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

Optimal. Leaf size=242 \[ \frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac{\sqrt [3]{b} \left (4 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}-\frac{x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

[Out]

-c/(2*a^2*x^2) - d/(a^2*x) - (x*(b*c + b*d*x + b*e*x^2))/(3*a^2*(a + b*x^3)) + (
b^(1/3)*(5*b^(1/3)*c + 4*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1
/3))])/(3*Sqrt[3]*a^(8/3)) + (e*Log[x])/a^2 - (b^(1/3)*(5*b^(1/3)*c - 4*a^(1/3)*
d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)) + (b^(1/3)*(5*b^(1/3)*c - 4*a^(1/3)*d)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)) - (e*Log[a + b*x^3]
)/(3*a^2)

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Rubi [A]  time = 0.686778, antiderivative size = 242, 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 \[ \frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac{\sqrt [3]{b} \left (4 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}-\frac{x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*a^2*x^2) - d/(a^2*x) - (x*(b*c + b*d*x + b*e*x^2))/(3*a^2*(a + b*x^3)) + (
b^(1/3)*(5*b^(1/3)*c + 4*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1
/3))])/(3*Sqrt[3]*a^(8/3)) + (e*Log[x])/a^2 - (b^(1/3)*(5*b^(1/3)*c - 4*a^(1/3)*
d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)) + (b^(1/3)*(5*b^(1/3)*c - 4*a^(1/3)*d)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)) - (e*Log[a + b*x^3]
)/(3*a^2)

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Rubi in Sympy [A]  time = 13.5804, size = 24, normalized size = 0.1 \[ \frac{x \left (\frac{c}{x^{3}} + \frac{d}{x^{2}} + \frac{e}{x}\right )}{3 a \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.328512, size = 221, normalized size = 0.91 \[ \frac{\sqrt [3]{b} \left (5 \sqrt [3]{a} \sqrt [3]{b} c-4 a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (4 a^{2/3} d-5 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{6 a (a e-b x (c+d x))}{a+b x^3}+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (4 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-6 a e \log \left (a+b x^3\right )-\frac{9 a c}{x^2}-\frac{18 a d}{x}+18 a e \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((-9*a*c)/x^2 - (18*a*d)/x + (6*a*(a*e - b*x*(c + d*x)))/(a + b*x^3) + 2*Sqrt[3]
*a^(1/3)*b^(1/3)*(5*b^(1/3)*c + 4*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/
Sqrt[3]] + 18*a*e*Log[x] + 2*b^(1/3)*(-5*a^(1/3)*b^(1/3)*c + 4*a^(2/3)*d)*Log[a^
(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*c - 4*a^(2/3)*d)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 6*a*e*Log[a + b*x^3])/(18*a^3)

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Maple [A]  time = 0.019, size = 276, normalized size = 1.1 \[ -{\frac{d}{{a}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-{\frac{c}{2\,{a}^{2}{x}^{2}}}-{\frac{b{x}^{2}d}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{bcx}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{e}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{5\,c}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,c}{18\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,c\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{4\,d}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,d}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,d\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216707, size = 344, normalized size = 1.42 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b} + \frac{{\left (4 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{2} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b} - \frac{8 \, b d x^{4} + 5 \, b c x^{3} - 2 \, a x^{2} e + 6 \, a d x + 3 \, a c}{6 \,{\left (b x^{3} + a\right )} a^{2} x^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*e*ln(abs(b*x^3 + a))/a^2 + e*ln(abs(x))/a^2 - 1/18*(5*(-a*b^2)^(1/3)*b*c +
4*(-a*b^2)^(2/3)*d)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) + 1/9*(4*a^2
*b^2*d*(-a/b)^(1/3) + 5*a^2*b^2*c)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^5*b
) - 1/6*(8*b*d*x^4 + 5*b*c*x^3 - 2*a*x^2*e + 6*a*d*x + 3*a*c)/((b*x^3 + a)*a^2*x
^2) - 1/9*sqrt(3)*(5*(-a*b^2)^(1/3)*a*b^3*c - 4*(-a*b^2)^(2/3)*a*b^2*d)*arctan(1
/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^3)