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

Optimal. Leaf size=289 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a g+2 b d)-\sqrt [3]{a} (2 a h+b e)\right )}{18 a^{5/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+2 b d)-\sqrt [3]{a} (2 a h+b e)\right )}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^2}+\frac{c \log (x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 1.10344, antiderivative size = 287, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (2 a h+b e)}{\sqrt [3]{b}}+a g+2 b d\right )}{18 a^{5/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+2 b d)-\sqrt [3]{a} (2 a h+b e)\right )}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{3 a^2 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^2}+\frac{c \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 149.517, size = 262, normalized size = 0.91 \[ \frac{f \log{\left (x \right )}}{a b} - \frac{f \log{\left (a + b x^{3} \right )}}{3 a b} - \frac{x \left (\frac{a f}{x} + a g + a h x - \frac{b c}{x} - b d - b e x\right )}{3 a b \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (2 a h + b e\right ) - \sqrt [3]{b} \left (a g + 2 b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} \left (2 a h + b e\right ) - \sqrt [3]{b} \left (a g + 2 b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (2 a^{\frac{4}{3}} h + \sqrt [3]{a} b e + a \sqrt [3]{b} g + 2 b^{\frac{4}{3}} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{5}{3}}} \]

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

[Out]

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

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Mathematica [A]  time = 0.373485, size = 269, normalized size = 0.93 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e-a \sqrt [3]{b} g-2 b^{4/3} d\right )}{b^{5/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{4/3} h+\sqrt [3]{a} b e+a \sqrt [3]{b} g+2 b^{4/3} d\right )}{b^{5/3}}-\frac{6 a (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left (a+b x^3\right )}-6 c \log \left (a+b x^3\right )+18 c \log (x)}{18 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 509, normalized size = 1.8 \[ \text{result too large to display} \]

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226775, size = 452, normalized size = 1.56 \[ -\frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} + \frac{a b c - a^{2} f -{\left (a^{2} h - a b e\right )} x^{2} +{\left (a b d - a^{2} g\right )} x}{3 \,{\left (b x^{3} + a\right )} a^{2} b} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} - \frac{{\left (2 \, a^{4} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 2 \, a^{3} b^{3} d + a^{4} b^{2} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b^{3}} \]

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