3.1349 \(\int \frac{(A+B x) (d+e x)^2}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=142 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a A e^2+2 a B d e+3 A c d^2\right )}{8 a^{5/2} c^{3/2}}-\frac{2 a e (a B e+2 A c d)-c x \left (-a A e^2+2 a B d e+3 A c d^2\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^2 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-((a*B - A*c*x)*(d + e*x)^2)/(4*a*c*(a + c*x^2)^2) - (2*a*e*(2*A*c*d + a*B*e) -
c*(3*A*c*d^2 + 2*a*B*d*e - a*A*e^2)*x)/(8*a^2*c^2*(a + c*x^2)) + ((3*A*c*d^2 + 2
*a*B*d*e + a*A*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.293111, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a A e^2+2 a B d e+3 A c d^2\right )}{8 a^{5/2} c^{3/2}}-\frac{2 a e (a B e+2 A c d)-c x \left (-a A e^2+2 a B d e+3 A c d^2\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^2 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a*B - A*c*x)*(d + e*x)^2)/(4*a*c*(a + c*x^2)^2) - (2*a*e*(2*A*c*d + a*B*e) -
c*(3*A*c*d^2 + 2*a*B*d*e - a*A*e^2)*x)/(8*a^2*c^2*(a + c*x^2)) + ((3*A*c*d^2 + 2
*a*B*d*e + a*A*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.4093, size = 119, normalized size = 0.84 \[ - \frac{\left (d + e x\right )^{2} \left (- A c x + B a\right )}{4 a c \left (a + c x^{2}\right )^{2}} - \frac{\left (a e - c d x\right ) \left (3 A c d + A c e x + 2 B a e\right )}{8 a^{2} c^{2} \left (a + c x^{2}\right )} + \frac{\left (A a e^{2} + d \left (3 A c d + 2 B a e\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**2*(-A*c*x + B*a)/(4*a*c*(a + c*x**2)**2) - (a*e - c*d*x)*(3*A*c*d +
A*c*e*x + 2*B*a*e)/(8*a**2*c**2*(a + c*x**2)) + (A*a*e**2 + d*(3*A*c*d + 2*B*a*e
))*atan(sqrt(c)*x/sqrt(a))/(8*a**(5/2)*c**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.200165, size = 158, normalized size = 1.11 \[ \frac{\frac{\sqrt{a} \left (-4 a^2 B e^2+a c e x (A e+2 B d)+3 A c^2 d^2 x\right )}{a+c x^2}+\frac{2 a^{3/2} \left (a^2 B e^2-a c (A e (2 d+e x)+B d (d+2 e x))+A c^2 d^2 x\right )}{\left (a+c x^2\right )^2}+\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a A e^2+2 a B d e+3 A c d^2\right )}{8 a^{5/2} c^2} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(-4*a^2*B*e^2 + 3*A*c^2*d^2*x + a*c*e*(2*B*d + A*e)*x))/(a + c*x^2) +
(2*a^(3/2)*(a^2*B*e^2 + A*c^2*d^2*x - a*c*(A*e*(2*d + e*x) + B*d*(d + 2*e*x))))/
(a + c*x^2)^2 + Sqrt[c]*(3*A*c*d^2 + 2*a*B*d*e + a*A*e^2)*ArcTan[(Sqrt[c]*x)/Sqr
t[a]])/(8*a^(5/2)*c^2)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 180, normalized size = 1.3 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Aa{e}^{2}+3\,Ac{d}^{2}+2\,aBde \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{B{e}^{2}{x}^{2}}{2\,c}}-{\frac{ \left ( Aa{e}^{2}-5\,Ac{d}^{2}+2\,aBde \right ) x}{8\,ac}}-{\frac{2\,Acde+aB{e}^{2}+Bc{d}^{2}}{4\,{c}^{2}}} \right ) }+{\frac{A{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{2}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Bde}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.279237, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, A a^{2} c^{2} d^{2} + 2 \, B a^{3} c d e + A a^{3} c e^{2} +{\left (3 \, A c^{4} d^{2} + 2 \, B a c^{3} d e + A a c^{3} e^{2}\right )} x^{4} + 2 \,{\left (3 \, A a c^{3} d^{2} + 2 \, B a^{2} c^{2} d e + A a^{2} c^{2} e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (4 \, B a^{2} c e^{2} x^{2} + 2 \, B a^{2} c d^{2} + 4 \, A a^{2} c d e + 2 \, B a^{3} e^{2} -{\left (3 \, A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{3} -{\left (5 \, A a c^{2} d^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{-a c}}, \frac{{\left (3 \, A a^{2} c^{2} d^{2} + 2 \, B a^{3} c d e + A a^{3} c e^{2} +{\left (3 \, A c^{4} d^{2} + 2 \, B a c^{3} d e + A a c^{3} e^{2}\right )} x^{4} + 2 \,{\left (3 \, A a c^{3} d^{2} + 2 \, B a^{2} c^{2} d e + A a^{2} c^{2} e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (4 \, B a^{2} c e^{2} x^{2} + 2 \, B a^{2} c d^{2} + 4 \, A a^{2} c d e + 2 \, B a^{3} e^{2} -{\left (3 \, A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{3} -{\left (5 \, A a c^{2} d^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{a c}}\right ] \]

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, algorithm="fricas")

[Out]

[1/16*((3*A*a^2*c^2*d^2 + 2*B*a^3*c*d*e + A*a^3*c*e^2 + (3*A*c^4*d^2 + 2*B*a*c^3
*d*e + A*a*c^3*e^2)*x^4 + 2*(3*A*a*c^3*d^2 + 2*B*a^2*c^2*d*e + A*a^2*c^2*e^2)*x^
2)*log((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(4*B*a^2*c*e^2*x^2 +
2*B*a^2*c*d^2 + 4*A*a^2*c*d*e + 2*B*a^3*e^2 - (3*A*c^3*d^2 + 2*B*a*c^2*d*e + A*a
*c^2*e^2)*x^3 - (5*A*a*c^2*d^2 - 2*B*a^2*c*d*e - A*a^2*c*e^2)*x)*sqrt(-a*c))/((a
^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-a*c)), 1/8*((3*A*a^2*c^2*d^2 + 2*B*a
^3*c*d*e + A*a^3*c*e^2 + (3*A*c^4*d^2 + 2*B*a*c^3*d*e + A*a*c^3*e^2)*x^4 + 2*(3*
A*a*c^3*d^2 + 2*B*a^2*c^2*d*e + A*a^2*c^2*e^2)*x^2)*arctan(sqrt(a*c)*x/a) - (4*B
*a^2*c*e^2*x^2 + 2*B*a^2*c*d^2 + 4*A*a^2*c*d*e + 2*B*a^3*e^2 - (3*A*c^3*d^2 + 2*
B*a*c^2*d*e + A*a*c^2*e^2)*x^3 - (5*A*a*c^2*d^2 - 2*B*a^2*c*d*e - A*a^2*c*e^2)*x
)*sqrt(a*c))/((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(a*c))]

_______________________________________________________________________________________

Sympy [A]  time = 32.9158, size = 274, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (A a e^{2} + 3 A c d^{2} + 2 B a d e\right ) \log{\left (- a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (A a e^{2} + 3 A c d^{2} + 2 B a d e\right ) \log{\left (a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{- 4 A a^{2} c d e - 2 B a^{3} e^{2} - 2 B a^{2} c d^{2} - 4 B a^{2} c e^{2} x^{2} + x^{3} \left (A a c^{2} e^{2} + 3 A c^{3} d^{2} + 2 B a c^{2} d e\right ) + x \left (- A a^{2} c e^{2} + 5 A a c^{2} d^{2} - 2 B a^{2} c d e\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \]

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]

-sqrt(-1/(a**5*c**3))*(A*a*e**2 + 3*A*c*d**2 + 2*B*a*d*e)*log(-a**3*c*sqrt(-1/(a
**5*c**3)) + x)/16 + sqrt(-1/(a**5*c**3))*(A*a*e**2 + 3*A*c*d**2 + 2*B*a*d*e)*lo
g(a**3*c*sqrt(-1/(a**5*c**3)) + x)/16 + (-4*A*a**2*c*d*e - 2*B*a**3*e**2 - 2*B*a
**2*c*d**2 - 4*B*a**2*c*e**2*x**2 + x**3*(A*a*c**2*e**2 + 3*A*c**3*d**2 + 2*B*a*
c**2*d*e) + x*(-A*a**2*c*e**2 + 5*A*a*c**2*d**2 - 2*B*a**2*c*d*e))/(8*a**4*c**2
+ 16*a**3*c**3*x**2 + 8*a**2*c**4*x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.3038, size = 228, normalized size = 1.61 \[ \frac{{\left (3 \, A c d^{2} + 2 \, B a d e + A a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{3 \, A c^{3} d^{2} x^{3} + 2 \, B a c^{2} d x^{3} e + A a c^{2} x^{3} e^{2} + 5 \, A a c^{2} d^{2} x - 4 \, B a^{2} c x^{2} e^{2} - 2 \, B a^{2} c d x e - 2 \, B a^{2} c d^{2} - A a^{2} c x e^{2} - 4 \, A a^{2} c d e - 2 \, B a^{3} e^{2}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]

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, algorithm="giac")

[Out]

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