Optimal. Leaf size=191 \[ \frac{e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac{2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac{2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac{B e^4 (a+b x)^3}{3 b^6} \]
[Out]
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Rubi [A] time = 0.550536, antiderivative size = 191, 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 \[ \frac{e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac{2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac{2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac{B e^4 (a+b x)^3}{3 b^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^4)/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 71.0256, size = 192, normalized size = 1.01 \[ \frac{B e^{4} \left (a + b x\right )^{3}}{3 b^{6}} - \frac{2 e^{2} x \left (a e - b d\right ) \left (2 A b e - 5 B a e + 3 B b d\right )}{b^{5}} + \frac{e^{3} \left (a + b x\right )^{2} \left (A b e - 5 B a e + 4 B b d\right )}{2 b^{6}} + \frac{2 e \left (a e - b d\right )^{2} \left (3 A b e - 5 B a e + 2 B b d\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{\left (a e - b d\right )^{3} \left (4 A b e - 5 B a e + B b d\right )}{b^{6} \left (a + b x\right )} - \frac{\left (A b - B a\right ) \left (a e - b d\right )^{4}}{2 b^{6} \left (a + b x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**4/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.190828, size = 187, normalized size = 0.98 \[ \frac{6 b e^2 x \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+3 b^2 e^3 x^2 (-3 a B e+A b e+4 b B d)-\frac{6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{a+b x}-\frac{3 (A b-a B) (b d-a e)^4}{(a+b x)^2}+12 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)+2 b^3 B e^4 x^3}{6 b^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x)^3,x]
[Out]
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Maple [B] time = 0.018, size = 601, normalized size = 3.2 \[ 6\,{\frac{{e}^{2}B{d}^{2}x}{{b}^{3}}}+6\,{\frac{{e}^{4}\ln \left ( bx+a \right ) A{a}^{2}}{{b}^{5}}}+6\,{\frac{{e}^{2}\ln \left ( bx+a \right ) A{d}^{2}}{{b}^{3}}}-10\,{\frac{{e}^{4}\ln \left ( bx+a \right ) B{a}^{3}}{{b}^{6}}}+4\,{\frac{e\ln \left ( bx+a \right ) B{d}^{3}}{{b}^{3}}}+4\,{\frac{{a}^{3}A{e}^{4}}{ \left ( bx+a \right ){b}^{5}}}-4\,{\frac{A{d}^{3}e}{ \left ( bx+a \right ){b}^{2}}}-5\,{\frac{B{a}^{4}{e}^{4}}{ \left ( bx+a \right ){b}^{6}}}-{\frac{A{a}^{4}{e}^{4}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{B{a}^{5}{e}^{4}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+{\frac{Ba{d}^{4}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{B{d}^{4}}{ \left ( bx+a \right ){b}^{2}}}-{\frac{A{d}^{4}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{B{e}^{4}{x}^{3}}{3\,{b}^{3}}}+{\frac{{e}^{4}A{x}^{2}}{2\,{b}^{3}}}-12\,{\frac{{e}^{3}\ln \left ( bx+a \right ) Aad}{{b}^{4}}}+24\,{\frac{{e}^{3}\ln \left ( bx+a \right ) B{a}^{2}d}{{b}^{5}}}-18\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ba{d}^{2}}{{b}^{4}}}-12\,{\frac{{a}^{2}Ad{e}^{3}}{ \left ( bx+a \right ){b}^{4}}}+12\,{\frac{Aa{d}^{2}{e}^{2}}{ \left ( bx+a \right ){b}^{3}}}+16\,{\frac{B{a}^{3}d{e}^{3}}{ \left ( bx+a \right ){b}^{5}}}-18\,{\frac{B{a}^{2}{d}^{2}{e}^{2}}{ \left ( bx+a \right ){b}^{4}}}+8\,{\frac{Ba{d}^{3}e}{ \left ( bx+a \right ){b}^{3}}}+2\,{\frac{{a}^{3}Ad{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}-3\,{\frac{{a}^{2}A{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}+2\,{\frac{Aa{d}^{3}e}{{b}^{2} \left ( bx+a \right ) ^{2}}}-2\,{\frac{B{a}^{4}d{e}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}+3\,{\frac{B{a}^{3}{d}^{2}{e}^{2}}{{b}^{4} \left ( bx+a \right ) ^{2}}}-2\,{\frac{B{a}^{2}{d}^{3}e}{{b}^{3} \left ( bx+a \right ) ^{2}}}-12\,{\frac{{e}^{3}Badx}{{b}^{4}}}+4\,{\frac{{e}^{3}Adx}{{b}^{3}}}+6\,{\frac{B{a}^{2}{e}^{4}x}{{b}^{5}}}-{\frac{3\,B{e}^{4}{x}^{2}a}{2\,{b}^{4}}}+2\,{\frac{{e}^{3}B{x}^{2}d}{{b}^{3}}}-3\,{\frac{{e}^{4}Aax}{{b}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^4/(b*x+a)^3,x)
[Out]
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Maxima [A] time = 1.39499, size = 572, normalized size = 2.99 \[ -\frac{{\left (B a b^{4} + A b^{5}\right )} d^{4} - 4 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} +{\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + 2 \,{\left (B b^{5} d^{4} - 4 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, B b^{2} e^{4} x^{3} + 3 \,{\left (4 \, B b^{2} d e^{3} -{\left (3 \, B a b - A b^{2}\right )} e^{4}\right )} x^{2} + 6 \,{\left (6 \, B b^{2} d^{2} e^{2} - 4 \,{\left (3 \, B a b - A b^{2}\right )} d e^{3} + 3 \,{\left (2 \, B a^{2} - A a b\right )} e^{4}\right )} x}{6 \, b^{5}} + \frac{2 \,{\left (2 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{3} -{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^4/(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216772, size = 902, normalized size = 4.72 \[ \frac{2 \, B b^{5} e^{4} x^{5} - 3 \,{\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \,{\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \,{\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} +{\left (12 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \,{\left (9 \, B b^{5} d^{2} e^{2} - 6 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} +{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \,{\left (24 \, B a b^{4} d^{2} e^{2} - 4 \,{\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} +{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \,{\left (B b^{5} d^{4} - 4 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} -{\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (2 \, B a^{2} b^{3} d^{3} e - 3 \,{\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} -{\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} +{\left (2 \, B b^{5} d^{3} e - 3 \,{\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} -{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \,{\left (2 \, B a b^{4} d^{3} e - 3 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} -{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^4/(b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 31.9834, size = 435, normalized size = 2.28 \[ \frac{B e^{4} x^{3}}{3 b^{3}} - \frac{- 7 A a^{4} b e^{4} + 20 A a^{3} b^{2} d e^{3} - 18 A a^{2} b^{3} d^{2} e^{2} + 4 A a b^{4} d^{3} e + A b^{5} d^{4} + 9 B a^{5} e^{4} - 28 B a^{4} b d e^{3} + 30 B a^{3} b^{2} d^{2} e^{2} - 12 B a^{2} b^{3} d^{3} e + B a b^{4} d^{4} + x \left (- 8 A a^{3} b^{2} e^{4} + 24 A a^{2} b^{3} d e^{3} - 24 A a b^{4} d^{2} e^{2} + 8 A b^{5} d^{3} e + 10 B a^{4} b e^{4} - 32 B a^{3} b^{2} d e^{3} + 36 B a^{2} b^{3} d^{2} e^{2} - 16 B a b^{4} d^{3} e + 2 B b^{5} d^{4}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac{x^{2} \left (- A b e^{4} + 3 B a e^{4} - 4 B b d e^{3}\right )}{2 b^{4}} + \frac{x \left (- 3 A a b e^{4} + 4 A b^{2} d e^{3} + 6 B a^{2} e^{4} - 12 B a b d e^{3} + 6 B b^{2} d^{2} e^{2}\right )}{b^{5}} - \frac{2 e \left (a e - b d\right )^{2} \left (- 3 A b e + 5 B a e - 2 B b d\right ) \log{\left (a + b x \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**4/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.229479, size = 567, normalized size = 2.97 \[ \frac{2 \,{\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \,{\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, B b^{6} x^{3} e^{4} + 12 \, B b^{6} d x^{2} e^{3} + 36 \, B b^{6} d^{2} x e^{2} - 9 \, B a b^{5} x^{2} e^{4} + 3 \, A b^{6} x^{2} e^{4} - 72 \, B a b^{5} d x e^{3} + 24 \, A b^{6} d x e^{3} + 36 \, B a^{2} b^{4} x e^{4} - 18 \, A a b^{5} x e^{4}}{6 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^4/(b*x + a)^3,x, algorithm="giac")
[Out]