3.1345 \(\int \frac{A+B x}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=290 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (c d^2-3 a e^2\right )-A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^3}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a B e^2-4 A c d e+3 B c d^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e \left (-3 a A e^2+4 a B d e+A c d^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^2}-\frac{e^2 \log (d+e x) \left (-a B e^2-4 A c d e+3 B c d^2\right )}{\left (a e^2+c d^2\right )^3} \]

[Out]

(e*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2))/(2*a*(c*d^2 + a*e^2)^2*(d + e*x)) - (a*(B*
d - A*e) - (A*c*d + a*B*e)*x)/(2*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)) - (Sqr
t[c]*(2*a*B*d*e*(c*d^2 - 3*a*e^2) - A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4))*Arc
Tan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*(c*d^2 + a*e^2)^3) - (e^2*(3*B*c*d^2 - 4*A*
c*d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^3 + (e^2*(3*B*c*d^2 - 4*A*c*d*e -
a*B*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)

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Rubi [A]  time = 0.851747, antiderivative size = 290, 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 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (c d^2-3 a e^2\right )-A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^3}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a B e^2-4 A c d e+3 B c d^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e \left (-3 a A e^2+4 a B d e+A c d^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^2}-\frac{e^2 \log (d+e x) \left (-a B e^2-4 A c d e+3 B c d^2\right )}{\left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(e*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2))/(2*a*(c*d^2 + a*e^2)^2*(d + e*x)) - (a*(B*
d - A*e) - (A*c*d + a*B*e)*x)/(2*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)) - (Sqr
t[c]*(2*a*B*d*e*(c*d^2 - 3*a*e^2) - A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4))*Arc
Tan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*(c*d^2 + a*e^2)^3) - (e^2*(3*B*c*d^2 - 4*A*
c*d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^3 + (e^2*(3*B*c*d^2 - 4*A*c*d*e -
a*B*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.807371, size = 251, normalized size = 0.87 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (a^2 B e^2-a c (A e (e x-2 d)+B d (d-2 e x))+A c^2 d^2 x\right )}{a \left (a+c x^2\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+2 a B d e \left (3 a e^2-c d^2\right )\right )}{a^{3/2}}-e^2 \log \left (a+c x^2\right ) \left (a B e^2+4 A c d e-3 B c d^2\right )-\frac{2 e^2 \left (a e^2+c d^2\right ) (A e-B d)}{d+e x}+2 e^2 \log (d+e x) \left (a B e^2+4 A c d e-3 B c d^2\right )}{2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

((-2*e^2*(-(B*d) + A*e)*(c*d^2 + a*e^2))/(d + e*x) + ((c*d^2 + a*e^2)*(a^2*B*e^2
+ A*c^2*d^2*x - a*c*(B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(a*(a + c*x^2)) + (S
qrt[c]*(2*a*B*d*e*(-(c*d^2) + 3*a*e^2) + A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)
)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*e^2*(-3*B*c*d^2 + 4*A*c*d*e + a*B*e^2
)*Log[d + e*x] - e^2*(-3*B*c*d^2 + 4*A*c*d*e + a*B*e^2)*Log[a + c*x^2])/(2*(c*d^
2 + a*e^2)^3)

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Maple [B]  time = 0.026, size = 667, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*e^3/(a*e^2+c*d^2)^3*ln(e*x+d)*A*c*d+e^4/(a*e^2+c*d^2)^3*ln(e*x+d)*a*B-3*e^2/(a
*e^2+c*d^2)^3*ln(e*x+d)*B*c*d^2-e^3/(a*e^2+c*d^2)^2/(e*x+d)*A+e^2/(a*e^2+c*d^2)^
2/(e*x+d)*B*d-1/2*c/(a*e^2+c*d^2)^3/(c*x^2+a)*a*x*A*e^4+1/2*c^3/(a*e^2+c*d^2)^3/
(c*x^2+a)/a*x*A*d^4+c/(a*e^2+c*d^2)^3/(c*x^2+a)*a*x*B*d*e^3+c^2/(a*e^2+c*d^2)^3/
(c*x^2+a)*x*B*d^3*e+c/(a*e^2+c*d^2)^3/(c*x^2+a)*A*a*d*e^3+c^2/(a*e^2+c*d^2)^3/(c
*x^2+a)*A*d^3*e+1/2/(a*e^2+c*d^2)^3/(c*x^2+a)*B*e^4*a^2-1/2*c^2/(a*e^2+c*d^2)^3/
(c*x^2+a)*B*d^4-2*c/(a*e^2+c*d^2)^3*ln(a*(c*x^2+a))*A*d*e^3-1/2/(a*e^2+c*d^2)^3*
a*ln(a*(c*x^2+a))*B*e^4+3/2*c/(a*e^2+c*d^2)^3*ln(a*(c*x^2+a))*B*d^2*e^2-3/2*c/(a
*e^2+c*d^2)^3*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*e^4+3*c^2/(a*e^2+c*d^2)^3/
(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^2*e^2+1/2*c^3/(a*e^2+c*d^2)^3/a/(a*c)^(1
/2)*arctan(c*x/(a*c)^(1/2))*A*d^4+3*c/(a*e^2+c*d^2)^3*a/(a*c)^(1/2)*arctan(c*x/(
a*c)^(1/2))*B*d*e^3-c^2/(a*e^2+c*d^2)^3/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*d^
3*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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 64.9968, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*B*a*c^2*d^5 - 4*A*a*c^2*d^4*e - 4*B*a^2*c*d^3*e^2 - 6*B*a^3*d*e^4 + 4*A
*a^3*e^5 - 2*(A*c^3*d^4*e + 4*B*a*c^2*d^3*e^2 - 2*A*a*c^2*d^2*e^3 + 4*B*a^2*c*d*
e^4 - 3*A*a^2*c*e^5)*x^2 + (A*a*c^2*d^5 - 2*B*a^2*c*d^4*e + 6*A*a^2*c*d^3*e^2 +
6*B*a^3*d^2*e^3 - 3*A*a^3*d*e^4 + (A*c^3*d^4*e - 2*B*a*c^2*d^3*e^2 + 6*A*a*c^2*d
^2*e^3 + 6*B*a^2*c*d*e^4 - 3*A*a^2*c*e^5)*x^3 + (A*c^3*d^5 - 2*B*a*c^2*d^4*e + 6
*A*a*c^2*d^3*e^2 + 6*B*a^2*c*d^2*e^3 - 3*A*a^2*c*d*e^4)*x^2 + (A*a*c^2*d^4*e - 2
*B*a^2*c*d^3*e^2 + 6*A*a^2*c*d^2*e^3 + 6*B*a^3*d*e^4 - 3*A*a^3*e^5)*x)*sqrt(-c/a
)*log((c*x^2 - 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) - 2*(A*c^3*d^5 + B*a*c^2*d^4*e
+ 2*A*a*c^2*d^3*e^2 + 2*B*a^2*c*d^2*e^3 + A*a^2*c*d*e^4 + B*a^3*e^5)*x - 2*(3*B
*a^2*c*d^3*e^2 - 4*A*a^2*c*d^2*e^3 - B*a^3*d*e^4 + (3*B*a*c^2*d^2*e^3 - 4*A*a*c^
2*d*e^4 - B*a^2*c*e^5)*x^3 + (3*B*a*c^2*d^3*e^2 - 4*A*a*c^2*d^2*e^3 - B*a^2*c*d*
e^4)*x^2 + (3*B*a^2*c*d^2*e^3 - 4*A*a^2*c*d*e^4 - B*a^3*e^5)*x)*log(c*x^2 + a) +
4*(3*B*a^2*c*d^3*e^2 - 4*A*a^2*c*d^2*e^3 - B*a^3*d*e^4 + (3*B*a*c^2*d^2*e^3 - 4
*A*a*c^2*d*e^4 - B*a^2*c*e^5)*x^3 + (3*B*a*c^2*d^3*e^2 - 4*A*a*c^2*d^2*e^3 - B*a
^2*c*d*e^4)*x^2 + (3*B*a^2*c*d^2*e^3 - 4*A*a^2*c*d*e^4 - B*a^3*e^5)*x)*log(e*x +
d))/(a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*e^6 + (a*c^4*d^6
*e + 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a*c^4*d^7 + 3*a^2
*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e + 3*a^3*c^2
*d^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*x), -1/2*(B*a*c^2*d^5 - 2*A*a*c^2*d^4*e -
2*B*a^2*c*d^3*e^2 - 3*B*a^3*d*e^4 + 2*A*a^3*e^5 - (A*c^3*d^4*e + 4*B*a*c^2*d^3*e
^2 - 2*A*a*c^2*d^2*e^3 + 4*B*a^2*c*d*e^4 - 3*A*a^2*c*e^5)*x^2 - (A*a*c^2*d^5 - 2
*B*a^2*c*d^4*e + 6*A*a^2*c*d^3*e^2 + 6*B*a^3*d^2*e^3 - 3*A*a^3*d*e^4 + (A*c^3*d^
4*e - 2*B*a*c^2*d^3*e^2 + 6*A*a*c^2*d^2*e^3 + 6*B*a^2*c*d*e^4 - 3*A*a^2*c*e^5)*x
^3 + (A*c^3*d^5 - 2*B*a*c^2*d^4*e + 6*A*a*c^2*d^3*e^2 + 6*B*a^2*c*d^2*e^3 - 3*A*
a^2*c*d*e^4)*x^2 + (A*a*c^2*d^4*e - 2*B*a^2*c*d^3*e^2 + 6*A*a^2*c*d^2*e^3 + 6*B*
a^3*d*e^4 - 3*A*a^3*e^5)*x)*sqrt(c/a)*arctan(c*x/(a*sqrt(c/a))) - (A*c^3*d^5 + B
*a*c^2*d^4*e + 2*A*a*c^2*d^3*e^2 + 2*B*a^2*c*d^2*e^3 + A*a^2*c*d*e^4 + B*a^3*e^5
)*x - (3*B*a^2*c*d^3*e^2 - 4*A*a^2*c*d^2*e^3 - B*a^3*d*e^4 + (3*B*a*c^2*d^2*e^3
- 4*A*a*c^2*d*e^4 - B*a^2*c*e^5)*x^3 + (3*B*a*c^2*d^3*e^2 - 4*A*a*c^2*d^2*e^3 -
B*a^2*c*d*e^4)*x^2 + (3*B*a^2*c*d^2*e^3 - 4*A*a^2*c*d*e^4 - B*a^3*e^5)*x)*log(c*
x^2 + a) + 2*(3*B*a^2*c*d^3*e^2 - 4*A*a^2*c*d^2*e^3 - B*a^3*d*e^4 + (3*B*a*c^2*d
^2*e^3 - 4*A*a*c^2*d*e^4 - B*a^2*c*e^5)*x^3 + (3*B*a*c^2*d^3*e^2 - 4*A*a*c^2*d^2
*e^3 - B*a^2*c*d*e^4)*x^2 + (3*B*a^2*c*d^2*e^3 - 4*A*a^2*c*d*e^4 - B*a^3*e^5)*x)
*log(e*x + d))/(a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*e^6 +
(a*c^4*d^6*e + 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a*c^4*d
^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e +
3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29152, size = 667, normalized size = 2.3 \[ \frac{{\left (A c^{3} d^{4} e^{2} - 2 \, B a c^{2} d^{3} e^{3} + 6 \, A a c^{2} d^{2} e^{4} + 6 \, B a^{2} c d e^{5} - 3 \, A a^{2} c e^{6}\right )} \arctan \left (\frac{{\left (c d - \frac{c d^{2}}{x e + d} - \frac{a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{a c}}\right ) e^{\left (-2\right )}}{2 \,{\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} \sqrt{a c}} + \frac{{\left (3 \, B c d^{2} e^{2} - 4 \, A c d e^{3} - B a e^{4}\right )}{\rm ln}\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac{\frac{B d e^{6}}{x e + d} - \frac{A e^{7}}{x e + d}}{c^{2} d^{4} e^{4} + 2 \, a c d^{2} e^{6} + a^{2} e^{8}} + \frac{\frac{A c^{3} d^{3} e + 3 \, B a c^{2} d^{2} e^{2} - 3 \, A a c^{2} d e^{3} - B a^{2} c e^{4}}{c d^{2} + a e^{2}} - \frac{{\left (A c^{3} d^{4} e^{2} + 4 \, B a c^{2} d^{3} e^{3} - 6 \, A a c^{2} d^{2} e^{4} - 4 \, B a^{2} c d e^{5} + A a^{2} c e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} + a e^{2}\right )}{\left (x e + d\right )}}}{2 \,{\left (c d^{2} + a e^{2}\right )}^{2} a{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(A*c^3*d^4*e^2 - 2*B*a*c^2*d^3*e^3 + 6*A*a*c^2*d^2*e^4 + 6*B*a^2*c*d*e^5 - 3
*A*a^2*c*e^6)*arctan((c*d - c*d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-1)/sqrt(a*c))
*e^(-2)/((a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(a*c))
+ 1/2*(3*B*c*d^2*e^2 - 4*A*c*d*e^3 - B*a*e^4)*ln(c - 2*c*d/(x*e + d) + c*d^2/(x*
e + d)^2 + a*e^2/(x*e + d)^2)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3
*e^6) + (B*d*e^6/(x*e + d) - A*e^7/(x*e + d))/(c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2
*e^8) + 1/2*((A*c^3*d^3*e + 3*B*a*c^2*d^2*e^2 - 3*A*a*c^2*d*e^3 - B*a^2*c*e^4)/(
c*d^2 + a*e^2) - (A*c^3*d^4*e^2 + 4*B*a*c^2*d^3*e^3 - 6*A*a*c^2*d^2*e^4 - 4*B*a^
2*c*d*e^5 + A*a^2*c*e^6)*e^(-1)/((c*d^2 + a*e^2)*(x*e + d)))/((c*d^2 + a*e^2)^2*
a*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2))