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

Optimal. Leaf size=192 \[ -\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt [3]{b}}+\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt [3]{b}}+\frac{\left (b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} \sqrt [3]{b}}-\frac{d \log \left (a+b x^3\right )}{3 a}-\frac{c}{a x}+\frac{d \log (x)}{a} \]

[Out]

-(c/(a*x)) + ((b^(2/3)*c - a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^
(1/3))])/(Sqrt[3]*a^(4/3)*b^(1/3)) + (d*Log[x])/a + ((b^(2/3)*c + a^(2/3)*e)*Log
[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)*b^(1/3)) - ((b^(2/3)*c + a^(2/3)*e)*Log[a^(2/3
) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*b^(1/3)) - (d*Log[a + b*x^3])/(
3*a)

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Rubi [A]  time = 0.460485, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt [3]{b}}+\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt [3]{b}}+\frac{\left (b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} \sqrt [3]{b}}-\frac{d \log \left (a+b x^3\right )}{3 a}-\frac{c}{a x}+\frac{d \log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a*x)) + ((b^(2/3)*c - a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^
(1/3))])/(Sqrt[3]*a^(4/3)*b^(1/3)) + (d*Log[x])/a + ((b^(2/3)*c + a^(2/3)*e)*Log
[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)*b^(1/3)) - ((b^(2/3)*c + a^(2/3)*e)*Log[a^(2/3
) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*b^(1/3)) - (d*Log[a + b*x^3])/(
3*a)

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Rubi in Sympy [A]  time = 77.108, size = 175, normalized size = 0.91 \[ - \frac{c}{a x} + \frac{d \log{\left (x \right )}}{a} - \frac{d \log{\left (a + b x^{3} \right )}}{3 a} - \frac{\sqrt{3} \left (a^{\frac{2}{3}} e - b^{\frac{2}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{4}{3}} \sqrt [3]{b}} + \frac{\left (a^{\frac{2}{3}} e + b^{\frac{2}{3}} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{4}{3}} \sqrt [3]{b}} - \frac{\left (a^{\frac{2}{3}} e + b^{\frac{2}{3}} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{4}{3}} \sqrt [3]{b}} \]

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),x)

[Out]

-c/(a*x) + d*log(x)/a - d*log(a + b*x**3)/(3*a) - sqrt(3)*(a**(2/3)*e - b**(2/3)
*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(4/3)*b**(1/3)) +
(a**(2/3)*e + b**(2/3)*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(4/3)*b**(1/3)) - (a
**(2/3)*e + b**(2/3)*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a
**(4/3)*b**(1/3))

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Mathematica [A]  time = 0.501373, size = 184, normalized size = 0.96 \[ -\frac{\frac{\left (a^{2/3} b^{2/3} c+a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac{2 \left (a^{2/3} b^{2/3} c+a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} a^{2/3} \left (a^{2/3} e-b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}+2 a d \log \left (a+b x^3\right )+\frac{6 a c}{x}-6 a d \log (x)}{6 a^2} \]

Antiderivative was successfully verified.

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

[Out]

-((6*a*c)/x + (2*Sqrt[3]*a^(2/3)*(-(b^(2/3)*c) + a^(2/3)*e)*ArcTan[(1 - (2*b^(1/
3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) - 6*a*d*Log[x] - (2*(a^(2/3)*b^(2/3)*c + a^(4/3
)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((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) + 2*a*d*Log[a + b*x^3])/(6*a^2
)

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Maple [A]  time = 0.01, size = 216, normalized size = 1.1 \[{\frac{d\ln \left ( x \right ) }{a}}-{\frac{c}{ax}}+{\frac{e}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{e}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}e}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*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)*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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216593, size = 279, normalized size = 1.45 \[ -\frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{c}{a x} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - \left (-a b^{2}\right )^{\frac{2}{3}} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b} + \frac{{\left (a b^{2} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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