Optimal. Leaf size=213 \[ \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {3 c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 \sqrt {g} (c d f-a e g)^{5/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {886, 888, 211}
\begin {gather*} \frac {3 c^2 d^2 \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 \sqrt {g} (c d f-a e g)^{5/2}}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt {d+e x} (f+g x) (c d f-a e g)^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 886
Rule 888
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {(3 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 (c d f-a e g)}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {\left (3 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 (c d f-a e g)^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {\left (3 c^2 d^2 e^2\right ) \text {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{4 (c d f-a e g)^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {3 c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 \sqrt {g} (c d f-a e g)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 163, normalized size = 0.77 \begin {gather*} \frac {\sqrt {d+e x} \left (\sqrt {g} \sqrt {c d f-a e g} (a e+c d x) (-2 a e g+c d (5 f+3 g x))+3 c^2 d^2 \sqrt {a e+c d x} (f+g x)^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 \sqrt {g} (c d f-a e g)^{5/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 275, normalized size = 1.29
method | result | size |
default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} g^{2} x^{2}+6 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f g x +3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2}-3 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d g x +2 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a e g -5 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d f \right )}{4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right )^{2} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\) | \(275\) |
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. 643 vs.
\(2 (197) = 394\).
time = 3.65, size = 1325, normalized size = 6.22 \begin {gather*} \left [-\frac {3 \, {\left (c^{2} d^{3} g^{2} x^{2} + 2 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + {\left (c^{2} d^{2} g^{2} x^{3} + 2 \, c^{2} d^{2} f g x^{2} + c^{2} d^{2} f^{2} x\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e - 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) - 2 \, {\left (3 \, c^{2} d^{2} f g^{2} x + 5 \, c^{2} d^{2} f^{2} g + 2 \, a^{2} g^{3} e^{2} - {\left (3 \, a c d g^{3} x + 7 \, a c d f g^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{8 \, {\left (c^{3} d^{4} f^{3} g^{3} x^{2} + 2 \, c^{3} d^{4} f^{4} g^{2} x + c^{3} d^{4} f^{5} g - {\left (a^{3} g^{6} x^{3} + 2 \, a^{3} f g^{5} x^{2} + a^{3} f^{2} g^{4} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{5} x^{3} - a^{3} d f^{2} g^{4} + {\left (6 \, a^{2} c d f^{2} g^{4} - a^{3} d g^{6}\right )} x^{2} + {\left (3 \, a^{2} c d f^{3} g^{3} - 2 \, a^{3} d f g^{5}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{4} x^{3} - a^{2} c d^{2} f^{3} g^{3} + {\left (2 \, a c^{2} d^{2} f^{3} g^{3} - a^{2} c d^{2} f g^{5}\right )} x^{2} + {\left (a c^{2} d^{2} f^{4} g^{2} - 2 \, a^{2} c d^{2} f^{2} g^{4}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{3} x^{3} - 3 \, a c^{2} d^{3} f^{4} g^{2} + {\left (2 \, c^{3} d^{3} f^{4} g^{2} - 3 \, a c^{2} d^{3} f^{2} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} f^{5} g - 6 \, a c^{2} d^{3} f^{3} g^{3}\right )} x\right )} e\right )}}, -\frac {3 \, {\left (c^{2} d^{3} g^{2} x^{2} + 2 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + {\left (c^{2} d^{2} g^{2} x^{3} + 2 \, c^{2} d^{2} f g x^{2} + c^{2} d^{2} f^{2} x\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) - {\left (3 \, c^{2} d^{2} f g^{2} x + 5 \, c^{2} d^{2} f^{2} g + 2 \, a^{2} g^{3} e^{2} - {\left (3 \, a c d g^{3} x + 7 \, a c d f g^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{4 \, {\left (c^{3} d^{4} f^{3} g^{3} x^{2} + 2 \, c^{3} d^{4} f^{4} g^{2} x + c^{3} d^{4} f^{5} g - {\left (a^{3} g^{6} x^{3} + 2 \, a^{3} f g^{5} x^{2} + a^{3} f^{2} g^{4} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{5} x^{3} - a^{3} d f^{2} g^{4} + {\left (6 \, a^{2} c d f^{2} g^{4} - a^{3} d g^{6}\right )} x^{2} + {\left (3 \, a^{2} c d f^{3} g^{3} - 2 \, a^{3} d f g^{5}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{4} x^{3} - a^{2} c d^{2} f^{3} g^{3} + {\left (2 \, a c^{2} d^{2} f^{3} g^{3} - a^{2} c d^{2} f g^{5}\right )} x^{2} + {\left (a c^{2} d^{2} f^{4} g^{2} - 2 \, a^{2} c d^{2} f^{2} g^{4}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{3} x^{3} - 3 \, a c^{2} d^{3} f^{4} g^{2} + {\left (2 \, c^{3} d^{3} f^{4} g^{2} - 3 \, a c^{2} d^{3} f^{2} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} f^{5} g - 6 \, a c^{2} d^{3} f^{3} g^{3}\right )} x\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F(-1)] Timed out
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{{\left (f+g\,x\right )}^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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