3.1340 \(\int \frac{(A+B x) (d+e x)^3}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=161 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \log \left (a+c x^2\right ) (A e+3 B d)}{2 c^2}-\frac{e^2 x (A c d-3 a B e)}{2 a c^2}-\frac{(d+e x)^2 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

[Out]

-(e^2*(A*c*d - 3*a*B*e)*x)/(2*a*c^2) - ((d + e*x)^2*(a*(B*d + A*e) - (A*c*d - a*
B*e)*x))/(2*a*c*(a + c*x^2)) + ((3*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)*c^(5/2)) + (e^2*(3*B*d + A*e)*Log[a
+ c*x^2])/(2*c^2)

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Rubi [A]  time = 0.45644, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \log \left (a+c x^2\right ) (A e+3 B d)}{2 c^2}-\frac{e^2 x (A c d-3 a B e)}{2 a c^2}-\frac{(d+e x)^2 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*(A*c*d - 3*a*B*e)*x)/(2*a*c^2) - ((d + e*x)^2*(a*(B*d + A*e) - (A*c*d - a*
B*e)*x))/(2*a*c*(a + c*x^2)) + ((3*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)*c^(5/2)) + (e^2*(3*B*d + A*e)*Log[a
+ c*x^2])/(2*c^2)

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Rubi in Sympy [A]  time = 107.708, size = 160, normalized size = 0.99 \[ \frac{e^{2} \left (A e + 3 B d\right ) \log{\left (a + c x^{2} \right )}}{2 c^{2}} - \frac{\left (d + e x\right )^{2} \left (2 a \left (A e + B d\right ) - x \left (2 A c d - 2 B a e\right )\right )}{4 a c \left (a + c x^{2}\right )} - \frac{e^{2} x \left (A c d - 3 B a e\right )}{2 a c^{2}} + \frac{\left (3 A a c d e^{2} + A c^{2} d^{3} - 3 B a^{2} e^{3} + 3 B a c d^{2} e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.221674, size = 171, normalized size = 1.06 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )}{a^{3/2}}+\frac{\sqrt{c} \left (a^2 e^2 (A e+3 B d+B e x)-a c d (3 A e (d+e x)+B d (d+3 e x))+A c^2 d^3 x\right )}{a \left (a+c x^2\right )}+\sqrt{c} e^2 \log \left (a+c x^2\right ) (A e+3 B d)+2 B \sqrt{c} e^3 x}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 300, normalized size = 1.9 \[{\frac{B{e}^{3}x}{{c}^{2}}}-{\frac{3\,xAd{e}^{2}}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{3}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}+{\frac{aBx{e}^{3}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{3\,Bx{d}^{2}e}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{aA{e}^{3}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{3\,A{d}^{2}e}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{3\,aBd{e}^{2}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{B{d}^{3}}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) A{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) Bd{e}^{2}}{2\,{c}^{2}}}+{\frac{3\,Ad{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{d}^{3}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{3\,B{e}^{3}a}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{d}^{2}e}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292368, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (A a c^{2} d^{3} + 3 \, B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} - 3 \, B a^{3} e^{3} +{\left (A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} - 3 \, B a^{2} c e^{3}\right )} x^{2}\right )} \log \left (-\frac{2 \, a c x -{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (2 \, B a c e^{3} x^{3} - B a c d^{3} - 3 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3} +{\left (A c^{2} d^{3} - 3 \, B a c d^{2} e - 3 \, A a c d e^{2} + 3 \, B a^{2} e^{3}\right )} x +{\left (3 \, B a^{2} d e^{2} + A a^{2} e^{3} +{\left (3 \, B a c d e^{2} + A a c e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-a c}}, \frac{{\left (A a c^{2} d^{3} + 3 \, B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} - 3 \, B a^{3} e^{3} +{\left (A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} - 3 \, B a^{2} c e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (2 \, B a c e^{3} x^{3} - B a c d^{3} - 3 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3} +{\left (A c^{2} d^{3} - 3 \, B a c d^{2} e - 3 \, A a c d e^{2} + 3 \, B a^{2} e^{3}\right )} x +{\left (3 \, B a^{2} d e^{2} + A a^{2} e^{3} +{\left (3 \, B a c d e^{2} + A a c e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 25.4015, size = 583, normalized size = 3.62 \[ \frac{B e^{3} x}{c^{2}} + \left (\frac{e^{2} \left (A e + 3 B d\right )}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e\right )}{4 a^{3} c^{5}}\right ) \log{\left (x + \frac{2 A a^{2} e^{3} + 6 B a^{2} d e^{2} - 4 a^{2} c^{2} \left (\frac{e^{2} \left (A e + 3 B d\right )}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e\right )}{4 a^{3} c^{5}}\right )}{- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e} \right )} + \left (\frac{e^{2} \left (A e + 3 B d\right )}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e\right )}{4 a^{3} c^{5}}\right ) \log{\left (x + \frac{2 A a^{2} e^{3} + 6 B a^{2} d e^{2} - 4 a^{2} c^{2} \left (\frac{e^{2} \left (A e + 3 B d\right )}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e\right )}{4 a^{3} c^{5}}\right )}{- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e} \right )} + \frac{A a^{2} e^{3} - 3 A a c d^{2} e + 3 B a^{2} d e^{2} - B a c d^{3} + x \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*e**3*x/c**2 + (e**2*(A*e + 3*B*d)/(2*c**2) - sqrt(-a**3*c**5)*(-3*A*a*c*d*e**2
- A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e)/(4*a**3*c**5))*log(x + (2*A*a**
2*e**3 + 6*B*a**2*d*e**2 - 4*a**2*c**2*(e**2*(A*e + 3*B*d)/(2*c**2) - sqrt(-a**3
*c**5)*(-3*A*a*c*d*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e)/(4*a**3*
c**5)))/(-3*A*a*c*d*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e)) + (e**
2*(A*e + 3*B*d)/(2*c**2) + sqrt(-a**3*c**5)*(-3*A*a*c*d*e**2 - A*c**2*d**3 + 3*B
*a**2*e**3 - 3*B*a*c*d**2*e)/(4*a**3*c**5))*log(x + (2*A*a**2*e**3 + 6*B*a**2*d*
e**2 - 4*a**2*c**2*(e**2*(A*e + 3*B*d)/(2*c**2) + sqrt(-a**3*c**5)*(-3*A*a*c*d*e
**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e)/(4*a**3*c**5)))/(-3*A*a*c*d*
e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e)) + (A*a**2*e**3 - 3*A*a*c*d
**2*e + 3*B*a**2*d*e**2 - B*a*c*d**3 + x*(-3*A*a*c*d*e**2 + A*c**2*d**3 + B*a**2
*e**3 - 3*B*a*c*d**2*e))/(2*a**2*c**2 + 2*a*c**3*x**2)

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GIAC/XCAS [A]  time = 0.298962, size = 242, normalized size = 1.5 \[ \frac{B x e^{3}}{c^{2}} + \frac{{\left (3 \, B d e^{2} + A e^{3}\right )}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{{\left (A c^{2} d^{3} + 3 \, B a c d^{2} e + 3 \, A a c d e^{2} - 3 \, B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} - \frac{B a c d^{3} + 3 \, A a c d^{2} e - 3 \, B a^{2} d e^{2} - A a^{2} e^{3} -{\left (A c^{2} d^{3} - 3 \, B a c d^{2} e - 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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