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

Optimal. Leaf size=306 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{18 a^{8/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a^2 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 1.14383, antiderivative size = 304, 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 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}-2 a f+5 b c\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a^2 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 110.265, size = 241, normalized size = 0.79 \[ - \frac{f}{2 a b x^{2}} - \frac{g}{a b x} + \frac{h \log{\left (x \right )}}{a b} - \frac{h \log{\left (a + b x^{3} \right )}}{3 a b} - \frac{x \left (\frac{a f}{x^{3}} + \frac{a g}{x^{2}} + \frac{a h}{x} - \frac{b c}{x^{3}} - \frac{b d}{x^{2}} - \frac{b e}{x}\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (\sqrt [3]{a} g - \sqrt [3]{b} f\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\left (\sqrt [3]{a} g - \sqrt [3]{b} f\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{2}{3}}} + \frac{\sqrt{3} \left (\sqrt [3]{a} g + \sqrt [3]{b} f\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]

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

[Out]

-f/(2*a*b*x**2) - g/(a*b*x) + h*log(x)/(a*b) - h*log(a + b*x**3)/(3*a*b) - x*(a*
f/x**3 + a*g/x**2 + a*h/x - b*c/x**3 - b*d/x**2 - b*e/x)/(3*a*b*(a + b*x**3)) +
(a**(1/3)*g - b**(1/3)*f)*log(a**(1/3) + b**(1/3)*x)/(3*a**(5/3)*b**(2/3)) - (a*
*(1/3)*g - b**(1/3)*f)*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)*g + b**(1/3)*f)*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 = 1.08419, size = 292, normalized size = 0.95 \[ -\frac{-\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g-4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} g-4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g-4 \sqrt [3]{a} b d+2 a \sqrt [3]{b} f-5 b^{4/3} c\right )}{b^{2/3}}+\frac{6 a \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{b \left (a+b x^3\right )}+6 a e \log \left (a+b x^3\right )+\frac{9 a c}{x^2}+\frac{18 a d}{x}-18 a e \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 527, 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^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/(b*x^3+a)*x^2*g-1/3/a^2*b*x^2/(b*x^3+a)
*d+1/3/a*x/(b*x^3+a)*f-1/3/a^2*b*x/(b*x^3+a)*c-1/3/(b*x^3+a)/b*h+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))+2/9/a*f/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/9/a*f/b/(a/b)^(2/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))+2/9/a*f/b/(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-1/9/a*g/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/18/a*g/b/(a/b)^(
1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/9/a*g*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*
3^(1/2)*(2/(a/b)^(1/3)*x-1))-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((h*x^5 + g*x^4 + f*x^3 + 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((h*x^5 + g*x^4 + f*x^3 + 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((h*x**5+g*x**4+f*x**3+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 = 0.226211, size = 483, normalized size = 1.58 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{2}} + \frac{{\left (4 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{2} c - 2 \, a^{3} b f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b} - \frac{2 \,{\left (4 \, b^{2} d - a b g\right )} x^{4} + 6 \, a b d x +{\left (5 \, b^{2} c - 2 \, a b f\right )} x^{3} + 3 \, a b c + 2 \,{\left (a^{2} h - a b e\right )} x^{2}}{6 \,{\left (b x^{3} + a\right )} a^{2} b x^{2}} \]

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^3),x, algorithm="giac")

[Out]

-1/3*e*ln(abs(b*x^3 + a))/a^2 + e*ln(abs(x))/a^2 - 1/9*sqrt(3)*(5*(-a*b^2)^(1/3)
*b^2*c - 2*(-a*b^2)^(1/3)*a*b*f - 4*(-a*b^2)^(2/3)*b*d + (-a*b^2)^(2/3)*a*g)*arc
tan(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*c - 2*(-a*b^2)^(1/3)*a*b*f + 4*(-a*b^2)^(2/3)*b*d - (-a*b^2)^(2/3)*a*g
)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2) + 1/9*(4*a^2*b^2*d*(-a/b)^(1
/3) - a^3*b*g*(-a/b)^(1/3) + 5*a^2*b^2*c - 2*a^3*b*f)*(-a/b)^(1/3)*ln(abs(x - (-
a/b)^(1/3)))/(a^5*b) - 1/6*(2*(4*b^2*d - a*b*g)*x^4 + 6*a*b*d*x + (5*b^2*c - 2*a
*b*f)*x^3 + 3*a*b*c + 2*(a^2*h - a*b*e)*x^2)/((b*x^3 + a)*a^2*b*x^2)