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

Optimal. Leaf size=218 \[ -\frac{\left (-2 a b c d+a c^2 e+b^3 d-b^2 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac{x (a d+b e)}{a^2 e^2}-\frac{\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac{x^2}{2 a e} \]

[Out]

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

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Rubi [A]  time = 0.793686, antiderivative size = 218, 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 \[ -\frac{\left (-2 a b c d+a c^2 e+b^3 d-b^2 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac{x (a d+b e)}{a^2 e^2}-\frac{\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac{x^2}{2 a e} \]

Antiderivative was successfully verified.

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

[Out]

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

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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),x)

[Out]

Timed out

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Mathematica [A]  time = 0.308177, size = 218, normalized size = 1. \[ \frac{\left (2 a b c d-a c^2 e+b^3 (-d)+b^2 c e\right ) \log (x (a x+b)+c)}{2 a^3 \left (a d^2+e (c e-b d)\right )}-\frac{x (a d+b e)}{a^2 e^2}+\frac{\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^3 \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2+e (c e-b d)\right )}+\frac{x^2}{2 a e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 512, normalized size = 2.4 \[{\frac{{x}^{2}}{2\,ae}}-{\frac{dx}{a{e}^{2}}}-{\frac{bx}{{a}^{2}e}}+{\frac{{d}^{4}\ln \left ( ex+d \right ) }{{e}^{3} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bcd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}e}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+2\,{\frac{{c}^{2}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}cd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{b{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/a/e-1/a/e^2*d*x-1/a^2/e*b*x+1/e^3*d^4/(a*d^2-b*d*e+c*e^2)*ln(e*x+d)+1/(a
*d^2-b*d*e+c*e^2)/a^2*ln(a*x^2+b*x+c)*b*c*d-1/2/(a*d^2-b*d*e+c*e^2)/a^2*ln(a*x^2
+b*x+c)*c^2*e-1/2/(a*d^2-b*d*e+c*e^2)/a^3*ln(a*x^2+b*x+c)*b^3*d+1/2/(a*d^2-b*d*e
+c*e^2)/a^3*ln(a*x^2+b*x+c)*b^2*c*e+2/(a*d^2-b*d*e+c*e^2)/a/(4*a*c-b^2)^(1/2)*ar
ctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*c^2*d-4/(a*d^2-b*d*e+c*e^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*d+3/(a*d^2-b*d*e+c*e^2)/a^2/(4*a*c
-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*e+1/(a*d^2-b*d*e+c*e^2)/a^
3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b^4*d-1/(a*d^2-b*d*e+c*e
^2)/a^3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*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(x^2/((e*x + d)*(a + b/x + c/x^2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.9436, size = 1, normalized size = 0. \[ \left [\frac{{\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d e^{3} -{\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x -{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right ) +{\left (2 \, a^{3} d^{4} \log \left (e x + d\right ) +{\left (a^{3} d^{2} e^{2} - a^{2} b d e^{3} + a^{2} c e^{4}\right )} x^{2} - 2 \,{\left (a^{3} d^{3} e + a b c e^{4} -{\left (a b^{2} - a^{2} c\right )} d e^{3}\right )} x -{\left ({\left (b^{3} - 2 \, a b c\right )} d e^{3} -{\left (b^{2} c - a c^{2}\right )} e^{4}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{4} d^{2} e^{3} - a^{3} b d e^{4} + a^{3} c e^{5}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d e^{3} -{\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, a^{3} d^{4} \log \left (e x + d\right ) +{\left (a^{3} d^{2} e^{2} - a^{2} b d e^{3} + a^{2} c e^{4}\right )} x^{2} - 2 \,{\left (a^{3} d^{3} e + a b c e^{4} -{\left (a b^{2} - a^{2} c\right )} d e^{3}\right )} x -{\left ({\left (b^{3} - 2 \, a b c\right )} d e^{3} -{\left (b^{2} c - a c^{2}\right )} e^{4}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{4} d^{2} e^{3} - a^{3} b d e^{4} + a^{3} c e^{5}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296006, size = 302, normalized size = 1.39 \[ \frac{d^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{2} e^{3} - b d e^{4} + c e^{5}} - \frac{{\left (b^{3} d - 2 \, a b c d - b^{2} c e + a c^{2} e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a^{4} d^{2} - a^{3} b d e + a^{3} c e^{2}\right )}} + \frac{{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} d^{2} - a^{3} b d e + a^{3} c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (a x^{2} e - 2 \, a d x - 2 \, b x e\right )} e^{\left (-2\right )}}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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