Optimal. Leaf size=86 \[ -\frac{\left (a e^2+c d^2\right ) (B d-A e) \log (d+e x)}{e^4}+\frac{x \left (a B e^2-A c d e+B c d^2\right )}{e^3}-\frac{c x^2 (B d-A e)}{2 e^2}+\frac{B c x^3}{3 e} \]
[Out]
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Rubi [A] time = 0.170059, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{\left (a e^2+c d^2\right ) (B d-A e) \log (d+e x)}{e^4}+\frac{x \left (a B e^2-A c d e+B c d^2\right )}{e^3}-\frac{c x^2 (B d-A e)}{2 e^2}+\frac{B c x^3}{3 e} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2))/(d + e*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B c x^{3}}{3 e} + \frac{c \left (A e - B d\right ) \int x\, dx}{e^{2}} + \left (- A c d e + B a e^{2} + B c d^{2}\right ) \int \frac{1}{e^{3}}\, dx + \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.0711511, size = 80, normalized size = 0.93 \[ \frac{e x \left (6 a B e^2+3 A c e (e x-2 d)+B c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (a e^2+c d^2\right ) (B d-A e) \log (d+e x)}{6 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2))/(d + e*x),x]
[Out]
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Maple [A] time = 0.006, size = 116, normalized size = 1.4 \[{\frac{Bc{x}^{3}}{3\,e}}+{\frac{Ac{x}^{2}}{2\,e}}-{\frac{Bc{x}^{2}d}{2\,{e}^{2}}}-{\frac{Acdx}{{e}^{2}}}+{\frac{aBx}{e}}+{\frac{Bc{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) aA}{e}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) Ac}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) aBd}{{e}^{2}}}-{\frac{{d}^{3}\ln \left ( ex+d \right ) Bc}{{e}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)/(e*x+d),x)
[Out]
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Maxima [A] time = 0.699799, size = 131, normalized size = 1.52 \[ \frac{2 \, B c e^{2} x^{3} - 3 \,{\left (B c d e - A c e^{2}\right )} x^{2} + 6 \,{\left (B c d^{2} - A c d e + B a e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270825, size = 132, normalized size = 1.53 \[ \frac{2 \, B c e^{3} x^{3} - 3 \,{\left (B c d e^{2} - A c e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e - A c d e^{2} + B a e^{3}\right )} x - 6 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.98883, size = 80, normalized size = 0.93 \[ \frac{B c x^{3}}{3 e} - \frac{x^{2} \left (- A c e + B c d\right )}{2 e^{2}} + \frac{x \left (- A c d e + B a e^{2} + B c d^{2}\right )}{e^{3}} - \frac{\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.27855, size = 131, normalized size = 1.52 \[ -{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c x^{3} e^{2} - 3 \, B c d x^{2} e + 6 \, B c d^{2} x + 3 \, A c x^{2} e^{2} - 6 \, A c d x e + 6 \, B a x e^{2}\right )} e^{\left (-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d),x, algorithm="giac")
[Out]