3.412 \(\int \frac{x^3}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx\)

Optimal. Leaf size=147 \[ \frac{\left (a^2 c^2-3 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac{b \left (5 a^2 c^2-5 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{b x \left (b^2-2 a c\right )}{c^4}+\frac{x^2 \left (b^2-a c\right )}{2 c^3}-\frac{b x^3}{3 c^2}+\frac{x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.276181, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\left (a^2 c^2-3 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac{b \left (5 a^2 c^2-5 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{b x \left (b^2-2 a c\right )}{c^4}+\frac{x^2 \left (b^2-a c\right )}{2 c^3}-\frac{b x^3}{3 c^2}+\frac{x^4}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(c + a/x^2 + b/x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b x^{3}}{3 c^{2}} + \frac{b \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{5} \sqrt{- 4 a c + b^{2}}} + \frac{x^{4}}{4 c} + \frac{\left (- a c + b^{2}\right ) \int x\, dx}{c^{3}} - \frac{\left (- 2 a c + b^{2}\right ) \int b\, dx}{c^{4}} + \frac{\left (a^{2} c^{2} - 3 a b^{2} c + b^{4}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c+a/x**2+b/x),x)

[Out]

-b*x**3/(3*c**2) + b*(5*a**2*c**2 - 5*a*b**2*c + b**4)*atanh((b + 2*c*x)/sqrt(-4
*a*c + b**2))/(c**5*sqrt(-4*a*c + b**2)) + x**4/(4*c) + (-a*c + b**2)*Integral(x
, x)/c**3 - (-2*a*c + b**2)*Integral(b, x)/c**4 + (a**2*c**2 - 3*a*b**2*c + b**4
)*log(a + b*x + c*x**2)/(2*c**5)

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Mathematica [A]  time = 0.215489, size = 140, normalized size = 0.95 \[ \frac{6 \left (a^2 c^2-3 a b^2 c+b^4\right ) \log (a+x (b+c x))-\frac{12 b \left (5 a^2 c^2-5 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c x \left (-4 b c \left (c x^2-6 a\right )+3 c^2 x \left (c x^2-2 a\right )-12 b^3+6 b^2 c x\right )}{12 c^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(c + a/x^2 + b/x),x]

[Out]

(c*x*(-12*b^3 + 6*b^2*c*x - 4*b*c*(-6*a + c*x^2) + 3*c^2*x*(-2*a + c*x^2)) - (12
*b*(b^4 - 5*a*b^2*c + 5*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b
^2 + 4*a*c] + 6*(b^4 - 3*a*b^2*c + a^2*c^2)*Log[a + x*(b + c*x)])/(12*c^5)

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Maple [A]  time = 0.009, size = 236, normalized size = 1.6 \[{\frac{{x}^{4}}{4\,c}}-{\frac{b{x}^{3}}{3\,{c}^{2}}}-{\frac{a{x}^{2}}{2\,{c}^{2}}}+{\frac{{x}^{2}{b}^{2}}{2\,{c}^{3}}}+2\,{\frac{abx}{{c}^{3}}}-{\frac{{b}^{3}x}{{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}}{2\,{c}^{3}}}-{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}}{2\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}}{2\,{c}^{5}}}-5\,{\frac{{a}^{2}b}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{a{b}^{3}}{{c}^{4}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}}{{c}^{5}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c+a/x^2+b/x),x)

[Out]

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

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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^3/(c + b/x + a/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262378, size = 1, normalized size = 0.01 \[ \left [\frac{6 \,{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (3 \, c^{4} x^{4} - 4 \, b c^{3} x^{3} + 6 \,{\left (b^{2} c^{2} - a c^{3}\right )} x^{2} - 12 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} x + 6 \,{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{12 \, \sqrt{b^{2} - 4 \, a c} c^{5}}, -\frac{12 \,{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (3 \, c^{4} x^{4} - 4 \, b c^{3} x^{3} + 6 \,{\left (b^{2} c^{2} - a c^{3}\right )} x^{2} - 12 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} x + 6 \,{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{12 \, \sqrt{-b^{2} + 4 \, a c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c + b/x + a/x^2),x, algorithm="fricas")

[Out]

[1/12*(6*(b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2
)*x + (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a))
+ (3*c^4*x^4 - 4*b*c^3*x^3 + 6*(b^2*c^2 - a*c^3)*x^2 - 12*(b^3*c - 2*a*b*c^2)*x
+ 6*(b^4 - 3*a*b^2*c + a^2*c^2)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/(sqrt(b
^2 - 4*a*c)*c^5), -1/12*(12*(b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*arctan(-sqrt(-b^2 +
4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (3*c^4*x^4 - 4*b*c^3*x^3 + 6*(b^2*c^2 - a*c^
3)*x^2 - 12*(b^3*c - 2*a*b*c^2)*x + 6*(b^4 - 3*a*b^2*c + a^2*c^2)*log(c*x^2 + b*
x + a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)]

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Sympy [A]  time = 3.93449, size = 600, normalized size = 4.08 \[ - \frac{b x^{3}}{3 c^{2}} + \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac{a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) \log{\left (x + \frac{2 a^{3} c^{2} - 4 a^{2} b^{2} c + a b^{4} - 4 a c^{5} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac{a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) + b^{2} c^{4} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac{a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right )}{5 a^{2} b c^{2} - 5 a b^{3} c + b^{5}} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac{a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) \log{\left (x + \frac{2 a^{3} c^{2} - 4 a^{2} b^{2} c + a b^{4} - 4 a c^{5} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac{a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) + b^{2} c^{4} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac{a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right )}{5 a^{2} b c^{2} - 5 a b^{3} c + b^{5}} \right )} + \frac{x^{4}}{4 c} - \frac{x^{2} \left (a c - b^{2}\right )}{2 c^{3}} + \frac{x \left (2 a b c - b^{3}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c+a/x**2+b/x),x)

[Out]

-b*x**3/(3*c**2) + (-b*sqrt(-4*a*c + b**2)*(5*a**2*c**2 - 5*a*b**2*c + b**4)/(2*
c**5*(4*a*c - b**2)) + (a**2*c**2 - 3*a*b**2*c + b**4)/(2*c**5))*log(x + (2*a**3
*c**2 - 4*a**2*b**2*c + a*b**4 - 4*a*c**5*(-b*sqrt(-4*a*c + b**2)*(5*a**2*c**2 -
5*a*b**2*c + b**4)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2 - 3*a*b**2*c + b**4)/(2
*c**5)) + b**2*c**4*(-b*sqrt(-4*a*c + b**2)*(5*a**2*c**2 - 5*a*b**2*c + b**4)/(2
*c**5*(4*a*c - b**2)) + (a**2*c**2 - 3*a*b**2*c + b**4)/(2*c**5)))/(5*a**2*b*c**
2 - 5*a*b**3*c + b**5)) + (b*sqrt(-4*a*c + b**2)*(5*a**2*c**2 - 5*a*b**2*c + b**
4)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2 - 3*a*b**2*c + b**4)/(2*c**5))*log(x + (
2*a**3*c**2 - 4*a**2*b**2*c + a*b**4 - 4*a*c**5*(b*sqrt(-4*a*c + b**2)*(5*a**2*c
**2 - 5*a*b**2*c + b**4)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2 - 3*a*b**2*c + b**
4)/(2*c**5)) + b**2*c**4*(b*sqrt(-4*a*c + b**2)*(5*a**2*c**2 - 5*a*b**2*c + b**4
)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2 - 3*a*b**2*c + b**4)/(2*c**5)))/(5*a**2*b
*c**2 - 5*a*b**3*c + b**5)) + x**4/(4*c) - x**2*(a*c - b**2)/(2*c**3) + x*(2*a*b
*c - b**3)/c**4

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GIAC/XCAS [A]  time = 0.278178, size = 196, normalized size = 1.33 \[ \frac{3 \, c^{3} x^{4} - 4 \, b c^{2} x^{3} + 6 \, b^{2} c x^{2} - 6 \, a c^{2} x^{2} - 12 \, b^{3} x + 24 \, a b c x}{12 \, c^{4}} + \frac{{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{5}} - \frac{{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c + b/x + a/x^2),x, algorithm="giac")

[Out]

1/12*(3*c^3*x^4 - 4*b*c^2*x^3 + 6*b^2*c*x^2 - 6*a*c^2*x^2 - 12*b^3*x + 24*a*b*c*
x)/c^4 + 1/2*(b^4 - 3*a*b^2*c + a^2*c^2)*ln(c*x^2 + b*x + a)/c^5 - (b^5 - 5*a*b^
3*c + 5*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^
5)