Optimal. Leaf size=200 \[ -\frac{b^2 (A b-a B)}{(a+b x) (b d-a e)^4}+\frac{b^2 \log (a+b x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}-\frac{b^2 \log (d+e x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}+\frac{b (2 a B e-3 A b e+b B d)}{(d+e x) (b d-a e)^4}+\frac{a B e-2 A b e+b B d}{2 (d+e x)^2 (b d-a e)^3}+\frac{B d-A e}{3 (d+e x)^3 (b d-a e)^2} \]
[Out]
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Rubi [A] time = 0.467711, antiderivative size = 200, 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 \[ -\frac{b^2 (A b-a B)}{(a+b x) (b d-a e)^4}+\frac{b^2 \log (a+b x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}-\frac{b^2 \log (d+e x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}+\frac{b (2 a B e-3 A b e+b B d)}{(d+e x) (b d-a e)^4}+\frac{a B e-2 A b e+b B d}{2 (d+e x)^2 (b d-a e)^3}+\frac{B d-A e}{3 (d+e x)^3 (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^2*(d + e*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 105.173, size = 189, normalized size = 0.94 \[ \frac{b^{2} \left (4 A b e - 3 B a e - B b d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{5}} - \frac{b^{2} \left (4 A b e - 3 B a e - B b d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{5}} - \frac{b^{2} \left (A b - B a\right )}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{b \left (3 A b e - 2 B a e - B b d\right )}{\left (d + e x\right ) \left (a e - b d\right )^{4}} + \frac{2 A b e - B a e - B b d}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{3}} - \frac{A e - B d}{3 \left (d + e x\right )^{3} \left (a e - b d\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**2/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.244681, size = 188, normalized size = 0.94 \[ \frac{-\frac{6 b^2 (A b-a B) (b d-a e)}{a+b x}+6 b^2 \log (a+b x) (3 a B e-4 A b e+b B d)-6 b^2 \log (d+e x) (3 a B e-4 A b e+b B d)+\frac{2 (b d-a e)^3 (B d-A e)}{(d+e x)^3}+\frac{3 (b d-a e)^2 (a B e-2 A b e+b B d)}{(d+e x)^2}+\frac{6 b (b d-a e) (2 a B e-3 A b e+b B d)}{d+e x}}{6 (b d-a e)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^4),x]
[Out]
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Maple [A] time = 0.024, size = 364, normalized size = 1.8 \[ -{\frac{Ae}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}+{\frac{Bd}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-3\,{\frac{{b}^{2}Ae}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+2\,{\frac{Bbae}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{{b}^{2}Bd}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{Abe}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Bae}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Bbd}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-4\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Ae}{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{{b}^{2}\ln \left ( ex+d \right ) Bae}{ \left ( ae-bd \right ) ^{5}}}+{\frac{{b}^{3}\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{5}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) Ae}{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) Bae}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{b}^{3}\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{b}^{3}A}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+{\frac{Ba{b}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^2/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 1.42145, size = 1027, normalized size = 5.14 \[ \frac{{\left (B b^{3} d +{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{{\left (B b^{3} d +{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{2 \, A a^{3} e^{3} -{\left (17 \, B a b^{2} - 6 \, A b^{3}\right )} d^{3} - 2 \,{\left (4 \, B a^{2} b - 13 \, A a b^{2}\right )} d^{2} e +{\left (B a^{3} - 10 \, A a^{2} b\right )} d e^{2} - 6 \,{\left (B b^{3} d e^{2} +{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e^{3}\right )} x^{3} - 3 \,{\left (5 \, B b^{3} d^{2} e + 4 \,{\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} d e^{2} +{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{3}\right )} x^{2} -{\left (11 \, B b^{3} d^{3} +{\left (41 \, B a b^{2} - 44 \, A b^{3}\right )} d^{2} e +{\left (23 \, B a^{2} b - 32 \, A a b^{2}\right )} d e^{2} -{\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} e^{3}\right )} x}{6 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*(e*x + d)^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235141, size = 1646, normalized size = 8.23 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*(e*x + d)^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 21.8248, size = 1445, normalized size = 7.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b*x+a)**2/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.240843, size = 571, normalized size = 2.86 \[ -\frac{{\left (B b^{4} d + 3 \, B a b^{3} e - 4 \, A b^{4} e\right )}{\rm ln}\left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac{\frac{B a b^{6}}{b x + a} - \frac{A b^{7}}{b x + a}}{b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}} - \frac{11 \, B b^{3} d e^{3} + 15 \, B a b^{2} e^{4} - 26 \, A b^{3} e^{4} + \frac{3 \,{\left (9 \, B b^{5} d^{2} e^{2} + 2 \, B a b^{4} d e^{3} - 20 \, A b^{5} d e^{3} - 11 \, B a^{2} b^{3} e^{4} + 20 \, A a b^{4} e^{4}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (B b^{7} d^{3} e - B a b^{6} d^{2} e^{2} - 2 \, A b^{7} d^{2} e^{2} - B a^{2} b^{5} d e^{3} + 4 \, A a b^{6} d e^{3} + B a^{3} b^{4} e^{4} - 2 \, A a^{2} b^{5} e^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \,{\left (b d - a e\right )}^{5}{\left (\frac{b d}{b x + a} - \frac{a e}{b x + a} + e\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*(e*x + d)^4),x, algorithm="giac")
[Out]