Optimal. Leaf size=187 \[ \frac{e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}+\frac{e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}-\frac{(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac{2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac{(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac{B e^4 (a+b x)^4}{4 b^6} \]
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Rubi [A] time = 0.265149, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ \frac{e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}+\frac{e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}-\frac{(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac{2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac{(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac{B e^4 (a+b x)^4}{4 b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(A+B x) (d+e x)^4}{(a+b x)^2} \, dx\\ &=\int \left (\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^5}+\frac{(A b-a B) (b d-a e)^4}{b^5 (a+b x)^2}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^5 (a+b x)}+\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)}{b^5}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)^2}{b^5}+\frac{B e^4 (a+b x)^3}{b^5}\right ) \, dx\\ &=\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) x}{b^5}-\frac{(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac{e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^2}{b^6}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)^3}{3 b^6}+\frac{B e^4 (a+b x)^4}{4 b^6}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e) \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.179351, size = 365, normalized size = 1.95 \[ \frac{-4 A b \left (6 a^2 b^2 e^2 \left (3 d^2+4 d e x-e^2 x^2\right )-3 a^3 b e^3 (4 d+3 e x)+3 a^4 e^4+2 a b^3 e \left (-9 d^2 e x-6 d^3+9 d e^2 x^2+e^3 x^3\right )+b^4 \left (-18 d^2 e^2 x^2+3 d^4-6 d e^3 x^3-e^4 x^4\right )\right )+B \left (6 a^3 b^2 e^2 \left (12 d^2+24 d e x-5 e^2 x^2\right )+2 a^2 b^3 e \left (-72 d^2 e x-24 d^3+48 d e^2 x^2+5 e^3 x^3\right )-48 a^4 b e^3 (d+e x)+12 a^5 e^4+a b^4 \left (-108 d^2 e^2 x^2+48 d^3 e x+12 d^4-32 d e^3 x^3-5 e^4 x^4\right )+b^5 e x^2 \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (a+b x) (b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{12 b^6 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 564, normalized size = 3. \begin{align*} 18\,{\frac{\ln \left ( bx+a \right ) B{d}^{2}{a}^{2}{e}^{2}}{{b}^{4}}}-6\,{\frac{A{a}^{2}{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-12\,{\frac{aB{e}^{2}{d}^{2}x}{{b}^{3}}}+4\,{\frac{A{a}^{3}d{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+12\,{\frac{Bd{a}^{2}{e}^{3}x}{{b}^{4}}}+4\,{\frac{aA{d}^{3}e}{{b}^{2} \left ( bx+a \right ) }}-4\,{\frac{{e}^{3}B{x}^{2}ad}{{b}^{3}}}-8\,{\frac{aA{e}^{3}dx}{{b}^{3}}}-4\,{\frac{B{a}^{2}{d}^{3}e}{{b}^{3} \left ( bx+a \right ) }}+12\,{\frac{\ln \left ( bx+a \right ) Ad{a}^{2}{e}^{3}}{{b}^{4}}}-12\,{\frac{\ln \left ( bx+a \right ) A{d}^{2}a{e}^{2}}{{b}^{3}}}+6\,{\frac{B{a}^{3}{d}^{2}{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}-4\,{\frac{B{a}^{4}d{e}^{3}}{{b}^{5} \left ( bx+a \right ) }}-8\,{\frac{\ln \left ( bx+a \right ) B{d}^{3}ae}{{b}^{3}}}-16\,{\frac{\ln \left ( bx+a \right ) Bd{a}^{3}{e}^{3}}{{b}^{5}}}+{\frac{\ln \left ( bx+a \right ) B{d}^{4}}{{b}^{2}}}+{\frac{A{e}^{4}{x}^{3}}{3\,{b}^{2}}}-{\frac{A{d}^{4}}{b \left ( bx+a \right ) }}+{\frac{B{e}^{4}{x}^{4}}{4\,{b}^{2}}}+{\frac{3\,B{e}^{4}{x}^{2}{a}^{2}}{2\,{b}^{4}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{{b}^{2}}}+3\,{\frac{A{e}^{4}{a}^{2}x}{{b}^{4}}}-{\frac{2\,B{e}^{4}{x}^{3}a}{3\,{b}^{3}}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,{b}^{2}}}+4\,{\frac{Be{d}^{3}x}{{b}^{2}}}-{\frac{A{e}^{4}{x}^{2}a}{{b}^{3}}}+2\,{\frac{{e}^{3}A{x}^{2}d}{{b}^{2}}}-4\,{\frac{B{a}^{3}{e}^{4}x}{{b}^{5}}}-{\frac{A{a}^{4}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+{\frac{B{a}^{5}{e}^{4}}{{b}^{6} \left ( bx+a \right ) }}+{\frac{aB{d}^{4}}{{b}^{2} \left ( bx+a \right ) }}+6\,{\frac{A{e}^{2}{d}^{2}x}{{b}^{2}}}+5\,{\frac{\ln \left ( bx+a \right ) B{e}^{4}{a}^{4}}{{b}^{6}}}-4\,{\frac{\ln \left ( bx+a \right ) A{a}^{3}{e}^{4}}{{b}^{5}}}+4\,{\frac{\ln \left ( bx+a \right ) A{d}^{3}e}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09146, size = 555, normalized size = 2.97 \begin{align*} \frac{{\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} +{\left (B a^{5} - A a^{4} b\right )} e^{4}}{b^{7} x + a b^{6}} + \frac{3 \, B b^{3} e^{4} x^{4} + 4 \,{\left (4 \, B b^{3} d e^{3} -{\left (2 \, B a b^{2} - A b^{3}\right )} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B b^{3} d^{2} e^{2} - 4 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d e^{3} +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B b^{3} d^{3} e - 6 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 4 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{3} -{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} x}{12 \, b^{5}} + \frac{{\left (B b^{4} d^{4} - 4 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6463, size = 1245, normalized size = 6.66 \begin{align*} \frac{3 \, B b^{5} e^{4} x^{5} + 12 \,{\left (B a b^{4} - A b^{5}\right )} d^{4} - 48 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 72 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 48 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + 12 \,{\left (B a^{5} - A a^{4} b\right )} e^{4} +{\left (16 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} e^{4}\right )} x^{4} + 2 \,{\left (18 \, B b^{5} d^{2} e^{2} - 4 \,{\left (4 \, B a b^{4} - 3 \, A b^{5}\right )} d e^{3} +{\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \,{\left (8 \, B b^{5} d^{3} e - 6 \,{\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} d^{2} e^{2} + 4 \,{\left (4 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} d e^{3} -{\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B a b^{4} d^{3} e - 6 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 4 \,{\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} -{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (B a b^{4} d^{4} - 4 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} +{\left (5 \, B a^{5} - 4 \, A a^{4} b\right )} e^{4} +{\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\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.68202, size = 384, normalized size = 2.05 \begin{align*} \frac{B e^{4} x^{4}}{4 b^{2}} + \frac{- A a^{4} b e^{4} + 4 A a^{3} b^{2} d e^{3} - 6 A a^{2} b^{3} d^{2} e^{2} + 4 A a b^{4} d^{3} e - A b^{5} d^{4} + B a^{5} e^{4} - 4 B a^{4} b d e^{3} + 6 B a^{3} b^{2} d^{2} e^{2} - 4 B a^{2} b^{3} d^{3} e + B a b^{4} d^{4}}{a b^{6} + b^{7} x} - \frac{x^{3} \left (- A b e^{4} + 2 B a e^{4} - 4 B b d e^{3}\right )}{3 b^{3}} + \frac{x^{2} \left (- 2 A a b e^{4} + 4 A b^{2} d e^{3} + 3 B a^{2} e^{4} - 8 B a b d e^{3} + 6 B b^{2} d^{2} e^{2}\right )}{2 b^{4}} - \frac{x \left (- 3 A a^{2} b e^{4} + 8 A a b^{2} d e^{3} - 6 A b^{3} d^{2} e^{2} + 4 B a^{3} e^{4} - 12 B a^{2} b d e^{3} + 12 B a b^{2} d^{2} e^{2} - 4 B b^{3} d^{3} e\right )}{b^{5}} + \frac{\left (a e - b d\right )^{3} \left (- 4 A b e + 5 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15844, size = 585, normalized size = 3.13 \begin{align*} \frac{{\left (B b^{4} d^{4} - 8 \, B a b^{3} d^{3} e + 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 16 \, B a^{3} b d e^{3} + 12 \, A a^{2} b^{2} d e^{3} + 5 \, B a^{4} e^{4} - 4 \, A a^{3} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}}{{\left (b x + a\right )} b^{6}} + \frac{3 \, B b^{6} x^{4} e^{4} + 16 \, B b^{6} d x^{3} e^{3} + 36 \, B b^{6} d^{2} x^{2} e^{2} + 48 \, B b^{6} d^{3} x e - 8 \, B a b^{5} x^{3} e^{4} + 4 \, A b^{6} x^{3} e^{4} - 48 \, B a b^{5} d x^{2} e^{3} + 24 \, A b^{6} d x^{2} e^{3} - 144 \, B a b^{5} d^{2} x e^{2} + 72 \, A b^{6} d^{2} x e^{2} + 18 \, B a^{2} b^{4} x^{2} e^{4} - 12 \, A a b^{5} x^{2} e^{4} + 144 \, B a^{2} b^{4} d x e^{3} - 96 \, A a b^{5} d x e^{3} - 48 \, B a^{3} b^{3} x e^{4} + 36 \, A a^{2} b^{4} x e^{4}}{12 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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