3.336 \(\int \frac{c+d x+e x^2}{x^2 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=231 \[ -\frac{\left (a^{2/3} e+2 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3} \sqrt [3]{b}}+\frac{2 \left (a^{2/3} e+2 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} \sqrt [3]{b}}+\frac{2 \left (2 b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} \sqrt [3]{b}}+\frac{x \left (a e-b c x-b d x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2} \]

[Out]

-(c/(a^2*x)) + (x*(a*e - b*c*x - b*d*x^2))/(3*a^2*(a + b*x^3)) + (2*(2*b^(2/3)*c
- a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7
/3)*b^(1/3)) + (d*Log[x])/a^2 + (2*(2*b^(2/3)*c + a^(2/3)*e)*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(7/3)*b^(1/3)) - ((2*b^(2/3)*c + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(9*a^(7/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^2)

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Rubi [A]  time = 0.680472, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{\left (a^{2/3} e+2 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3} \sqrt [3]{b}}+\frac{2 \left (a^{2/3} e+2 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} \sqrt [3]{b}}+\frac{2 \left (2 b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} \sqrt [3]{b}}+\frac{x \left (a e-b c x-b d x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^2*x)) + (x*(a*e - b*c*x - b*d*x^2))/(3*a^2*(a + b*x^3)) + (2*(2*b^(2/3)*c
- a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7
/3)*b^(1/3)) + (d*Log[x])/a^2 + (2*(2*b^(2/3)*c + a^(2/3)*e)*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(7/3)*b^(1/3)) - ((2*b^(2/3)*c + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(9*a^(7/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^2)

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Rubi in Sympy [A]  time = 49.6303, size = 139, normalized size = 0.6 \[ \frac{x \left (\frac{c}{x^{2}} + \frac{d}{x} + e\right )}{3 a \left (a + b x^{3}\right )} + \frac{2 e \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{e \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} e \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c/x**2 + d/x + e)/(3*a*(a + b*x**3)) + 2*e*log(a**(1/3) + b**(1/3)*x)/(9*a**(
5/3)*b**(1/3)) - e*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(9*a**(5/
3)*b**(1/3)) - 2*sqrt(3)*e*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/
(9*a**(5/3)*b**(1/3))

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Mathematica [A]  time = 0.522439, size = 213, normalized size = 0.92 \[ -\frac{\frac{\left (2 a^{2/3} b^{2/3} c+a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac{2 \left (2 a^{2/3} b^{2/3} c+a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} a^{2/3} \left (a^{2/3} e-2 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}-\frac{3 a \left (a (d+e x)-b c x^2\right )}{a+b x^3}+3 a d \log \left (a+b x^3\right )+\frac{9 a c}{x}-9 a d \log (x)}{9 a^3} \]

Antiderivative was successfully verified.

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

[Out]

-((9*a*c)/x - (3*a*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3) + (2*Sqrt[3]*a^(2/3)*
(-2*b^(2/3)*c + a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)
- 9*a*d*Log[x] - (2*(2*a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/
b^(1/3) + ((2*a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/b^(1/3) + 3*a*d*Log[a + b*x^3])/(9*a^3)

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Maple [A]  time = 0.021, size = 275, normalized size = 1.2 \[{\frac{d\ln \left ( x \right ) }{{a}^{2}}}-{\frac{c}{{a}^{2}x}}-{\frac{b{x}^{2}c}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{ex}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{d}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2\,e}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}e}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{4\,c}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,c}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,c\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*ln(x)/a^2-1/a^2*c/x-1/3/a^2*b*x^2/(b*x^3+a)*c+1/3/a*x/(b*x^3+a)*e+1/3/a/(b*x^3
+a)*d+2/9/a/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-1/9/a/b/(a/b)^(2/3)*ln(x^2-x*(a/b)
^(1/3)+(a/b)^(2/3))*e+2/9/a/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1
/3)*x-1))*e+4/9/a^2*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-2/9/a^2*c/(a/b)^(1/3)*ln(x^2
-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9/a^2*c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/
(a/b)^(1/3)*x-1))-1/3*d*ln(b*x^3+a)/a^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((e*x^2 + d*x + c)/((b*x^3 + a)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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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((e*x**2+d*x+c)/x**2/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216959, size = 328, normalized size = 1.42 \[ -\frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{4 \, b c x^{3} - a x^{2} e - a d x + 3 \, a c}{3 \,{\left (b x^{4} + a x\right )} a^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{3} b} + \frac{2 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{3}} + \frac{2 \,{\left (2 \, a^{2} b^{2} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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