^{2} - b d^{2} f^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} a f^{2} e + 2 \, a d f^{2} e - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )}^{2} d e - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d^{2} e\right )}} - \frac {2 \, {\left (b^{2} d^{2} f^{4} - 2 \, a b d f^{4} e + a^{2} f^{4} e^{2} - 2 \, b d^{3} f^{2} e + 2 \, a d^{2} f^{2} e^{2} + d^{4} e^{2}\right )}}{{\left (b f^{2} x + a f^{2} - 2 \, d x e - d^{2}\right )} {\left (b f^{2} - 2 \, d e\right )}^{3}} \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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