3.1347 \(\int \frac{(A+B x) (d+e x)^4}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=216 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}+\frac{B e^4 \log \left (a+c x^2\right )}{2 c^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.500055, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x) \left (x \left (4 a^2 B e^3-c d \left (a e (3 A e+4 B d)+3 A c d^2\right )\right )+a e \left (3 A \left (a e^2+c d^2\right )+8 a B d e\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2}+\frac{B e^4 \log \left (a+c x^2\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a + c*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 115.492, size = 236, normalized size = 1.09 \[ \frac{B e^{4} \log{\left (a + c x^{2} \right )}}{2 c^{3}} - \frac{\left (d + e x\right )^{3} \left (2 a \left (A e + B d\right ) - x \left (2 A c d - 2 B a e\right )\right )}{8 a c \left (a + c x^{2}\right )^{2}} - \frac{\left (d + e x\right ) \left (4 a e \left (3 A a e^{2} + 3 A c d^{2} + 8 B a d e\right ) + x \left (16 B a^{2} e^{3} - 4 c d \left (3 A c d^{2} + a e \left (3 A e + 4 B d\right )\right )\right )\right )}{32 a^{2} c^{2} \left (a + c x^{2}\right )} + \frac{\left (3 A a^{2} e^{4} + 6 A a c d^{2} e^{2} + 3 A c^{2} d^{4} + 12 B a^{2} d e^{3} + 4 B a c d^{3} e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(c*x**2+a)**3,x)

[Out]

B*e**4*log(a + c*x**2)/(2*c**3) - (d + e*x)**3*(2*a*(A*e + B*d) - x*(2*A*c*d - 2
*B*a*e))/(8*a*c*(a + c*x**2)**2) - (d + e*x)*(4*a*e*(3*A*a*e**2 + 3*A*c*d**2 + 8
*B*a*d*e) + x*(16*B*a**2*e**3 - 4*c*d*(3*A*c*d**2 + a*e*(3*A*e + 4*B*d))))/(32*a
**2*c**2*(a + c*x**2)) + (3*A*a**2*e**4 + 6*A*a*c*d**2*e**2 + 3*A*c**2*d**4 + 12
*B*a**2*d*e**3 + 4*B*a*c*d**3*e)*atan(sqrt(c)*x/sqrt(a))/(8*a**(5/2)*c**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.390246, size = 263, normalized size = 1.22 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A \left (a e^2+c d^2\right )^2+4 a B d e \left (3 a e^2+c d^2\right )\right )}{a^{5/2}}+\frac{-2 a^3 B e^4+2 a^2 c e^2 (A e (4 d+e x)+2 B d (3 d+2 e x))-2 a c^2 d^2 (2 A e (2 d+3 e x)+B d (d+4 e x))+2 A c^3 d^4 x}{a \left (a+c x^2\right )^2}+\frac{8 a^3 B e^4-a^2 c e^2 (A e (16 d+5 e x)+4 B d (6 d+5 e x))+2 a c^2 d^2 e x (3 A e+2 B d)+3 A c^3 d^4 x}{a^2 \left (a+c x^2\right )}+4 B e^4 \log \left (a+c x^2\right )}{8 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(a + c*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 366, normalized size = 1.7 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 5\,A{a}^{2}{e}^{4}-6\,Aac{d}^{2}{e}^{2}-3\,A{d}^{4}{c}^{2}+20\,B{a}^{2}d{e}^{3}-4\,Bac{d}^{3}e \right ){x}^{3}}{8\,{a}^{2}c}}-{\frac{{e}^{2} \left ( 2\,Acde-aB{e}^{2}+3\,Bc{d}^{2} \right ){x}^{2}}{{c}^{2}}}-{\frac{ \left ( 3\,A{a}^{2}{e}^{4}+6\,Aac{d}^{2}{e}^{2}-5\,A{d}^{4}{c}^{2}+12\,B{a}^{2}d{e}^{3}+4\,Bac{d}^{3}e \right ) x}{8\,a{c}^{2}}}-{\frac{4\,Aacd{e}^{3}+4\,A{c}^{2}{d}^{3}e-3\,B{e}^{4}{a}^{2}+6\,Bac{d}^{2}{e}^{2}+B{c}^{2}{d}^{4}}{4\,{c}^{3}}} \right ) }+{\frac{B{e}^{4}\ln \left ({a}^{2}{c}^{2} \left ( c{x}^{2}+a \right ) \right ) }{2\,{c}^{3}}}+{\frac{3\,A{e}^{4}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{2}{e}^{2}}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{4}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,Bd{e}^{3}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{B{d}^{3}e}{2\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(c*x^2+a)^3,x)

[Out]

(-1/8*(5*A*a^2*e^4-6*A*a*c*d^2*e^2-3*A*c^2*d^4+20*B*a^2*d*e^3-4*B*a*c*d^3*e)/a^2
/c*x^3-e^2*(2*A*c*d*e-B*a*e^2+3*B*c*d^2)/c^2*x^2-1/8*(3*A*a^2*e^4+6*A*a*c*d^2*e^
2-5*A*c^2*d^4+12*B*a^2*d*e^3+4*B*a*c*d^3*e)/a/c^2*x-1/4*(4*A*a*c*d*e^3+4*A*c^2*d
^3*e-3*B*a^2*e^4+6*B*a*c*d^2*e^2+B*c^2*d^4)/c^3)/(c*x^2+a)^2+1/2*B*e^4/c^3*ln(a^
2*c^2*(c*x^2+a))+3/8/c^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*e^4+3/4/a/c/(a*c)
^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^2*e^2+3/8/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1
/2))*A*d^4+3/2/c^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*d*e^3+1/2/a/c/(a*c)^(1/
2)*arctan(c*x/(a*c)^(1/2))*B*d^3*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)^4/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.301714, 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)*(e*x + d)^4/(c*x^2 + a)^3,x, algorithm="fricas")

[Out]

[1/16*((3*A*a^2*c^3*d^4 + 4*B*a^3*c^2*d^3*e + 6*A*a^3*c^2*d^2*e^2 + 12*B*a^4*c*d
*e^3 + 3*A*a^4*c*e^4 + (3*A*c^5*d^4 + 4*B*a*c^4*d^3*e + 6*A*a*c^4*d^2*e^2 + 12*B
*a^2*c^3*d*e^3 + 3*A*a^2*c^3*e^4)*x^4 + 2*(3*A*a*c^4*d^4 + 4*B*a^2*c^3*d^3*e + 6
*A*a^2*c^3*d^2*e^2 + 12*B*a^3*c^2*d*e^3 + 3*A*a^3*c^2*e^4)*x^2)*log((2*a*c*x + (
c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(2*B*a^2*c^2*d^4 + 8*A*a^2*c^2*d^3*e + 1
2*B*a^3*c*d^2*e^2 + 8*A*a^3*c*d*e^3 - 6*B*a^4*e^4 - (3*A*c^4*d^4 + 4*B*a*c^3*d^3
*e + 6*A*a*c^3*d^2*e^2 - 20*B*a^2*c^2*d*e^3 - 5*A*a^2*c^2*e^4)*x^3 + 8*(3*B*a^2*
c^2*d^2*e^2 + 2*A*a^2*c^2*d*e^3 - B*a^3*c*e^4)*x^2 - (5*A*a*c^3*d^4 - 4*B*a^2*c^
2*d^3*e - 6*A*a^2*c^2*d^2*e^2 - 12*B*a^3*c*d*e^3 - 3*A*a^3*c*e^4)*x - 4*(B*a^2*c
^2*e^4*x^4 + 2*B*a^3*c*e^4*x^2 + B*a^4*e^4)*log(c*x^2 + a))*sqrt(-a*c))/((a^2*c^
5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)*sqrt(-a*c)), 1/8*((3*A*a^2*c^3*d^4 + 4*B*a^3*c^
2*d^3*e + 6*A*a^3*c^2*d^2*e^2 + 12*B*a^4*c*d*e^3 + 3*A*a^4*c*e^4 + (3*A*c^5*d^4
+ 4*B*a*c^4*d^3*e + 6*A*a*c^4*d^2*e^2 + 12*B*a^2*c^3*d*e^3 + 3*A*a^2*c^3*e^4)*x^
4 + 2*(3*A*a*c^4*d^4 + 4*B*a^2*c^3*d^3*e + 6*A*a^2*c^3*d^2*e^2 + 12*B*a^3*c^2*d*
e^3 + 3*A*a^3*c^2*e^4)*x^2)*arctan(sqrt(a*c)*x/a) - (2*B*a^2*c^2*d^4 + 8*A*a^2*c
^2*d^3*e + 12*B*a^3*c*d^2*e^2 + 8*A*a^3*c*d*e^3 - 6*B*a^4*e^4 - (3*A*c^4*d^4 + 4
*B*a*c^3*d^3*e + 6*A*a*c^3*d^2*e^2 - 20*B*a^2*c^2*d*e^3 - 5*A*a^2*c^2*e^4)*x^3 +
8*(3*B*a^2*c^2*d^2*e^2 + 2*A*a^2*c^2*d*e^3 - B*a^3*c*e^4)*x^2 - (5*A*a*c^3*d^4
- 4*B*a^2*c^2*d^3*e - 6*A*a^2*c^2*d^2*e^2 - 12*B*a^3*c*d*e^3 - 3*A*a^3*c*e^4)*x
- 4*(B*a^2*c^2*e^4*x^4 + 2*B*a^3*c*e^4*x^2 + B*a^4*e^4)*log(c*x^2 + a))*sqrt(a*c
))/((a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)*sqrt(a*c))]

_______________________________________________________________________________________

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)**4/(c*x**2+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.301292, size = 421, normalized size = 1.95 \[ \frac{B e^{4}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (3 \, A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} + 12 \, B a^{2} d e^{3} + 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{{\left (3 \, A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 20 \, B a^{2} c d e^{3} - 5 \, A a^{2} c e^{4}\right )} x^{3} - 8 \,{\left (3 \, B a^{2} c d^{2} e^{2} + 2 \, A a^{2} c d e^{3} - B a^{3} e^{4}\right )} x^{2} +{\left (5 \, A a c^{2} d^{4} - 4 \, B a^{2} c d^{3} e - 6 \, A a^{2} c d^{2} e^{2} - 12 \, B a^{3} d e^{3} - 3 \, A a^{3} e^{4}\right )} x - \frac{2 \,{\left (B a^{2} c^{2} d^{4} + 4 \, A a^{2} c^{2} d^{3} e + 6 \, B a^{3} c d^{2} e^{2} + 4 \, A a^{3} c d e^{3} - 3 \, B a^{4} e^{4}\right )}}{c}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + a)^3,x, algorithm="giac")

[Out]

1/2*B*e^4*ln(c*x^2 + a)/c^3 + 1/8*(3*A*c^2*d^4 + 4*B*a*c*d^3*e + 6*A*a*c*d^2*e^2
+ 12*B*a^2*d*e^3 + 3*A*a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2) + 1/8
*((3*A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*A*a*c^2*d^2*e^2 - 20*B*a^2*c*d*e^3 - 5*A*a^
2*c*e^4)*x^3 - 8*(3*B*a^2*c*d^2*e^2 + 2*A*a^2*c*d*e^3 - B*a^3*e^4)*x^2 + (5*A*a*
c^2*d^4 - 4*B*a^2*c*d^3*e - 6*A*a^2*c*d^2*e^2 - 12*B*a^3*d*e^3 - 3*A*a^3*e^4)*x
- 2*(B*a^2*c^2*d^4 + 4*A*a^2*c^2*d^3*e + 6*B*a^3*c*d^2*e^2 + 4*A*a^3*c*d*e^3 - 3
*B*a^4*e^4)/c)/((c*x^2 + a)^2*a^2*c^2)