3.1348 \(\int \frac{(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=125 \[ \frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{8 a^{5/2} c^{5/2}}-\frac{3 (d+e x) (a e-c d x) (a B e+A c d)}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.168976, antiderivative size = 125, 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 \[ \frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{8 a^{5/2} c^{5/2}}-\frac{3 (d+e x) (a e-c d x) (a B e+A c d)}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.9016, size = 116, normalized size = 0.93 \[ - \frac{\left (d + e x\right )^{3} \left (- A c x + B a\right )}{4 a c \left (a + c x^{2}\right )^{2}} - \frac{3 \left (d + e x\right ) \left (a e - c d x\right ) \left (A c d + B a e\right )}{8 a^{2} c^{2} \left (a + c x^{2}\right )} + \frac{3 \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.190277, size = 186, normalized size = 1.49 \[ \frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{8 a^{5/2} c^{5/2}}+\frac{-a^2 e^2 (4 A e+12 B d+5 B e x)+3 a c d e x (A e+B d)+3 A c^2 d^3 x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{a^2 e^2 (A e+3 B d+B e x)-a c d (3 A e (d+e x)+B d (d+3 e x))+A c^2 d^3 x}{4 a c^2 \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*A*c^2*d^3*x + 3*a*c*d*e*(B*d + A*e)*x - a^2*e^2*(12*B*d + 4*A*e + 5*B*e*x))/(
8*a^2*c^2*(a + c*x^2)) + (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)))/(4*a*c^2*(a + c*x^2)^2) + (3*(A*c*d + a*B*e)*
(c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2))

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Maple [B]  time = 0.013, size = 260, normalized size = 2.1 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( 3\,Aacd{e}^{2}+3\,A{d}^{3}{c}^{2}-5\,{a}^{2}B{e}^{3}+3\,Bac{d}^{2}e \right ){x}^{3}}{8\,{a}^{2}c}}-{\frac{{e}^{2} \left ( Ae+3\,Bd \right ){x}^{2}}{2\,c}}-{\frac{ \left ( 3\,Aacd{e}^{2}-5\,A{d}^{3}{c}^{2}+3\,{a}^{2}B{e}^{3}+3\,Bac{d}^{2}e \right ) x}{8\,a{c}^{2}}}-{\frac{aA{e}^{3}+3\,Ac{d}^{2}e+3\,aBd{e}^{2}+Bc{d}^{3}}{4\,{c}^{2}}} \right ) }+{\frac{3\,Ad{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{3}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{e}^{3}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{d}^{2}e}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/8*(3*A*a*c*d*e^2+3*A*c^2*d^3-5*B*a^2*e^3+3*B*a*c*d^2*e)/a^2/c*x^3-1/2*e^2*(A*
e+3*B*d)*x^2/c-1/8*(3*A*a*c*d*e^2-5*A*c^2*d^3+3*B*a^2*e^3+3*B*a*c*d^2*e)/a/c^2*x
-1/4*(A*a*e^3+3*A*c*d^2*e+3*B*a*d*e^2+B*c*d^3)/c^2)/(c*x^2+a)^2+3/8/a/c/(a*c)^(1
/2)*arctan(c*x/(a*c)^(1/2))*A*d*e^2+3/8/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*
A*d^3+3/8/c^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*e^3+3/8/a/c/(a*c)^(1/2)*arct
an(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303531, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (A a^{2} c^{2} d^{3} + B a^{3} c d^{2} e + A a^{3} c d e^{2} + B a^{4} e^{3} +{\left (A c^{4} d^{3} + B a c^{3} d^{2} e + A a c^{3} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{4} + 2 \,{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (2 \, B a^{2} c d^{3} + 6 \, A a^{2} c d^{2} e + 6 \, B a^{3} d e^{2} + 2 \, A a^{3} e^{3} -{\left (3 \, A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} - 5 \, B a^{2} c e^{3}\right )} x^{3} + 4 \,{\left (3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{2} -{\left (5 \, A a c^{2} d^{3} - 3 \, B a^{2} c d^{2} e - 3 \, A a^{2} c d e^{2} - 3 \, B a^{3} e^{3}\right )} x\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{-a c}}, \frac{3 \,{\left (A a^{2} c^{2} d^{3} + B a^{3} c d^{2} e + A a^{3} c d e^{2} + B a^{4} e^{3} +{\left (A c^{4} d^{3} + B a c^{3} d^{2} e + A a c^{3} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{4} + 2 \,{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (2 \, B a^{2} c d^{3} + 6 \, A a^{2} c d^{2} e + 6 \, B a^{3} d e^{2} + 2 \, A a^{3} e^{3} -{\left (3 \, A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} - 5 \, B a^{2} c e^{3}\right )} x^{3} + 4 \,{\left (3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{2} -{\left (5 \, A a c^{2} d^{3} - 3 \, B a^{2} c d^{2} e - 3 \, A a^{2} c d e^{2} - 3 \, B a^{3} e^{3}\right )} x\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(A*a^2*c^2*d^3 + B*a^3*c*d^2*e + A*a^3*c*d*e^2 + B*a^4*e^3 + (A*c^4*d^3
+ B*a*c^3*d^2*e + A*a*c^3*d*e^2 + B*a^2*c^2*e^3)*x^4 + 2*(A*a*c^3*d^3 + B*a^2*c
^2*d^2*e + A*a^2*c^2*d*e^2 + B*a^3*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^2*c*d^3 + 6*A*a^2*c*d^2*e + 6*B*a^3*d*e^2 + 2*A*a^
3*e^3 - (3*A*c^3*d^3 + 3*B*a*c^2*d^2*e + 3*A*a*c^2*d*e^2 - 5*B*a^2*c*e^3)*x^3 +
4*(3*B*a^2*c*d*e^2 + A*a^2*c*e^3)*x^2 - (5*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)*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^3 + B*a^3*c*d^2*e + A*a^3*c*d*e^2 + B*a^4*e^3
+ (A*c^4*d^3 + B*a*c^3*d^2*e + A*a*c^3*d*e^2 + B*a^2*c^2*e^3)*x^4 + 2*(A*a*c^3*
d^3 + B*a^2*c^2*d^2*e + A*a^2*c^2*d*e^2 + B*a^3*c*e^3)*x^2)*arctan(sqrt(a*c)*x/a
) - (2*B*a^2*c*d^3 + 6*A*a^2*c*d^2*e + 6*B*a^3*d*e^2 + 2*A*a^3*e^3 - (3*A*c^3*d^
3 + 3*B*a*c^2*d^2*e + 3*A*a*c^2*d*e^2 - 5*B*a^2*c*e^3)*x^3 + 4*(3*B*a^2*c*d*e^2
+ A*a^2*c*e^3)*x^2 - (5*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)*x)*sqrt(a*c))/((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(a*c))]

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Sympy [A]  time = 92.4157, size = 466, normalized size = 3.73 \[ - \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \log{\left (- \frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right )}{3 A a c d e^{2} + 3 A c^{2} d^{3} + 3 B a^{2} e^{3} + 3 B a c d^{2} e} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \log{\left (\frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right )}{3 A a c d e^{2} + 3 A c^{2} d^{3} + 3 B a^{2} e^{3} + 3 B a c d^{2} e} + x \right )}}{16} - \frac{2 A a^{3} e^{3} + 6 A a^{2} c d^{2} e + 6 B a^{3} d e^{2} + 2 B a^{2} c d^{3} + x^{3} \left (- 3 A a c^{2} d e^{2} - 3 A c^{3} d^{3} + 5 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e\right ) + x^{2} \left (4 A a^{2} c e^{3} + 12 B a^{2} c d e^{2}\right ) + x \left (3 A a^{2} c d e^{2} - 5 A a c^{2} d^{3} + 3 B a^{3} e^{3} + 3 B a^{2} c d^{2} e\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)*(A*c*d + B*a*e)*log(-3*a**3*c**2*sqrt(
-1/(a**5*c**5))*(a*e**2 + c*d**2)*(A*c*d + B*a*e)/(3*A*a*c*d*e**2 + 3*A*c**2*d**
3 + 3*B*a**2*e**3 + 3*B*a*c*d**2*e) + x)/16 + 3*sqrt(-1/(a**5*c**5))*(a*e**2 + c
*d**2)*(A*c*d + B*a*e)*log(3*a**3*c**2*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)*(A
*c*d + B*a*e)/(3*A*a*c*d*e**2 + 3*A*c**2*d**3 + 3*B*a**2*e**3 + 3*B*a*c*d**2*e)
+ x)/16 - (2*A*a**3*e**3 + 6*A*a**2*c*d**2*e + 6*B*a**3*d*e**2 + 2*B*a**2*c*d**3
+ x**3*(-3*A*a*c**2*d*e**2 - 3*A*c**3*d**3 + 5*B*a**2*c*e**3 - 3*B*a*c**2*d**2*
e) + x**2*(4*A*a**2*c*e**3 + 12*B*a**2*c*d*e**2) + x*(3*A*a**2*c*d*e**2 - 5*A*a*
c**2*d**3 + 3*B*a**3*e**3 + 3*B*a**2*c*d**2*e))/(8*a**4*c**2 + 16*a**3*c**3*x**2
+ 8*a**2*c**4*x**4)

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GIAC/XCAS [A]  time = 0.318325, size = 315, normalized size = 2.52 \[ \frac{3 \,{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{3 \, A c^{3} d^{3} x^{3} + 3 \, B a c^{2} d^{2} x^{3} e + 3 \, A a c^{2} d x^{3} e^{2} + 5 \, A a c^{2} d^{3} x - 5 \, B a^{2} c x^{3} e^{3} - 12 \, B a^{2} c d x^{2} e^{2} - 3 \, B a^{2} c d^{2} x e - 2 \, B a^{2} c d^{3} - 4 \, A a^{2} c x^{2} e^{3} - 3 \, A a^{2} c d x e^{2} - 6 \, A a^{2} c d^{2} e - 3 \, B a^{3} x e^{3} - 6 \, B a^{3} d e^{2} - 2 \, A a^{3} e^{3}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*(A*c^2*d^3 + B*a*c*d^2*e + A*a*c*d*e^2 + B*a^2*e^3)*arctan(c*x/sqrt(a*c))/(s
qrt(a*c)*a^2*c^2) + 1/8*(3*A*c^3*d^3*x^3 + 3*B*a*c^2*d^2*x^3*e + 3*A*a*c^2*d*x^3
*e^2 + 5*A*a*c^2*d^3*x - 5*B*a^2*c*x^3*e^3 - 12*B*a^2*c*d*x^2*e^2 - 3*B*a^2*c*d^
2*x*e - 2*B*a^2*c*d^3 - 4*A*a^2*c*x^2*e^3 - 3*A*a^2*c*d*x*e^2 - 6*A*a^2*c*d^2*e
- 3*B*a^3*x*e^3 - 6*B*a^3*d*e^2 - 2*A*a^3*e^3)/((c*x^2 + a)^2*a^2*c^2)