3.1344 \(\int \frac{A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=195 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

[Out]

-(a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) - ((a*B*e
*(c*d^2 - a*e^2) - A*c*d*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3
/2)*Sqrt[c]*(c*d^2 + a*e^2)^2) - (e^2*(B*d - A*e)*Log[d + e*x])/(c*d^2 + a*e^2)^
2 + (e^2*(B*d - A*e)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)

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Rubi [A]  time = 0.58716, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) - ((a*B*e
*(c*d^2 - a*e^2) - A*c*d*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3
/2)*Sqrt[c]*(c*d^2 + a*e^2)^2) - (e^2*(B*d - A*e)*Log[d + e*x])/(c*d^2 + a*e^2)^
2 + (e^2*(B*d - A*e)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)

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Rubi in Sympy [A]  time = 77.1174, size = 177, normalized size = 0.91 \[ - \frac{e^{2} \left (A e - B d\right ) \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} + \frac{e^{2} \left (A e - B d\right ) \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{2}} + \frac{a \left (A e - B d\right ) + x \left (A c d + B a e\right )}{2 a \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\left (3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - B a c d^{2} e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{c} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**2*(A*e - B*d)*log(a + c*x**2)/(2*(a*e**2 + c*d**2)**2) + e**2*(A*e - B*d)*lo
g(d + e*x)/(a*e**2 + c*d**2)**2 + (a*(A*e - B*d) + x*(A*c*d + B*a*e))/(2*a*(a +
c*x**2)*(a*e**2 + c*d**2)) + (3*A*a*c*d*e**2 + A*c**2*d**3 + B*a**2*e**3 - B*a*c
*d**2*e)*atan(sqrt(c)*x/sqrt(a))/(2*a**(3/2)*sqrt(c)*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 0.305727, size = 158, normalized size = 0.81 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a B e \left (a e^2-c d^2\right )\right )}{a^{3/2} \sqrt{c}}+\frac{\left (a e^2+c d^2\right ) (a (A e-B d+B e x)+A c d x)}{a \left (a+c x^2\right )}+e^2 \log \left (a+c x^2\right ) (B d-A e)+2 e^2 (A e-B d) \log (d+e x)}{2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(((c*d^2 + a*e^2)*(A*c*d*x + a*(-(B*d) + A*e + B*e*x)))/(a*(a + c*x^2)) + ((a*B*
e*(-(c*d^2) + a*e^2) + A*c*d*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^
(3/2)*Sqrt[c]) + 2*e^2*(-(B*d) + A*e)*Log[d + e*x] + e^2*(B*d - A*e)*Log[a + c*x
^2])/(2*(c*d^2 + a*e^2)^2)

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Maple [B]  time = 0.022, size = 499, normalized size = 2.6 \[{\frac{{e}^{3}\ln \left ( ex+d \right ) A}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{{e}^{2}\ln \left ( ex+d \right ) Bd}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{Acxd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{3}{c}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) a}}+{\frac{aBx{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{Bcx{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{aA{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{Ac{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{aBd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{Bc{d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) A{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) Bd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{3\,Ad{e}^{2}c}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{d}^{3}{c}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{B{e}^{3}a}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{Bc{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^3/(a*e^2+c*d^2)^2*ln(e*x+d)*A-e^2/(a*e^2+c*d^2)^2*ln(e*x+d)*B*d+1/2/(a*e^2+c*d
^2)^2/(c*x^2+a)*x*A*c*d*e^2+1/2/(a*e^2+c*d^2)^2/(c*x^2+a)/a*x*A*d^3*c^2+1/2/(a*e
^2+c*d^2)^2/(c*x^2+a)*a*x*B*e^3+1/2/(a*e^2+c*d^2)^2/(c*x^2+a)*x*B*c*d^2*e+1/2/(a
*e^2+c*d^2)^2/(c*x^2+a)*a*A*e^3+1/2/(a*e^2+c*d^2)^2/(c*x^2+a)*A*c*d^2*e-1/2/(a*e
^2+c*d^2)^2/(c*x^2+a)*a*B*d*e^2-1/2/(a*e^2+c*d^2)^2/(c*x^2+a)*B*c*d^3-1/2/(a*e^2
+c*d^2)^2*ln(a*(c*x^2+a))*A*e^3+1/2/(a*e^2+c*d^2)^2*ln(a*(c*x^2+a))*B*d*e^2+3/2/
(a*e^2+c*d^2)^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*c*d*e^2+1/2/(a*e^2+c*d^2)^
2/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^3*c^2+1/2/(a*e^2+c*d^2)^2*a/(a*c)^(1
/2)*arctan(c*x/(a*c)^(1/2))*B*e^3-1/2/(a*e^2+c*d^2)^2/(a*c)^(1/2)*arctan(c*x/(a*
c)^(1/2))*B*c*d^2*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)^2*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.9966, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} +{\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (B a c d^{3} - A a c d^{2} e + B a^{2} d e^{2} - A a^{2} e^{3} -{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} x -{\left (B a^{2} d e^{2} - A a^{2} e^{3} +{\left (B a c d e^{2} - A a c e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (B a^{2} d e^{2} - A a^{2} e^{3} +{\left (B a c d e^{2} - A a c e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )\right )} \sqrt{-a c}}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )} \sqrt{-a c}}, \frac{{\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} +{\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (B a c d^{3} - A a c d^{2} e + B a^{2} d e^{2} - A a^{2} e^{3} -{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} x -{\left (B a^{2} d e^{2} - A a^{2} e^{3} +{\left (B a c d e^{2} - A a c e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (B a^{2} d e^{2} - A a^{2} e^{3} +{\left (B a c d e^{2} - A a c e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )\right )} \sqrt{a c}}{2 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)/(c*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.305341, size = 362, normalized size = 1.86 \[ \frac{{\left (B d e^{2} - A e^{3}\right )}{\rm ln}\left (c x^{2} + a\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac{{\left (B d e^{3} - A e^{4}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac{{\left (A c^{2} d^{3} - B a c d^{2} e + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} - \frac{B a c d^{3} - A a c d^{2} e + B a^{2} d e^{2} - A a^{2} e^{3} -{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (c x^{2} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*d*e^2 - A*e^3)*ln(c*x^2 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - (B*d*e
^3 - A*e^4)*ln(abs(x*e + d))/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + 1/2*(A*c^2*
d^3 - B*a*c*d^2*e + 3*A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/sqrt(a*c))/((a*c^2*d^4
+ 2*a^2*c*d^2*e^2 + a^3*e^4)*sqrt(a*c)) - 1/2*(B*a*c*d^3 - A*a*c*d^2*e + B*a^2*
d*e^2 - A*a^2*e^3 - (A*c^2*d^3 + B*a*c*d^2*e + A*a*c*d*e^2 + B*a^2*e^3)*x)/((c*d
^2 + a*e^2)^2*(c*x^2 + a)*a)