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

Optimal. Leaf size=338 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{18 a^{8/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 1.45272, antiderivative size = 336, 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 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}-2 a g+5 b d\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 128.198, size = 255, normalized size = 0.75 \[ - \frac{f}{3 a b x^{3}} - \frac{g}{2 a b x^{2}} - \frac{h}{a b x} - \frac{x \left (\frac{a f}{x^{4}} + \frac{a g}{x^{3}} + \frac{a h}{x^{2}} - \frac{b c}{x^{4}} - \frac{b d}{x^{3}} - \frac{b e}{x^{2}}\right )}{3 a b \left (a + b x^{3}\right )} - \frac{f \log{\left (x \right )}}{a^{2}} + \frac{f \log{\left (a + b x^{3} \right )}}{3 a^{2}} + \frac{\left (\sqrt [3]{a} h - \sqrt [3]{b} g\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\left (\sqrt [3]{a} h - \sqrt [3]{b} g\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{2}{3}}} + \frac{\sqrt{3} \left (\sqrt [3]{a} h + \sqrt [3]{b} g\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]

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

[Out]

-f/(3*a*b*x**3) - g/(2*a*b*x**2) - h/(a*b*x) - x*(a*f/x**4 + a*g/x**3 + a*h/x**2
- b*c/x**4 - b*d/x**3 - b*e/x**2)/(3*a*b*(a + b*x**3)) - f*log(x)/a**2 + f*log(
a + b*x**3)/(3*a**2) + (a**(1/3)*h - b**(1/3)*g)*log(a**(1/3) + b**(1/3)*x)/(3*a
**(5/3)*b**(2/3)) - (a**(1/3)*h - b**(1/3)*g)*log(a**(2/3) - a**(1/3)*b**(1/3)*x
+ b**(2/3)*x**2)/(6*a**(5/3)*b**(2/3)) + sqrt(3)*(a**(1/3)*h + b**(1/3)*g)*atan
(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(5/3)*b**(2/3))

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Mathematica [A]  time = 1.1456, size = 303, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{2/3}}+\frac{a (6 a (f+x (g+h x))-6 b (c+x (d+e x)))}{a+b x^3}+6 (2 b c-a f) \log \left (a+b x^3\right )+18 \log (x) (a f-2 b c)-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}}{18 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.022, size = 561, 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^4/(b*x^3+a)^2,x)

[Out]

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

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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^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((h*x^5 + g*x^4 + f*x^3 + 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((h*x**5+g*x**4+f*x**3+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 = 0.226904, size = 520, normalized size = 1.54 \[ \frac{{\left (2 \, b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{{\left (2 \, b c - a f\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{2}} - \frac{{\left (a^{5} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{4} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - 5 \, a^{4} b^{2} d + 2 \, a^{5} b g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{7} b} + \frac{2 \,{\left (a^{2} h - 4 \, a b e\right )} x^{5} -{\left (5 \, a b d - 2 \, a^{2} g\right )} x^{4} - 6 \, a^{2} x^{2} e - 3 \, a^{2} d x - 2 \,{\left (2 \, a b c - a^{2} f\right )} x^{3} - 2 \, a^{2} c}{6 \,{\left (b x^{3} + a\right )} a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/((b*x^3 + a)^2*x^4),x, algorithm="giac")

[Out]

1/3*(2*b*c - a*f)*ln(abs(b*x^3 + a))/a^3 - (2*b*c - a*f)*ln(abs(x))/a^3 - 1/9*sq
rt(3)*(5*(-a*b^2)^(1/3)*b^2*d - 2*(-a*b^2)^(1/3)*a*b*g + (-a*b^2)^(2/3)*a*h - 4*
(-a*b^2)^(2/3)*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b
^2) - 1/18*(5*(-a*b^2)^(1/3)*b^2*d - 2*(-a*b^2)^(1/3)*a*b*g - (-a*b^2)^(2/3)*a*h
+ 4*(-a*b^2)^(2/3)*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2) - 1/9
*(a^5*b*h*(-a/b)^(1/3) - 4*a^4*b^2*(-a/b)^(1/3)*e - 5*a^4*b^2*d + 2*a^5*b*g)*(-a
/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^7*b) + 1/6*(2*(a^2*h - 4*a*b*e)*x^5 - (5*
a*b*d - 2*a^2*g)*x^4 - 6*a^2*x^2*e - 3*a^2*d*x - 2*(2*a*b*c - a^2*f)*x^3 - 2*a^2
*c)/((b*x^3 + a)*a^3*x^3)