3.406 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=301 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{18 a^{7/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{9 a^{7/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+4 b^{5/3} c\right )}{3 \sqrt{3} a^{7/3} b^{4/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2} \]

[Out]

-(c/(a^2*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a^2*
b*(a + b*x^3)) + ((4*b^(5/3)*c - 2*a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*ArcTan
[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(4/3)) + (d*Lo
g[x])/a^2 + ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(7/3)*b^(4/3)) - ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(7/3)*b^(4/3)) - (d*Log[a +
b*x^3])/(3*a^2)

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Rubi [A]  time = 1.18186, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{18 a^{7/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right )}{9 a^{7/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+4 b^{5/3} c\right )}{3 \sqrt{3} a^{7/3} b^{4/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^2*(a + b*x^3)^2),x]

[Out]

-(c/(a^2*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a^2*
b*(a + b*x^3)) + ((4*b^(5/3)*c - 2*a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*ArcTan
[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(4/3)) + (d*Lo
g[x])/a^2 + ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/
3)*x])/(9*a^(7/3)*b^(4/3)) - ((b^(2/3)*(4*b*c - a*f) + a^(2/3)*(2*b*e + a*h))*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(7/3)*b^(4/3)) - (d*Log[a +
b*x^3])/(3*a^2)

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Rubi in Sympy [A]  time = 138.149, size = 250, normalized size = 0.83 \[ - \frac{f}{a b x} + \frac{g \log{\left (x \right )}}{a b} - \frac{g \log{\left (a + b x^{3} \right )}}{3 a b} - \frac{x \left (\frac{a f}{x^{2}} + \frac{a g}{x} + a h - \frac{b c}{x^{2}} - \frac{b d}{x} - b e\right )}{3 a b \left (a + b x^{3}\right )} - \frac{\sqrt{3} \left (- 3 \sqrt [3]{a} b^{\frac{2}{3}} f + a h + 2 b e\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{4}{3}}} + \frac{\left (3 \sqrt [3]{a} b^{\frac{2}{3}} f + a h + 2 b e\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\left (3 \sqrt [3]{a} b^{\frac{2}{3}} f + a h + 2 b e\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**2/(b*x**3+a)**2,x)

[Out]

-f/(a*b*x) + g*log(x)/(a*b) - g*log(a + b*x**3)/(3*a*b) - x*(a*f/x**2 + a*g/x +
a*h - b*c/x**2 - b*d/x - b*e)/(3*a*b*(a + b*x**3)) - sqrt(3)*(-3*a**(1/3)*b**(2/
3)*f + a*h + 2*b*e)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(
5/3)*b**(4/3)) + (3*a**(1/3)*b**(2/3)*f + a*h + 2*b*e)*log(a**(1/3) + b**(1/3)*x
)/(9*a**(5/3)*b**(4/3)) - (3*a**(1/3)*b**(2/3)*f + a*h + 2*b*e)*log(a**(2/3) - a
**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(5/3)*b**(4/3))

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Mathematica [A]  time = 0.557731, size = 285, normalized size = 0.95 \[ -\frac{\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{2/3} b e+a^{5/3} h-a b^{2/3} f+4 b^{5/3} c\right )}{b^{4/3}}-\frac{2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{2/3} b e+a^{5/3} h-a b^{2/3} f+4 b^{5/3} c\right )}{b^{4/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{2/3} b e+a^{5/3} h+a b^{2/3} f-4 b^{5/3} c\right )}{b^{4/3}}+\frac{6 a \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{b \left (a+b x^3\right )}+6 a d \log \left (a+b x^3\right )+\frac{18 a c}{x}-18 a d \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^2*(a + b*x^3)^2),x]

[Out]

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

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Maple [B]  time = 0.019, size = 519, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((h*x^5+g*x^4+f*x^3+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*x^2/(b*x^3+a)*f-1/3/a^2*b*x^2/(b*x^3+a)*c-1/3/(b*x^3
+a)*x/b*h+1/3/a*x/(b*x^3+a)*e-1/3/(b*x^3+a)/b*g+1/3/a/(b*x^3+a)*d+1/9*h/b^2/(a/b
)^(2/3)*ln(x+(a/b)^(1/3))-1/18*h/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3
))+1/9*h/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+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-1
/9/a*f/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/18/a*f/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/
3)+(a/b)^(2/3))+1/9/a*f*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*
x-1))+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/a^2*d*ln(b*(b*x^3+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((h*x^5 + g*x^4 + f*x^3 + 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((h*x^5 + g*x^4 + f*x^3 + 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((h*x**5+g*x**4+f*x**3+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 = 0.226488, size = 473, normalized size = 1.57 \[ -\frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} - \frac{4 \, b^{2} c x^{3} - a b f x^{3} + a^{2} h x^{2} - a b x^{2} e - a b d x + a^{2} g x + 3 \, a b c}{3 \,{\left (b x^{4} + a x\right )} a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{2}} + \frac{{\left (4 \, a^{2} b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{4} b^{2} h - 2 \, a^{3} b^{3} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + 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 + 1/9*sqrt(3)*((-a*b^2)^(1/3)*a
^2*h + 2*(-a*b^2)^(1/3)*a*b*e + 4*(-a*b^2)^(2/3)*b*c - (-a*b^2)^(2/3)*a*f)*arcta
n(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^2) - 1/3*(4*b^2*c*x^3 -
a*b*f*x^3 + a^2*h*x^2 - a*b*x^2*e - a*b*d*x + a^2*g*x + 3*a*b*c)/((b*x^4 + a*x)*
a^2*b) + 1/18*((-a*b^2)^(1/3)*a^2*h + 2*(-a*b^2)^(1/3)*a*b*e - 4*(-a*b^2)^(2/3)*
b*c + (-a*b^2)^(2/3)*a*f)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2) + 1/
9*(4*a^2*b^4*c*(-a/b)^(1/3) - a^3*b^3*f*(-a/b)^(1/3) - a^4*b^2*h - 2*a^3*b^3*e)*
(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^5*b^3)