3.78 \(\int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^5 (d+e x)^2} \, dx\)

Optimal. Leaf size=372 \[ \frac{\left (a^3 c d^2-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a b^2 e (2 b d+3 c e)+b^4 \left (-e^2\right )\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-a^3 c d (3 b d+4 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )-a b^3 e (2 b d+5 c e)+b^5 e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x) \left (-c \left (a d^2-3 c e^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}-\frac{e^4 \log (d+e x) \left (5 a d^2-e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac{e^4}{d^3 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{b d+2 c e}{c^2 d^3 x}-\frac{1}{2 c d^2 x^2} \]

[Out]

-1/(2*c*d^2*x^2) + (b*d + 2*c*e)/(c^2*d^3*x) + e^4/(d^3*(a*d^2 - e*(b*d - c*e))*
(d + e*x)) + ((b^5*e^2 - a^3*c*d*(3*b*d + 4*c*e) - a*b^3*e*(2*b*d + 5*c*e) + a^2
*b*(b^2*d^2 + 8*b*c*d*e + 5*c^2*e^2))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(c
^3*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) + ((b^2*d^2 + 2*b*c*d*e - c*(a*d
^2 - 3*c*e^2))*Log[x])/(c^3*d^4) - (e^4*(5*a*d^2 - e*(4*b*d - 3*c*e))*Log[d + e*
x])/(d^4*(a*d^2 - e*(b*d - c*e))^2) + ((a^3*c*d^2 - b^4*e^2 + a*b^2*e*(2*b*d + 3
*c*e) - a^2*(b^2*d^2 + 4*b*c*d*e + c^2*e^2))*Log[c + b*x + a*x^2])/(2*c^3*(a*d^2
- e*(b*d - c*e))^2)

_______________________________________________________________________________________

Rubi [A]  time = 1.88739, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (a^3 c d^2-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a b^2 e (2 b d+3 c e)+b^4 \left (-e^2\right )\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-a^3 c d (3 b d+4 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )-a b^3 e (2 b d+5 c e)+b^5 e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x) \left (-c \left (a d^2-3 c e^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}-\frac{e^4 \log (d+e x) \left (5 a d^2-e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac{e^4}{d^3 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{b d+2 c e}{c^2 d^3 x}-\frac{1}{2 c d^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + c/x^2 + b/x)*x^5*(d + e*x)^2),x]

[Out]

-1/(2*c*d^2*x^2) + (b*d + 2*c*e)/(c^2*d^3*x) + e^4/(d^3*(a*d^2 - e*(b*d - c*e))*
(d + e*x)) + ((b^5*e^2 - a^3*c*d*(3*b*d + 4*c*e) - a*b^3*e*(2*b*d + 5*c*e) + a^2
*b*(b^2*d^2 + 8*b*c*d*e + 5*c^2*e^2))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(c
^3*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) + ((b^2*d^2 + 2*b*c*d*e - c*(a*d
^2 - 3*c*e^2))*Log[x])/(c^3*d^4) - (e^4*(5*a*d^2 - e*(4*b*d - 3*c*e))*Log[d + e*
x])/(d^4*(a*d^2 - e*(b*d - c*e))^2) + ((a^3*c*d^2 - b^4*e^2 + a*b^2*e*(2*b*d + 3
*c*e) - a^2*(b^2*d^2 + 4*b*c*d*e + c^2*e^2))*Log[c + b*x + a*x^2])/(2*c^3*(a*d^2
- e*(b*d - c*e))^2)

_______________________________________________________________________________________

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(1/(a+c/x**2+b/x)/x**5/(e*x+d)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.779386, size = 370, normalized size = 0.99 \[ -\frac{\left (-a^3 c d^2+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )-a b^2 e (2 b d+3 c e)+b^4 e^2\right ) \log (x (a x+b)+c)}{2 c^3 \left (a d^2+e (c e-b d)\right )^2}+\frac{\left (a^3 c d (3 b d+4 c e)-a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )+a b^3 e (2 b d+5 c e)+b^5 \left (-e^2\right )\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c^3 \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac{\log (x) \left (c \left (3 c e^2-a d^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}-\frac{e^4 \log (d+e x) \left (5 a d^2+e (3 c e-4 b d)\right )}{d^4 \left (a d^2+e (c e-b d)\right )^2}+\frac{e^4}{d^3 (d+e x) \left (a d^2+e (c e-b d)\right )}+\frac{b d+2 c e}{c^2 d^3 x}-\frac{1}{2 c d^2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + c/x^2 + b/x)*x^5*(d + e*x)^2),x]

[Out]

-1/(2*c*d^2*x^2) + (b*d + 2*c*e)/(c^2*d^3*x) + e^4/(d^3*(a*d^2 + e*(-(b*d) + c*e
))*(d + e*x)) + ((-(b^5*e^2) + a^3*c*d*(3*b*d + 4*c*e) + a*b^3*e*(2*b*d + 5*c*e)
- a^2*b*(b^2*d^2 + 8*b*c*d*e + 5*c^2*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c
]])/(c^3*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b^2*d^2 + 2*b*c*d*
e + c*(-(a*d^2) + 3*c*e^2))*Log[x])/(c^3*d^4) - (e^4*(5*a*d^2 + e*(-4*b*d + 3*c*
e))*Log[d + e*x])/(d^4*(a*d^2 + e*(-(b*d) + c*e))^2) - ((-(a^3*c*d^2) + b^4*e^2
- a*b^2*e*(2*b*d + 3*c*e) + a^2*(b^2*d^2 + 4*b*c*d*e + c^2*e^2))*Log[c + x*(b +
a*x)])/(2*c^3*(a*d^2 + e*(-(b*d) + c*e))^2)

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 993, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)/x^5/(e*x+d)^2,x)

[Out]

2/c^3/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2)
)*a*b^4*d*e-8/c^2/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*
c-b^2)^(1/2))*a^2*b^2*d*e+3/c^2/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((
2*a*x+b)/(4*a*c-b^2)^(1/2))*a^3*b*d^2+4/c/(a*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2
)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^3*d*e-1/c^3/(a*d^2-b*d*e+c*e^2)^2/(4*a*c
-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^3*d^2-5/c/(a*d^2-b*d*e+c*e
^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*e^2+5/c^2/(a*d
^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^
2+1/x/c^2/d^2*b+2/x/c/d^3*e-1/c^2/d^2*ln(x)*a+1/c^3/d^2*ln(x)*b^2+3/c/d^4*ln(x)*
e^2+e^4/d^3/(a*d^2-b*d*e+c*e^2)/(e*x+d)-1/2/x^2/c/d^2+2/c^2/d^3*ln(x)*b*e-5*e^4/
d^2/(a*d^2-b*d*e+c*e^2)^2*ln(e*x+d)*a-1/2/c^3/(a*d^2-b*d*e+c*e^2)^2*a^2*ln(a*x^2
+b*x+c)*b^2*d^2+3/2/c^2/(a*d^2-b*d*e+c*e^2)^2*a*ln(a*x^2+b*x+c)*b^2*e^2-1/c^3/(a
*d^2-b*d*e+c*e^2)^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b^5*e^
2-2/c^2/(a*d^2-b*d*e+c*e^2)^2*a^2*ln(a*x^2+b*x+c)*b*d*e+1/c^3/(a*d^2-b*d*e+c*e^2
)^2*a*ln(a*x^2+b*x+c)*b^3*d*e+4*e^5/d^3/(a*d^2-b*d*e+c*e^2)^2*ln(e*x+d)*b-3*e^6/
d^4/(a*d^2-b*d*e+c*e^2)^2*ln(e*x+d)*c+1/2/c^2/(a*d^2-b*d*e+c*e^2)^2*a^3*ln(a*x^2
+b*x+c)*d^2-1/2/c/(a*d^2-b*d*e+c*e^2)^2*a^2*ln(a*x^2+b*x+c)*e^2-1/2/c^3/(a*d^2-b
*d*e+c*e^2)^2*ln(a*x^2+b*x+c)*b^4*e^2

_______________________________________________________________________________________

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(1/((e*x + d)^2*(a + b/x + c/x^2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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(1/((e*x + d)^2*(a + b/x + c/x^2)*x^5),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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(1/(a+c/x**2+b/x)/x**5/(e*x+d)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.289699, size = 792, normalized size = 2.13 \[ \frac{{\left (a^{2} b^{3} d^{2} e^{2} - 3 \, a^{3} b c d^{2} e^{2} - 2 \, a b^{4} d e^{3} + 8 \, a^{2} b^{2} c d e^{3} - 4 \, a^{3} c^{2} d e^{3} + b^{5} e^{4} - 5 \, a b^{3} c e^{4} + 5 \, a^{2} b c^{2} e^{4}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} c^{3} d^{4} - 2 \, a b c^{3} d^{3} e + b^{2} c^{3} d^{2} e^{2} + 2 \, a c^{4} d^{2} e^{2} - 2 \, b c^{4} d e^{3} + c^{5} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (a^{2} b^{2} d^{2} - a^{3} c d^{2} - 2 \, a b^{3} d e + 4 \, a^{2} b c d e + b^{4} e^{2} - 3 \, a b^{2} c e^{2} + a^{2} c^{2} e^{2}\right )}{\rm ln}\left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (a^{2} c^{3} d^{4} - 2 \, a b c^{3} d^{3} e + b^{2} c^{3} d^{2} e^{2} + 2 \, a c^{4} d^{2} e^{2} - 2 \, b c^{4} d e^{3} + c^{5} e^{4}\right )}} + \frac{e^{9}}{{\left (a d^{5} e^{5} - b d^{4} e^{6} + c d^{3} e^{7}\right )}{\left (x e + d\right )}} + \frac{{\left (b^{2} d^{2} e - a c d^{2} e + 2 \, b c d e^{2} + 3 \, c^{2} e^{3}\right )} e^{\left (-1\right )}{\rm ln}\left ({\left | -\frac{d}{x e + d} + 1 \right |}\right )}{c^{3} d^{4}} + \frac{2 \, b c d e + 5 \, c^{2} e^{2} - \frac{2 \,{\left (b c d^{2} e^{2} + 3 \, c^{2} d e^{3}\right )} e^{\left (-1\right )}}{x e + d}}{2 \, c^{3} d^{4}{\left (\frac{d}{x e + d} - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x + d)^2*(a + b/x + c/x^2)*x^5),x, algorithm="giac")

[Out]

(a^2*b^3*d^2*e^2 - 3*a^3*b*c*d^2*e^2 - 2*a*b^4*d*e^3 + 8*a^2*b^2*c*d*e^3 - 4*a^3
*c^2*d*e^3 + b^5*e^4 - 5*a*b^3*c*e^4 + 5*a^2*b*c^2*e^4)*arctan(-(2*a*d - 2*a*d^2
/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*c*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a
*c))*e^(-2)/((a^2*c^3*d^4 - 2*a*b*c^3*d^3*e + b^2*c^3*d^2*e^2 + 2*a*c^4*d^2*e^2
- 2*b*c^4*d*e^3 + c^5*e^4)*sqrt(-b^2 + 4*a*c)) - 1/2*(a^2*b^2*d^2 - a^3*c*d^2 -
2*a*b^3*d*e + 4*a^2*b*c*d*e + b^4*e^2 - 3*a*b^2*c*e^2 + a^2*c^2*e^2)*ln(-a + 2*a
*d/(x*e + d) - a*d^2/(x*e + d)^2 - b*e/(x*e + d) + b*d*e/(x*e + d)^2 - c*e^2/(x*
e + d)^2)/(a^2*c^3*d^4 - 2*a*b*c^3*d^3*e + b^2*c^3*d^2*e^2 + 2*a*c^4*d^2*e^2 - 2
*b*c^4*d*e^3 + c^5*e^4) + e^9/((a*d^5*e^5 - b*d^4*e^6 + c*d^3*e^7)*(x*e + d)) +
(b^2*d^2*e - a*c*d^2*e + 2*b*c*d*e^2 + 3*c^2*e^3)*e^(-1)*ln(abs(-d/(x*e + d) + 1
))/(c^3*d^4) + 1/2*(2*b*c*d*e + 5*c^2*e^2 - 2*(b*c*d^2*e^2 + 3*c^2*d*e^3)*e^(-1)
/(x*e + d))/(c^3*d^4*(d/(x*e + d) - 1)^2)