Optimal. Leaf size=149 \[ \frac {2 (3 C d-B e) \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2}+\frac {(3 C d-B e) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1653, 807, 677,
223, 209} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{3 d e^3 (d+e x)^3}+\frac {\text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) (3 C d-B e)}{e^3}+\frac {2 \sqrt {d^2-e^2 x^2} (3 C d-B e)}{e^3 (d+e x)}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 677
Rule 807
Rule 1653
Rubi steps
\begin {align*} \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^3} \, dx &=-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2}-\frac {\int \frac {\left (e^2 \left (2 C d^2-A e^2\right )+e^3 (3 C d-B e) x\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^3} \, dx}{e^4}\\ &=-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2}-\frac {(3 C d-B e) \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx}{e^2}\\ &=\frac {2 (3 C d-B e) \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2}+\frac {(3 C d-B e) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}\\ &=\frac {2 (3 C d-B e) \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2}+\frac {(3 C d-B e) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ &=\frac {2 (3 C d-B e) \sqrt {d^2-e^2 x^2}}{e^3 (d+e x)}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{3 d e^3 (d+e x)^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^2}+\frac {(3 C d-B e) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.63, size = 133, normalized size = 0.89 \begin {gather*} \frac {\frac {e \sqrt {d^2-e^2 x^2} \left (C d \left (14 d^2+19 d e x+3 e^2 x^2\right )+e (A e (-d+e x)-B d (5 d+7 e x))\right )}{d (d+e x)^2}-3 \sqrt {-e^2} (-3 C d+B e) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{3 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 278, normalized size = 1.87
method | result | size |
default | \(\frac {C \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{3}}-\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{6} d \left (x +\frac {d}{e}\right )^{3}}+\frac {\left (B e -2 C d \right ) \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{e^{4}}\) | \(278\) |
risch | \(\frac {C \sqrt {-e^{2} x^{2}+d^{2}}}{e^{3}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) B}{e \sqrt {e^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) C d}{e^{2} \sqrt {e^{2}}}-\frac {2 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}\, A}{3 e^{3} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}\, B}{3 e^{4} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}\, C}{3 e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}\, A}{3 e^{2} d \left (x +\frac {d}{e}\right )}-\frac {7 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}\, B}{3 e^{3} \left (x +\frac {d}{e}\right )}+\frac {13 d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}\, C}{3 e^{4} \left (x +\frac {d}{e}\right )}\) | \(354\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 0.36, size = 244, normalized size = 1.64 \begin {gather*} \frac {14 \, C d^{4} - A x^{2} e^{4} - 6 \, {\left (3 \, C d^{4} - B d x^{2} e^{3} + {\left (3 \, C d^{2} x^{2} - 2 \, B d^{2} x\right )} e^{2} + {\left (6 \, C d^{3} x - B d^{3}\right )} e\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (5 \, B d x^{2} + 2 \, A d x\right )} e^{3} + {\left (14 \, C d^{2} x^{2} - 10 \, B d^{2} x - A d^{2}\right )} e^{2} + {\left (28 \, C d^{3} x - 5 \, B d^{3}\right )} e + {\left (14 \, C d^{3} + A x e^{3} + {\left (3 \, C d x^{2} - 7 \, B d x - A d\right )} e^{2} + {\left (19 \, C d^{2} x - 5 \, B d^{2}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d x^{2} e^{5} + 2 \, d^{2} x e^{4} + d^{3} e^{3}\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 {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (A + B x + C x^{2}\right )}{\left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.48, size = 256, normalized size = 1.72 \begin {gather*} {\left (3 \, C d - B e\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \sqrt {-x^{2} e^{2} + d^{2}} C e^{\left (-3\right )} - \frac {2 \, {\left (\frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} C d^{2} e^{\left (-2\right )}}{x} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} C d^{2} e^{\left (-4\right )}}{x^{2}} + 11 \, C d^{2} - 5 \, B d e - \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} B d e^{\left (-1\right )}}{x} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} B d e^{\left (-3\right )}}{x^{2}} - A e^{2} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} A e^{\left (-2\right )}}{x^{2}}\right )} e^{\left (-3\right )}}{3 \, d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\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 = -0.01 \begin {gather*} \int \frac {\sqrt {d^2-e^2\,x^2}\,\left (C\,x^2+B\,x+A\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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