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

Optimal. Leaf size=222 \[ -\frac{\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^2}+\frac{c \log (x)}{a^2} \]

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(3*a^2*(a + b*x^3)) - ((2*b^(1/3)*d + a^(1/3)*e)*Arc
Tan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(2/3)) + (c
*Log[x])/a^2 + ((2*b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b
^(2/3)) - ((2*b^(1/3)*d - a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x
^2])/(18*a^(5/3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^2)

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Rubi [A]  time = 0.616815, antiderivative size = 222, 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 \[ -\frac{\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^2}+\frac{c \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(3*a^2*(a + b*x^3)) - ((2*b^(1/3)*d + a^(1/3)*e)*Arc
Tan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(2/3)) + (c
*Log[x])/a^2 + ((2*b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b
^(2/3)) - ((2*b^(1/3)*d - a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x
^2])/(18*a^(5/3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^2)

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Rubi in Sympy [A]  time = 58.7766, size = 175, normalized size = 0.79 \[ \frac{x \left (\frac{c}{x} + d + e x\right )}{3 a \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} e - 2 \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} + \frac{\left (\sqrt [3]{a} e - 2 \sqrt [3]{b} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (\sqrt [3]{a} e + 2 \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c/x + d + e*x)/(3*a*(a + b*x**3)) - (a**(1/3)*e - 2*b**(1/3)*d)*log(a**(1/3)
+ b**(1/3)*x)/(9*a**(5/3)*b**(2/3)) + (a**(1/3)*e - 2*b**(1/3)*d)*log(a**(2/3) -
a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(5/3)*b**(2/3)) - sqrt(3)*(a**(1/3)
*e + 2*b**(1/3)*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(5
/3)*b**(2/3))

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Mathematica [A]  time = 0.329429, size = 199, normalized size = 0.9 \[ \frac{\frac{\left (a^{2/3} e-2 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 \left (2 \sqrt [3]{a} \sqrt [3]{b} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 a (c+x (d+e x))}{a+b x^3}-6 c \log \left (a+b x^3\right )+18 c \log (x)}{18 a^2} \]

Antiderivative was successfully verified.

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

[Out]

((6*a*(c + x*(d + e*x)))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(2*b^(1/3)*d + a^(1/3)
*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 18*c*Log[x] + (2*(2*a
^(1/3)*b^(1/3)*d - a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-2*a^(1/3)*b
^(1/3)*d + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) -
6*c*Log[a + b*x^3])/(18*a^2)

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Maple [A]  time = 0.017, size = 274, normalized size = 1.2 \[{\frac{c\ln \left ( x \right ) }{{a}^{2}}}+{\frac{e{x}^{2}}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{dx}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{c}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2\,d}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,d\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}e}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(x)/a^2+1/3/a*x^2/(b*x^3+a)*e+1/3/a*x/(b*x^3+a)*d+1/3/a/(b*x^3+a)*c+2/9/a/b*
d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/9/a/b*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)
^(2/3))+2/9/a/b*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/
9/a/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e+1/18/a/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+
(a/b)^(2/3))*e+1/9/a/b*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))*e-1/3*c*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),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),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/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218173, size = 311, normalized size = 1.4 \[ -\frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{a x^{2} e + a d x + a c}{3 \,{\left (b x^{3} + a\right )} a^{2}} - \frac{{\left (a^{3} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 2 \, a^{3} b d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c*ln(abs(b*x^3 + a))/a^2 + c*ln(abs(x))/a^2 + 1/9*sqrt(3)*(2*(-a*b^2)^(1/3)
*b*d - (-a*b^2)^(2/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(
a^2*b^2) + 1/3*(a*x^2*e + a*d*x + a*c)/((b*x^3 + a)*a^2) - 1/9*(a^3*b*(-a/b)^(1/
3)*e + 2*a^3*b*d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^5*b) + 1/18*(2*(-a*b
^2)^(1/3)*a*b^3*d + (-a*b^2)^(2/3)*a*b^2*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/
3))/(a^3*b^4)