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

Optimal. Leaf size=274 \[ -\frac{(b d-c e) \left (-2 a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (-b^2 c \left (4 a d^2-c e^2\right )+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{d^4}{e^3 (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x) \left (2 a d^2-e (3 b d-4 c e)\right )}{e^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{x}{a e^2} \]

[Out]

x/(a*e^2) - d^4/(e^3*(a*d^2 - e*(b*d - c*e))*(d + e*x)) - ((b^4*d^2 - 2*b^3*c*d*
e + 6*a*b*c^2*d*e + 2*a*c^2*(a*d^2 - c*e^2) - b^2*c*(4*a*d^2 - c*e^2))*ArcTanh[(
b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2)
- (d^3*(2*a*d^2 - e*(3*b*d - 4*c*e))*Log[d + e*x])/(e^3*(a*d^2 - e*(b*d - c*e))
^2) - ((b*d - c*e)*(b^2*d - 2*a*c*d - b*c*e)*Log[c + b*x + a*x^2])/(2*a^2*(a*d^2
- e*(b*d - c*e))^2)

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Rubi [A]  time = 1.10399, antiderivative size = 274, 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 \[ -\frac{(b d-c e) \left (-2 a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (-b^2 c \left (4 a d^2-c e^2\right )+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{d^4}{e^3 (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x) \left (2 a d^2-e (3 b d-4 c e)\right )}{e^3 \left (a d^2-e (b d-c e)\right )^2}+\frac{x}{a e^2} \]

Antiderivative was successfully verified.

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

[Out]

x/(a*e^2) - d^4/(e^3*(a*d^2 - e*(b*d - c*e))*(d + e*x)) - ((b^4*d^2 - 2*b^3*c*d*
e + 6*a*b*c^2*d*e + 2*a*c^2*(a*d^2 - c*e^2) - b^2*c*(4*a*d^2 - c*e^2))*ArcTanh[(
b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2)
- (d^3*(2*a*d^2 - e*(3*b*d - 4*c*e))*Log[d + e*x])/(e^3*(a*d^2 - e*(b*d - c*e))
^2) - ((b*d - c*e)*(b^2*d - 2*a*c*d - b*c*e)*Log[c + b*x + a*x^2])/(2*a^2*(a*d^2
- e*(b*d - c*e))^2)

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

[Out]

Timed out

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Mathematica [A]  time = 0.545471, size = 269, normalized size = 0.98 \[ \frac{(b d-c e) \left (2 a c d+b^2 (-d)+b c e\right ) \log (x (a x+b)+c)}{2 a^2 \left (a d^2+e (c e-b d)\right )^2}+\frac{\left (b^2 c \left (c e^2-4 a d^2\right )+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^2 \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}-\frac{d^4}{e^3 (d+e x) \left (a d^2+e (c e-b d)\right )}-\frac{\log (d+e x) \left (2 a d^5+d^3 e (4 c e-3 b d)\right )}{e^3 \left (a d^2+e (c e-b d)\right )^2}+\frac{x}{a e^2} \]

Antiderivative was successfully verified.

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

[Out]

x/(a*e^2) - d^4/(e^3*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((b^4*d^2 - 2*b^3*c
*d*e + 6*a*b*c^2*d*e + 2*a*c^2*(a*d^2 - c*e^2) + b^2*c*(-4*a*d^2 + c*e^2))*ArcTa
n[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^2*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) +
c*e))^2) - ((2*a*d^5 + d^3*e*(-3*b*d + 4*c*e))*Log[d + e*x])/(e^3*(a*d^2 + e*(-(
b*d) + c*e))^2) + ((b*d - c*e)*(-(b^2*d) + 2*a*c*d + b*c*e)*Log[c + x*(b + a*x)]
)/(2*a^2*(a*d^2 + e*(-(b*d) + c*e))^2)

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Maple [B]  time = 0.016, size = 765, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 50.4761, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3*e^3 - 2*(b^3*c - 3*a*b*c^2)*d^2*e^4 +
(b^2*c^2 - 2*a*c^3)*d*e^5 + ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*e^4 - 2*(b^3*c -
3*a*b*c^2)*d*e^5 + (b^2*c^2 - 2*a*c^3)*e^6)*x)*log((b^3 - 4*a*b*c + 2*(a*b^2 - 4
*a^2*c)*x + (2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(a*x^2 + b*x
+ c)) + (2*a^3*d^6 - 2*a^2*b*d^5*e + 2*a^2*c*d^4*e^2 - 2*(a^3*d^4*e^2 - 2*a^2*b*
d^3*e^3 - 2*a*b*c*d*e^5 + a*c^2*e^6 + (a*b^2 + 2*a^2*c)*d^2*e^4)*x^2 - 2*(a^3*d^
5*e - 2*a^2*b*d^4*e^2 - 2*a*b*c*d^2*e^4 + a*c^2*d*e^5 + (a*b^2 + 2*a^2*c)*d^3*e^
3)*x + (b*c^2*d*e^5 + (b^3 - 2*a*b*c)*d^3*e^3 - 2*(b^2*c - a*c^2)*d^2*e^4 + (b*c
^2*e^6 + (b^3 - 2*a*b*c)*d^2*e^4 - 2*(b^2*c - a*c^2)*d*e^5)*x)*log(a*x^2 + b*x +
c) + 2*(2*a^3*d^6 - 3*a^2*b*d^5*e + 4*a^2*c*d^4*e^2 + (2*a^3*d^5*e - 3*a^2*b*d^
4*e^2 + 4*a^2*c*d^3*e^3)*x)*log(e*x + d))*sqrt(b^2 - 4*a*c))/((a^4*d^5*e^3 - 2*a
^3*b*d^4*e^4 - 2*a^2*b*c*d^2*e^6 + a^2*c^2*d*e^7 + (a^2*b^2 + 2*a^3*c)*d^3*e^5 +
(a^4*d^4*e^4 - 2*a^3*b*d^3*e^5 - 2*a^2*b*c*d*e^7 + a^2*c^2*e^8 + (a^2*b^2 + 2*a
^3*c)*d^2*e^6)*x)*sqrt(b^2 - 4*a*c)), 1/2*(2*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3*
e^3 - 2*(b^3*c - 3*a*b*c^2)*d^2*e^4 + (b^2*c^2 - 2*a*c^3)*d*e^5 + ((b^4 - 4*a*b^
2*c + 2*a^2*c^2)*d^2*e^4 - 2*(b^3*c - 3*a*b*c^2)*d*e^5 + (b^2*c^2 - 2*a*c^3)*e^6
)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) - (2*a^3*d^6 - 2*a^2*
b*d^5*e + 2*a^2*c*d^4*e^2 - 2*(a^3*d^4*e^2 - 2*a^2*b*d^3*e^3 - 2*a*b*c*d*e^5 + a
*c^2*e^6 + (a*b^2 + 2*a^2*c)*d^2*e^4)*x^2 - 2*(a^3*d^5*e - 2*a^2*b*d^4*e^2 - 2*a
*b*c*d^2*e^4 + a*c^2*d*e^5 + (a*b^2 + 2*a^2*c)*d^3*e^3)*x + (b*c^2*d*e^5 + (b^3
- 2*a*b*c)*d^3*e^3 - 2*(b^2*c - a*c^2)*d^2*e^4 + (b*c^2*e^6 + (b^3 - 2*a*b*c)*d^
2*e^4 - 2*(b^2*c - a*c^2)*d*e^5)*x)*log(a*x^2 + b*x + c) + 2*(2*a^3*d^6 - 3*a^2*
b*d^5*e + 4*a^2*c*d^4*e^2 + (2*a^3*d^5*e - 3*a^2*b*d^4*e^2 + 4*a^2*c*d^3*e^3)*x)
*log(e*x + d))*sqrt(-b^2 + 4*a*c))/((a^4*d^5*e^3 - 2*a^3*b*d^4*e^4 - 2*a^2*b*c*d
^2*e^6 + a^2*c^2*d*e^7 + (a^2*b^2 + 2*a^3*c)*d^3*e^5 + (a^4*d^4*e^4 - 2*a^3*b*d^
3*e^5 - 2*a^2*b*c*d*e^7 + a^2*c^2*e^8 + (a^2*b^2 + 2*a^3*c)*d^2*e^6)*x)*sqrt(-b^
2 + 4*a*c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.299698, size = 643, normalized size = 2.35 \[ -\frac{d^{4} e^{3}}{{\left (a d^{2} e^{6} - b d e^{7} + c e^{8}\right )}{\left (x e + d\right )}} - \frac{{\left (b^{4} d^{2} e^{2} - 4 \, a b^{2} c d^{2} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{2} - 2 \, b^{3} c d e^{3} + 6 \, a b c^{2} d e^{3} + b^{2} c^{2} e^{4} - 2 \, a c^{3} e^{4}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{4} d^{4} - 2 \, a^{3} b d^{3} e + a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} c d^{2} e^{2} - 2 \, a^{2} b c d e^{3} + a^{2} c^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (x e + d\right )} e^{\left (-3\right )}}{a} - \frac{{\left (b^{3} d^{2} - 2 \, a b c d^{2} - 2 \, b^{2} c d e + 2 \, a c^{2} d e + b c^{2} e^{2}\right )}{\rm ln}\left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (a^{4} d^{4} - 2 \, a^{3} b d^{3} e + a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} c d^{2} e^{2} - 2 \, a^{2} b c d e^{3} + a^{2} c^{2} e^{4}\right )}} + \frac{{\left (2 \, a d + b e\right )} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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