Optimal. Leaf size=170 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{d e^3 (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2} \left (5 C d^2-2 e (2 B d-A e)\right )}{2 d e^3}-\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (5 C d^2-2 e (2 B d-A e)\right )}{2 e^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)} \]
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Rubi [A] time = 0.20, antiderivative size = 170, 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 = {1639, 793, 665, 217, 203} \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2} \left (A e^2-B d e+C d^2\right )}{d e^3 (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2} \left (5 C d^2-2 e (2 B d-A e)\right )}{2 d e^3}-\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (5 C d^2-2 e (2 B d-A e)\right )}{2 e^3}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 665
Rule 793
Rule 1639
Rubi steps
\begin {align*} \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx &=-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}-\frac {\int \frac {\left (e^2 \left (C d^2-2 A e^2\right )+e^3 (3 C d-2 B e) x\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx}{2 e^4}\\ &=-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{d e^3 (d+e x)^2}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}-\frac {\left (-3 e^5 \left (C d^2-2 A e^2\right )-2 \left (-d e^5 (3 C d-2 B e)-e^5 \left (C d^2-2 A e^2\right )\right )\right ) \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx}{2 d e^7}\\ &=-\frac {\left (5 C d^2-2 e (2 B d-A e)\right ) \sqrt {d^2-e^2 x^2}}{2 d e^3}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{d e^3 (d+e x)^2}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}-\frac {\left (-3 e^5 \left (C d^2-2 A e^2\right )-2 \left (-d e^5 (3 C d-2 B e)-e^5 \left (C d^2-2 A e^2\right )\right )\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=-\frac {\left (5 C d^2-2 e (2 B d-A e)\right ) \sqrt {d^2-e^2 x^2}}{2 d e^3}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{d e^3 (d+e x)^2}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}-\frac {\left (-3 e^5 \left (C d^2-2 A e^2\right )-2 \left (-d e^5 (3 C d-2 B e)-e^5 \left (C d^2-2 A e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^7}\\ &=-\frac {\left (5 C d^2-2 e (2 B d-A e)\right ) \sqrt {d^2-e^2 x^2}}{2 d e^3}-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{d e^3 (d+e x)^2}-\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{2 e^3 (d+e x)}-\frac {\left (5 C d^2-2 e (2 B d-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 = 0.23, size = 109, normalized size = 0.64 \[ \frac {\frac {\sqrt {d^2-e^2 x^2} \left (2 e (-2 A e+3 B d+B e x)+C \left (-8 d^2-3 d e x+e^2 x^2\right )\right )}{d+e x}-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (2 e (A e-2 B d)+5 C d^2\right )}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 190, normalized size = 1.12 \[ -\frac {8 \, C d^{3} - 6 \, B d^{2} e + 4 \, A d e^{2} + 2 \, {\left (4 \, C d^{2} e - 3 \, B d e^{2} + 2 \, A e^{3}\right )} x - 2 \, {\left (5 \, C d^{3} - 4 \, B d^{2} e + 2 \, A d e^{2} + {\left (5 \, C d^{2} e - 4 \, B d e^{2} + 2 \, A e^{3}\right )} x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (C e^{2} x^{2} - 8 \, C d^{2} + 6 \, B d e - 4 \, A e^{2} - {\left (3 \, C d e - 2 \, B e^{2}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, {\left (e^{4} x + d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 439, normalized size = 2.58 \[ -\frac {A \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}}+\frac {2 B d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e}-\frac {3 C \,d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e^{2}}+\frac {C \,d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{2}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, C x}{2 e^{2}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, A}{d e}+\frac {2 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, B}{e^{2}}-\frac {3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, C d}{e^{3}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} A}{\left (x +\frac {d}{e}\right )^{2} d \,e^{3}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} B}{\left (x +\frac {d}{e}\right )^{2} e^{4}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} C d}{\left (x +\frac {d}{e}\right )^{2} e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 197, normalized size = 1.16 \[ -\frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d^{2}}{e^{4} x + d e^{3}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} B d}{e^{3} x + d e^{2}} - \frac {5 \, C d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{3}} + \frac {2 \, B d \arcsin \left (\frac {e x}{d}\right )}{e^{2}} - \frac {A \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} A}{e^{2} x + d e} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} C x}{2 \, e^{2}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d}{e^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} B}{e^{2}} \]
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 \[ \int \frac {\sqrt {d^2-e^2\,x^2}\,\left (C\,x^2+B\,x+A\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (A + B x + C x^{2}\right )}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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