Optimal. Leaf size=185 \[ -\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (d+e x)}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{2 e^6 (d+e x)^2}+\frac{c x \left (2 a B e^2-3 A c d e+6 B c d^2\right )}{e^5}-\frac{2 c \log (d+e x) \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6}-\frac{c^2 x^2 (3 B d-A e)}{2 e^4}+\frac{B c^2 x^3}{3 e^3} \]
[Out]
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Rubi [A] time = 0.446698, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (d+e x)}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{2 e^6 (d+e x)^2}+\frac{c x \left (2 a B e^2-3 A c d e+6 B c d^2\right )}{e^5}-\frac{2 c \log (d+e x) \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6}-\frac{c^2 x^2 (3 B d-A e)}{2 e^4}+\frac{B c^2 x^3}{3 e^3} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B c^{2} x^{3}}{3 e^{3}} + \frac{c^{2} \left (A e - 3 B d\right ) \int x\, dx}{e^{4}} + \frac{2 c \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right ) \log{\left (d + e x \right )}}{e^{6}} + \frac{\left (- 3 A c d e + 2 B a e^{2} + 6 B c d^{2}\right ) \int c\, dx}{e^{5}} - \frac{\left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \left (d + e x\right )} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{2 e^{6} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.339537, size = 174, normalized size = 0.94 \[ \frac{6 c e x \left (2 a B e^2-3 A c d e+6 B c d^2\right )-\frac{6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{d+e x}+\frac{3 \left (a e^2+c d^2\right )^2 (B d-A e)}{(d+e x)^2}+12 c \log (d+e x) \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )+3 c^2 e^2 x^2 (A e-3 B d)+2 B c^2 e^3 x^3}{6 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^3,x]
[Out]
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Maple [A] time = 0.014, size = 331, normalized size = 1.8 \[{\frac{B{c}^{2}{x}^{3}}{3\,{e}^{3}}}+{\frac{A{c}^{2}{x}^{2}}{2\,{e}^{3}}}-{\frac{3\,B{c}^{2}{x}^{2}d}{2\,{e}^{4}}}-3\,{\frac{A{c}^{2}dx}{{e}^{4}}}+2\,{\frac{aBcx}{{e}^{3}}}+6\,{\frac{B{c}^{2}{d}^{2}x}{{e}^{5}}}-{\frac{A{a}^{2}}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{A{d}^{2}ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{A{d}^{4}{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{Bd{a}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{aBc{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+2\,{\frac{c\ln \left ( ex+d \right ) aA}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{2}}{{e}^{5}}}-6\,{\frac{c\ln \left ( ex+d \right ) aBd}{{e}^{4}}}-10\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{3}}{{e}^{6}}}+4\,{\frac{Adac}{{e}^{3} \left ( ex+d \right ) }}+4\,{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}B}{{e}^{2} \left ( ex+d \right ) }}-6\,{\frac{aBc{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}-5\,{\frac{B{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^2/(e*x+d)^3,x)
[Out]
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Maxima [A] time = 0.711168, size = 348, normalized size = 1.88 \[ -\frac{9 \, B c^{2} d^{5} - 7 \, A c^{2} d^{4} e + 10 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + A a^{2} e^{5} + 2 \,{\left (5 \, B c^{2} d^{4} e - 4 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} + B a^{2} e^{5}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B c^{2} e^{2} x^{3} - 3 \,{\left (3 \, B c^{2} d e - A c^{2} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B c^{2} d^{2} - 3 \, A c^{2} d e + 2 \, B a c e^{2}\right )} x}{6 \, e^{5}} - \frac{2 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269868, size = 532, normalized size = 2.88 \[ \frac{2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} + 21 \, A c^{2} d^{4} e - 30 \, B a c d^{3} e^{2} + 18 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} -{\left (5 \, B c^{2} d e^{4} - 3 \, A c^{2} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B c^{2} d^{2} e^{3} - 3 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 3 \,{\left (21 \, B c^{2} d^{3} e^{2} - 11 \, A c^{2} d^{2} e^{3} + 8 \, B a c d e^{4}\right )} x^{2} + 6 \,{\left (B c^{2} d^{4} e + A c^{2} d^{3} e^{2} - 4 \, B a c d^{2} e^{3} + 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} x - 12 \,{\left (5 \, B c^{2} d^{5} - 3 \, A c^{2} d^{4} e + 3 \, B a c d^{3} e^{2} - A a c d^{2} e^{3} +{\left (5 \, B c^{2} d^{3} e^{2} - 3 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 2 \,{\left (5 \, B c^{2} d^{4} e - 3 \, A c^{2} d^{3} e^{2} + 3 \, B a c d^{2} e^{3} - A a c d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.62, size = 277, normalized size = 1.5 \[ \frac{B c^{2} x^{3}}{3 e^{3}} - \frac{2 c \left (- A a e^{3} - 3 A c d^{2} e + 3 B a d e^{2} + 5 B c d^{3}\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{A a^{2} e^{5} - 6 A a c d^{2} e^{3} - 7 A c^{2} d^{4} e + B a^{2} d e^{4} + 10 B a c d^{3} e^{2} + 9 B c^{2} d^{5} + x \left (- 8 A a c d e^{4} - 8 A c^{2} d^{3} e^{2} + 2 B a^{2} e^{5} + 12 B a c d^{2} e^{3} + 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} - \frac{x^{2} \left (- A c^{2} e + 3 B c^{2} d\right )}{2 e^{4}} + \frac{x \left (- 3 A c^{2} d e + 2 B a c e^{2} + 6 B c^{2} d^{2}\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.278665, size = 320, normalized size = 1.73 \[ -2 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 3 \, A c^{2} x^{2} e^{6} - 18 \, A c^{2} d x e^{5} + 12 \, B a c x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B c^{2} d^{5} - 7 \, A c^{2} d^{4} e + 10 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + A a^{2} e^{5} + 2 \,{\left (5 \, B c^{2} d^{4} e - 4 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} + B a^{2} e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^3,x, algorithm="giac")
[Out]