Optimal. Leaf size=113 \[ -\frac{A b-a B}{2 b (a+b x)^2 (b d-a e)}-\frac{B d-A e}{(a+b x) (b d-a e)^2}-\frac{e \log (a+b x) (B d-A e)}{(b d-a e)^3}+\frac{e (B d-A e) \log (d+e x)}{(b d-a e)^3} \]
[Out]
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Rubi [A] time = 0.201936, antiderivative size = 113, 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 \[ -\frac{A b-a B}{2 b (a+b x)^2 (b d-a e)}-\frac{B d-A e}{(a+b x) (b d-a e)^2}-\frac{e \log (a+b x) (B d-A e)}{(b d-a e)^3}+\frac{e (B d-A e) \log (d+e x)}{(b d-a e)^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^3*(d + e*x)),x]
[Out]
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Rubi in Sympy [A] time = 29.7402, size = 90, normalized size = 0.8 \[ - \frac{e \left (A e - B d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{3}} + \frac{e \left (A e - B d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{3}} + \frac{A e - B d}{\left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{A b - B a}{2 b \left (a + b x\right )^{2} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**3/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.111014, size = 103, normalized size = 0.91 \[ \frac{\frac{(a B-A b) (b d-a e)^2}{b (a+b x)^2}+\frac{2 (b d-a e) (A e-B d)}{a+b x}+2 e \log (a+b x) (A e-B d)+2 e (B d-A e) \log (d+e x)}{2 (b d-a e)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)),x]
[Out]
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Maple [A] time = 0.016, size = 171, normalized size = 1.5 \[{\frac{{e}^{2}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{3}}}-{\frac{e\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{3}}}+{\frac{A}{ \left ( 2\,ae-2\,bd \right ) \left ( bx+a \right ) ^{2}}}-{\frac{Ba}{ \left ( 2\,ae-2\,bd \right ) b \left ( bx+a \right ) ^{2}}}+{\frac{Ae}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}-{\frac{Bd}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}-{\frac{{e}^{2}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{3}}}+{\frac{e\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^3/(e*x+d),x)
[Out]
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Maxima [A] time = 1.34555, size = 340, normalized size = 3.01 \[ -\frac{{\left (B d e - A e^{2}\right )} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac{{\left (B d e - A e^{2}\right )} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{{\left (B a b + A b^{2}\right )} d +{\left (B a^{2} - 3 \, A a b\right )} e + 2 \,{\left (B b^{2} d - A b^{2} e\right )} x}{2 \,{\left (a^{2} b^{3} d^{2} - 2 \, a^{3} b^{2} d e + a^{4} b e^{2} +{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} x^{2} + 2 \,{\left (a b^{4} d^{2} - 2 \, a^{2} b^{3} d e + a^{3} b^{2} e^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227024, size = 487, normalized size = 4.31 \[ \frac{4 \, A a b^{2} d e -{\left (B a b^{2} + A b^{3}\right )} d^{2} +{\left (B a^{3} - 3 \, A a^{2} b\right )} e^{2} - 2 \,{\left (B b^{3} d^{2} + A a b^{2} e^{2} -{\left (B a b^{2} + A b^{3}\right )} d e\right )} x - 2 \,{\left (B a^{2} b d e - A a^{2} b e^{2} +{\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 2 \,{\left (B a b^{2} d e - A a b^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left (B a^{2} b d e - A a^{2} b e^{2} +{\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 2 \,{\left (B a b^{2} d e - A a b^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.19897, size = 558, normalized size = 4.94 \[ - \frac{e \left (- A e + B d\right ) \log{\left (x + \frac{- A a e^{3} - A b d e^{2} + B a d e^{2} + B b d^{2} e - \frac{a^{4} e^{5} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b d e^{4} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{2} d^{2} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac{4 a b^{3} d^{3} e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac{b^{4} d^{4} e \left (- A e + B d\right )}{\left (a e - b d\right )^{3}}}{- 2 A b e^{3} + 2 B b d e^{2}} \right )}}{\left (a e - b d\right )^{3}} + \frac{e \left (- A e + B d\right ) \log{\left (x + \frac{- A a e^{3} - A b d e^{2} + B a d e^{2} + B b d^{2} e + \frac{a^{4} e^{5} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b d e^{4} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{2} d^{2} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac{4 a b^{3} d^{3} e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac{b^{4} d^{4} e \left (- A e + B d\right )}{\left (a e - b d\right )^{3}}}{- 2 A b e^{3} + 2 B b d e^{2}} \right )}}{\left (a e - b d\right )^{3}} - \frac{- 3 A a b e + A b^{2} d + B a^{2} e + B a b d + x \left (- 2 A b^{2} e + 2 B b^{2} d\right )}{2 a^{4} b e^{2} - 4 a^{3} b^{2} d e + 2 a^{2} b^{3} d^{2} + x^{2} \left (2 a^{2} b^{3} e^{2} - 4 a b^{4} d e + 2 b^{5} d^{2}\right ) + x \left (4 a^{3} b^{2} e^{2} - 8 a^{2} b^{3} d e + 4 a b^{4} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b*x+a)**3/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.230339, size = 309, normalized size = 2.73 \[ -\frac{{\left (B b d e - A b e^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac{{\left (B d e^{2} - A e^{3}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac{B a b^{2} d^{2} + A b^{3} d^{2} - 4 \, A a b^{2} d e - B a^{3} e^{2} + 3 \, A a^{2} b e^{2} + 2 \,{\left (B b^{3} d^{2} - B a b^{2} d e - A b^{3} d e + A a b^{2} e^{2}\right )} x}{2 \,{\left (b d - a e\right )}^{3}{\left (b x + a\right )}^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*(e*x + d)),x, algorithm="giac")
[Out]