3.338 \(\int \frac{c+d x+e x^2}{x^4 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=262 \[ \frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac{\sqrt [3]{b} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b c \log (x)}{a^3}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.792762, antiderivative size = 262, 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 \[ \frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac{\sqrt [3]{b} \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac{\sqrt [3]{b} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{2 b c \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b c \log (x)}{a^3}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.7963, size = 26, normalized size = 0.1 \[ \frac{x \left (\frac{c}{x^{4}} + \frac{d}{x^{3}} + \frac{e}{x^{2}}\right )}{3 a \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d*x+c)/x**4/(b*x**3+a)**2,x)

[Out]

x*(c/x**4 + d/x**3 + e/x**2)/(3*a*(a + b*x**3))

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Mathematica [A]  time = 0.3521, size = 225, normalized size = 0.86 \[ \frac{\sqrt [3]{b} \left (5 \sqrt [3]{a} \sqrt [3]{b} d-4 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (4 a^{2/3} e-5 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\frac{6 a b (c+x (d+e x))}{a+b x^3}+12 b c \log \left (a+b x^3\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}-36 b c \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^2),x]

[Out]

((-6*a*c)/x^3 - (9*a*d)/x^2 - (18*a*e)/x - (6*a*b*(c + x*(d + e*x)))/(a + b*x^3)
+ 2*Sqrt[3]*a^(1/3)*b^(1/3)*(5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*
x)/a^(1/3))/Sqrt[3]] - 36*b*c*Log[x] + 2*b^(1/3)*(-5*a^(1/3)*b^(1/3)*d + 4*a^(2/
3)*e)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*d - 4*a^(2/3)*e)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 12*b*c*Log[a + b*x^3])/(18*a^3)

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Maple [A]  time = 0.02, size = 289, normalized size = 1.1 \[ -{\frac{d}{2\,{a}^{2}{x}^{2}}}-{\frac{e}{{a}^{2}x}}-{\frac{c}{3\,{a}^{2}{x}^{3}}}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}-{\frac{be{x}^{2}}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{bxd}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{bc}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{5\,d}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,d}{18\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,d\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{4\,e}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,e}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,e\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,bc\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d*x+c)/x^4/(b*x^3+a)^2,x)

[Out]

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

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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^4),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^4),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**4/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21582, size = 373, normalized size = 1.42 \[ \frac{2 \, b c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{2 \, b c{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4} b^{3}} + \frac{{\left (4 \, a^{4} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, a^{4} b^{2} d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{7} b} - \frac{8 \, a b x^{5} e + 5 \, a b d x^{4} + 4 \, a b c x^{3} + 6 \, a^{2} x^{2} e + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \,{\left (b x^{3} + a\right )} a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)/((b*x^3 + a)^2*x^4),x, algorithm="giac")

[Out]

2/3*b*c*ln(abs(b*x^3 + a))/a^3 - 2*b*c*ln(abs(x))/a^3 - 1/18*(5*(-a*b^2)^(1/3)*b
*d + 4*(-a*b^2)^(2/3)*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) - 1/9*s
qrt(3)*(5*(-a*b^2)^(1/3)*a*b^3*d - 4*(-a*b^2)^(2/3)*a*b^2*e)*arctan(1/3*sqrt(3)*
(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^3) + 1/9*(4*a^4*b^2*(-a/b)^(1/3)*e + 5
*a^4*b^2*d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^7*b) - 1/6*(8*a*b*x^5*e +
5*a*b*d*x^4 + 4*a*b*c*x^3 + 6*a^2*x^2*e + 3*a^2*d*x + 2*a^2*c)/((b*x^3 + a)*a^3*
x^3)