Optimal. Leaf size=148 \[ \frac {\left (C d^2-e (B d-2 A e)\right ) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac {d \left (C d^2-e (B d-2 A e)\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 148, 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, 809, 679,
223, 209} \begin {gather*} \frac {d \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (C d^2-e (B d-2 A e)\right )}{2 e^3}+\frac {\sqrt {d^2-e^2 x^2} \left (C d^2-e (B d-2 A e)\right )}{2 e^3}+\frac {\left (d^2-e^2 x^2\right )^{3/2} (C d-B e)}{2 e^3 (d+e x)}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 679
Rule 809
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} \, dx &=-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}-\frac {\int \frac {\left (-3 A e^4+3 e^3 (C d-B e) x\right ) \sqrt {d^2-e^2 x^2}}{d+e x} \, dx}{3 e^4}\\ &=-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac {\left (C d^2-e (B d-2 A e)\right ) \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx}{2 e^2}\\ &=\frac {\left (C d^2-e (B d-2 A e)\right ) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac {\left (d \left (C d^2-e (B d-2 A e)\right )\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=\frac {\left (C d^2-e (B d-2 A e)\right ) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac {\left (d \left (C d^2-e (B d-2 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2}\\ &=\frac {\left (C d^2-e (B d-2 A e)\right ) \sqrt {d^2-e^2 x^2}}{2 e^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac {(C d-B e) \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}+\frac {d \left (C d^2-e (B d-2 A e)\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 123, normalized size = 0.83 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (3 e (-2 B d+2 A e+B e x)+C \left (4 d^2-3 d e x+2 e^2 x^2\right )\right )+3 \sqrt {-e^2} \left (C d^3+d e (-B d+2 A e)\right ) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{6 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 227, normalized size = 1.53
method | result | size |
risch | \(\frac {\left (2 C \,e^{2} x^{2}+3 B \,e^{2} x -3 C d e x +6 A \,e^{2}-6 B d e +4 C \,d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e^{3}}+\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) A}{\sqrt {e^{2}}}-\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) B}{2 e \sqrt {e^{2}}}+\frac {d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) C}{2 e^{2} \sqrt {e^{2}}}\) | \(163\) |
default | \(\frac {-\frac {C \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e}+B e \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )-C d \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{e^{2}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{3}}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 159, normalized size = 1.07 \begin {gather*} \frac {1}{2} \, C d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} - \frac {1}{2} \, B d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} C d x e^{\left (-2\right )} + \sqrt {-x^{2} e^{2} + d^{2}} C d^{2} e^{\left (-3\right )} + A d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} B x e^{\left (-1\right )} - \sqrt {-x^{2} e^{2} + d^{2}} B d e^{\left (-2\right )} - \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} C e^{\left (-3\right )} + \sqrt {-x^{2} e^{2} + d^{2}} A e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 104, normalized size = 0.70 \begin {gather*} -\frac {1}{6} \, {\left (6 \, {\left (C d^{3} - B d^{2} e + 2 \, A d e^{2}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (4 \, C d^{2} + {\left (2 \, C x^{2} + 3 \, B x + 6 \, A\right )} e^{2} - 3 \, {\left (C d x + 2 \, B d\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-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 )}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.29, size = 99, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, {\left (C d^{3} - B d^{2} e + 2 \, A d e^{2}\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left (2 \, C x e^{\left (-1\right )} - 3 \, {\left (C d e^{3} - B e^{4}\right )} e^{\left (-5\right )}\right )} x + 2 \, {\left (2 \, C d^{2} e^{2} - 3 \, B d e^{3} + 3 \, A e^{4}\right )} e^{\left (-5\right )}\right )} \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 )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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