3.1336 \(\int \frac{A+B x}{(d+e x)^2 \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=173 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 A c d e+B c d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{B d-A e}{(d+e x) \left (a e^2+c d^2\right )}-\frac{\log (d+e x) \left (-a B e^2-2 A c d e+B c d^2\right )}{\left (a e^2+c d^2\right )^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2+2 a B d e+A c d^2\right )}{\sqrt{a} \left (a e^2+c d^2\right )^2} \]

[Out]

(B*d - A*e)/((c*d^2 + a*e^2)*(d + e*x)) + (Sqrt[c]*(A*c*d^2 + 2*a*B*d*e - a*A*e^
2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2)^2) - ((B*c*d^2 - 2*A*c*
d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^2 + ((B*c*d^2 - 2*A*c*d*e - a*B*e^2
)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)

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Rubi [A]  time = 0.359809, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 A c d e+B c d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{B d-A e}{(d+e x) \left (a e^2+c d^2\right )}-\frac{\log (d+e x) \left (-a B e^2-2 A c d e+B c d^2\right )}{\left (a e^2+c d^2\right )^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2+2 a B d e+A c d^2\right )}{\sqrt{a} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^2*(a + c*x^2)),x]

[Out]

(B*d - A*e)/((c*d^2 + a*e^2)*(d + e*x)) + (Sqrt[c]*(A*c*d^2 + 2*a*B*d*e - a*A*e^
2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2)^2) - ((B*c*d^2 - 2*A*c*
d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^2 + ((B*c*d^2 - 2*A*c*d*e - a*B*e^2
)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)

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Rubi in Sympy [A]  time = 65.0757, size = 163, normalized size = 0.94 \[ - \frac{\left (2 A c d e + B a e^{2} - B c d^{2}\right ) \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} + \frac{\left (2 A c d e + B a e^{2} - B c d^{2}\right ) \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{2}} - \frac{A e - B d}{\left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt{c} \left (A a e^{2} - A c d^{2} - 2 B a d e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**2/(c*x**2+a),x)

[Out]

-(2*A*c*d*e + B*a*e**2 - B*c*d**2)*log(a + c*x**2)/(2*(a*e**2 + c*d**2)**2) + (2
*A*c*d*e + B*a*e**2 - B*c*d**2)*log(d + e*x)/(a*e**2 + c*d**2)**2 - (A*e - B*d)/
((d + e*x)*(a*e**2 + c*d**2)) - sqrt(c)*(A*a*e**2 - A*c*d**2 - 2*B*a*d*e)*atan(s
qrt(c)*x/sqrt(a))/(sqrt(a)*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 0.36679, size = 148, normalized size = 0.86 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 A c d e+B c d^2\right )+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{d+e x}+\log (d+e x) \left (2 a B e^2+4 A c d e-2 B c d^2\right )+\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2+2 a B d e+A c d^2\right )}{\sqrt{a}}}{2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^2*(a + c*x^2)),x]

[Out]

((2*(B*d - A*e)*(c*d^2 + a*e^2))/(d + e*x) + (2*Sqrt[c]*(A*c*d^2 + 2*a*B*d*e - a
*A*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + (-2*B*c*d^2 + 4*A*c*d*e + 2*a*B*e
^2)*Log[d + e*x] + (B*c*d^2 - 2*A*c*d*e - a*B*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a
*e^2)^2)

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Maple [A]  time = 0.015, size = 312, normalized size = 1.8 \[ -{\frac{Ae}{ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( ex+d \right ) }}+{\frac{Bd}{ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( ex+d \right ) }}+2\,{\frac{\ln \left ( ex+d \right ) Acde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ) aB{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( ex+d \right ) Bc{d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{c\ln \left ( c{x}^{2}+a \right ) Ade}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( c{x}^{2}+a \right ) aB{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{c\ln \left ( c{x}^{2}+a \right ) B{d}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{aAc{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{c}^{2}{d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+2\,{\frac{aBcde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^2/(c*x^2+a),x)

[Out]

-1/(a*e^2+c*d^2)/(e*x+d)*A*e+1/(a*e^2+c*d^2)/(e*x+d)*B*d+2/(a*e^2+c*d^2)^2*ln(e*
x+d)*A*c*d*e+1/(a*e^2+c*d^2)^2*ln(e*x+d)*a*B*e^2-1/(a*e^2+c*d^2)^2*ln(e*x+d)*B*c
*d^2-c/(a*e^2+c*d^2)^2*ln(c*x^2+a)*A*d*e-1/2/(a*e^2+c*d^2)^2*ln(c*x^2+a)*a*B*e^2
+1/2*c/(a*e^2+c*d^2)^2*ln(c*x^2+a)*B*d^2-c/(a*e^2+c*d^2)^2/(a*c)^(1/2)*arctan(c*
x/(a*c)^(1/2))*A*a*e^2+c^2/(a*e^2+c*d^2)^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A
*d^2+2*c/(a*e^2+c*d^2)^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*a*B*d*e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.75656, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, B c d^{3} - 2 \, A c d^{2} e + 2 \, B a d e^{2} - 2 \, A a e^{3} -{\left (A c d^{3} + 2 \, B a d^{2} e - A a d e^{2} +{\left (A c d^{2} e + 2 \, B a d e^{2} - A a e^{3}\right )} x\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) +{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )}}, \frac{2 \, B c d^{3} - 2 \, A c d^{2} e + 2 \, B a d e^{2} - 2 \, A a e^{3} + 2 \,{\left (A c d^{3} + 2 \, B a d^{2} e - A a d e^{2} +{\left (A c d^{2} e + 2 \, B a d e^{2} - A a e^{3}\right )} x\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{c x}{a \sqrt{\frac{c}{a}}}\right ) +{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (B c d^{3} - 2 \, A c d^{2} e - B a d e^{2} +{\left (B c d^{2} e - 2 \, A c d e^{2} - B a e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*(e*x + d)^2),x, algorithm="fricas")

[Out]

[1/2*(2*B*c*d^3 - 2*A*c*d^2*e + 2*B*a*d*e^2 - 2*A*a*e^3 - (A*c*d^3 + 2*B*a*d^2*e
- A*a*d*e^2 + (A*c*d^2*e + 2*B*a*d*e^2 - A*a*e^3)*x)*sqrt(-c/a)*log((c*x^2 - 2*
a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + (B*c*d^3 - 2*A*c*d^2*e - B*a*d*e^2 + (B*c*d^2
*e - 2*A*c*d*e^2 - B*a*e^3)*x)*log(c*x^2 + a) - 2*(B*c*d^3 - 2*A*c*d^2*e - B*a*d
*e^2 + (B*c*d^2*e - 2*A*c*d*e^2 - B*a*e^3)*x)*log(e*x + d))/(c^2*d^5 + 2*a*c*d^3
*e^2 + a^2*d*e^4 + (c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*x), 1/2*(2*B*c*d^3 - 2*
A*c*d^2*e + 2*B*a*d*e^2 - 2*A*a*e^3 + 2*(A*c*d^3 + 2*B*a*d^2*e - A*a*d*e^2 + (A*
c*d^2*e + 2*B*a*d*e^2 - A*a*e^3)*x)*sqrt(c/a)*arctan(c*x/(a*sqrt(c/a))) + (B*c*d
^3 - 2*A*c*d^2*e - B*a*d*e^2 + (B*c*d^2*e - 2*A*c*d*e^2 - B*a*e^3)*x)*log(c*x^2
+ a) - 2*(B*c*d^3 - 2*A*c*d^2*e - B*a*d*e^2 + (B*c*d^2*e - 2*A*c*d*e^2 - B*a*e^3
)*x)*log(e*x + d))/(c^2*d^5 + 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e + 2*a*c*d^2
*e^3 + a^2*e^5)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x+d)**2/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283422, size = 309, normalized size = 1.79 \[ \frac{{\left (A c^{2} d^{2} e^{2} + 2 \, B a c d e^{3} - A a c e^{4}\right )} \arctan \left (\frac{{\left (c d - \frac{c d^{2}}{x e + d} - \frac{a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{a c}}\right ) e^{\left (-2\right )}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{a c}} + \frac{{\left (B c d^{2} - 2 \, A c d e - B a e^{2}\right )}{\rm ln}\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{\frac{B d e^{2}}{x e + d} - \frac{A e^{3}}{x e + d}}{c d^{2} e^{2} + a e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*(e*x + d)^2),x, algorithm="giac")

[Out]

(A*c^2*d^2*e^2 + 2*B*a*c*d*e^3 - A*a*c*e^4)*arctan((c*d - c*d^2/(x*e + d) - a*e^
2/(x*e + d))*e^(-1)/sqrt(a*c))*e^(-2)/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(
a*c)) + 1/2*(B*c*d^2 - 2*A*c*d*e - B*a*e^2)*ln(c - 2*c*d/(x*e + d) + c*d^2/(x*e
+ d)^2 + a*e^2/(x*e + d)^2)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + (B*d*e^2/(x*e
+ d) - A*e^3/(x*e + d))/(c*d^2*e^2 + a*e^4)