3.1353 \(\int \frac{A+B x}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=443 \[ -\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (-5 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{x \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (3 a e^2+c d^2\right )\right )+a e \left (-5 a A e^2+6 a B d e+A c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-3 A \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac{a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}+\frac{e^4 \log \left (a+c x^2\right ) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{2 \left (a e^2+c d^2\right )^4}-\frac{e^4 \log (d+e x) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{\left (a e^2+c d^2\right )^4} \]

[Out]

-(e*(2*a*B*d*e*(c*d^2 - 11*a*e^2) - 3*A*(c^2*d^4 + 4*a*c*d^2*e^2 - 5*a^2*e^4)))/
(8*a^2*(c*d^2 + a*e^2)^3*(d + e*x)) - (a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(4*a*(
c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^2) - (a*e*(A*c*d^2 + 6*a*B*d*e - 5*a*A*e^2)
+ (2*a*B*e*(c*d^2 - 2*a*e^2) - 3*A*c*d*(c*d^2 + 3*a*e^2))*x)/(8*a^2*(c*d^2 + a*
e^2)^2*(d + e*x)*(a + c*x^2)) - (Sqrt[c]*(2*a*B*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 -
15*a^2*e^4) - 3*A*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6))*Ar
cTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c*d^2 + a*e^2)^4) - (e^4*(5*B*c*d^2 - 6*A
*c*d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^4 + (e^4*(5*B*c*d^2 - 6*A*c*d*e
- a*B*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

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Rubi [A]  time = 1.53839, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (-5 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{x \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (3 a e^2+c d^2\right )\right )+a e \left (-5 a A e^2+6 a B d e+A c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-3 A \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac{a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}+\frac{e^4 \log \left (a+c x^2\right ) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{2 \left (a e^2+c d^2\right )^4}-\frac{e^4 \log (d+e x) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{\left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^2*(a + c*x^2)^3),x]

[Out]

-(e*(2*a*B*d*e*(c*d^2 - 11*a*e^2) - 3*A*(c^2*d^4 + 4*a*c*d^2*e^2 - 5*a^2*e^4)))/
(8*a^2*(c*d^2 + a*e^2)^3*(d + e*x)) - (a*(B*d - A*e) - (A*c*d + a*B*e)*x)/(4*a*(
c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^2) - (a*e*(A*c*d^2 + 6*a*B*d*e - 5*a*A*e^2)
+ (2*a*B*e*(c*d^2 - 2*a*e^2) - 3*A*c*d*(c*d^2 + 3*a*e^2))*x)/(8*a^2*(c*d^2 + a*
e^2)^2*(d + e*x)*(a + c*x^2)) - (Sqrt[c]*(2*a*B*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 -
15*a^2*e^4) - 3*A*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6))*Ar
cTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c*d^2 + a*e^2)^4) - (e^4*(5*B*c*d^2 - 6*A
*c*d*e - a*B*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^4 + (e^4*(5*B*c*d^2 - 6*A*c*d*e
- a*B*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

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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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.996255, size = 378, normalized size = 0.85 \[ \frac{\frac{2 \left (a e^2+c d^2\right )^2 \left (a^2 B e^2-a c (A e (e x-2 d)+B d (d-2 e x))+A c^2 d^2 x\right )}{a \left (a+c x^2\right )^2}+\frac{\left (a e^2+c d^2\right ) \left (4 a^3 B e^4+a^2 c e^2 (A e (16 d-7 e x)-2 B d (6 d-7 e x))-2 a c^2 d^2 e x (B d-6 A e)+3 A c^3 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (2 a B d e \left (15 a^2 e^4-10 a c d^2 e^2-c^2 d^4\right )+3 A \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{a^{5/2}}-4 e^4 \log \left (a+c x^2\right ) \left (a B e^2+6 A c d e-5 B c d^2\right )-\frac{8 e^4 \left (a e^2+c d^2\right ) (A e-B d)}{d+e x}+8 e^4 \log (d+e x) \left (a B e^2+6 A c d e-5 B c d^2\right )}{8 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^2*(a + c*x^2)^3),x]

[Out]

((-8*e^4*(-(B*d) + A*e)*(c*d^2 + a*e^2))/(d + e*x) + ((c*d^2 + a*e^2)*(4*a^3*B*e
^4 + 3*A*c^3*d^4*x - 2*a*c^2*d^2*e*(B*d - 6*A*e)*x + a^2*c*e^2*(-2*B*d*(6*d - 7*
e*x) + A*e*(16*d - 7*e*x))))/(a^2*(a + c*x^2)) + (2*(c*d^2 + a*e^2)^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)^2) +
(Sqrt[c]*(2*a*B*d*e*(-(c^2*d^4) - 10*a*c*d^2*e^2 + 15*a^2*e^4) + 3*A*(c^3*d^6 +
5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^
(5/2) + 8*e^4*(-5*B*c*d^2 + 6*A*c*d*e + a*B*e^2)*Log[d + e*x] - 4*e^4*(-5*B*c*d^
2 + 6*A*c*d*e + a*B*e^2)*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^4)

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Maple [B]  time = 0.036, size = 1422, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^2/(c*x^2+a)^3,x)

[Out]

3/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a^3*e^6+e^6/(a*e^2+c*d^2)^4*ln(e*x+d)*a*B-1/4*
c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*d^6+e^4/(a*e^2+c*d^2)^3/(e*x+d)*B*d-1/4*c^3/(a
*e^2+c*d^2)^4/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*d^5*e-1/4*c^4/(a*e^2+c*d^2
)^4/(c*x^2+a)^2/a*x^3*B*d^5*e+2*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*A*a*d*e^5-c^
2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*B*a*d^2*e^4+3/8*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^
2*a*x*A*d^2*e^4+5/2*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x*B*d^3*e^3+9/4*c/(a*e^2+c
*d^2)^4/(c*x^2+a)^2*a^2*x*B*d*e^5+15/4*c/(a*e^2+c*d^2)^4*a/(a*c)^(1/2)*arctan(c*
x/(a*c)^(1/2))*B*d*e^5+15/8*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x^3*A*d^4*e^2+7/4*
c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x^3*B*d*e^5-e^5/(a*e^2+c*d^2)^3/(e*x+d)*A+1/2*
c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*B*a^2*e^6-9/8*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a^
2*x*A*e^6+5/2*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*d*a^2*e^5-7/8*c^2/(a*e^2+c*d^2)^4/
(c*x^2+a)^2*a*x^3*A*e^6+1/2*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*d^5*e-3*c/(a*e^2+c
*d^2)^4*ln(a^2*(c*x^2+a))*A*d*e^5-5*e^4/(a*e^2+c*d^2)^4*ln(e*x+d)*B*c*d^2+6*e^5/
(a*e^2+c*d^2)^4*ln(e*x+d)*A*c*d+5/2*c/(a*e^2+c*d^2)^4*ln(a^2*(c*x^2+a))*B*d^2*e^
4-1/2/(a*e^2+c*d^2)^4*a*ln(a^2*(c*x^2+a))*B*e^6+3/8*c^5/(a*e^2+c*d^2)^4/(c*x^2+a
)^2/a^2*x^3*A*d^6+2*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*A*d^3*e^3-3/2*c^3/(a*e^2
+c*d^2)^4/(c*x^2+a)^2*x^2*B*d^4*e^2+5/8*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x*A*d^
6+3*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*d^3*a*e^3-7/4*c^2/(a*e^2+c*d^2)^4/(c*x^2+a
)^2*B*a*d^4*e^2+5/8*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^3*A*d^2*e^4+3/2*c^3/(a*e^2
+c*d^2)^4/(c*x^2+a)^2*x^3*B*d^3*e^3-15/8*c/(a*e^2+c*d^2)^4*a/(a*c)^(1/2)*arctan(
c*x/(a*c)^(1/2))*A*e^6+3/8*c^4/(a*e^2+c*d^2)^4/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^
(1/2))*A*d^6+17/8*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x*A*d^4*e^2+1/4*c^3/(a*e^2+c*d
^2)^4/(c*x^2+a)^2*x*B*d^5*e+45/8*c^2/(a*e^2+c*d^2)^4/(a*c)^(1/2)*arctan(c*x/(a*c
)^(1/2))*A*d^2*e^4-5/2*c^2/(a*e^2+c*d^2)^4/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B
*d^3*e^3-3/4*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a^2*d^2*e^4+15/8*c^3/(a*e^2+c*d^2)^
4/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^4*e^2

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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)^3*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)/((c*x^2 + a)^3*(e*x + d)^2),x, algorithm="fricas")

[Out]

Timed out

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.314842, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^3*(e*x + d)^2),x, algorithm="giac")

[Out]

Done