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

Optimal. Leaf size=313 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}-\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*a*b*(a + b*x^3)^2) - (3*a*(
b*e + a*h) - b*x*(5*b*c + a*f + 2*(2*b*d + a*g)*x))/(18*a^2*b^2*(a + b*x^3)) - (
(5*b^(4/3)*c + 2*a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(5/3)) + ((b^(1/3)*(5*b*c + a*f)
- a^(1/3)*(2*b*d + a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(5/3)) - ((b^(
1/3)*(5*b*c + a*f) - a^(1/3)*(2*b*d + a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^
(2/3)*x^2])/(54*a^(8/3)*b^(5/3))

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Rubi [A]  time = 0.916225, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}-\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*a*b*(a + b*x^3)^2) - (3*a*(
b*e + a*h) - b*x*(5*b*c + a*f + 2*(2*b*d + a*g)*x))/(18*a^2*b^2*(a + b*x^3)) - (
(5*b^(4/3)*c + 2*a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(5/3)) + ((b^(1/3)*(5*b*c + a*f)
- a^(1/3)*(2*b*d + a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(5/3)) - ((b^(
1/3)*(5*b*c + a*f) - a^(1/3)*(2*b*d + a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^
(2/3)*x^2])/(54*a^(8/3)*b^(5/3))

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Rubi in Sympy [A]  time = 120.495, size = 287, normalized size = 0.92 \[ - \frac{x \left (a f - b c + x^{2} \left (a h - b e\right ) + x \left (a g - b d\right )\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{3 a \left (a h + b e\right ) - b x \left (a f + 5 b c + x \left (2 a g + 4 b d\right )\right )}{18 a^{2} b^{2} \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (a g + 2 b d\right ) - \sqrt [3]{b} \left (a f + 5 b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} \left (a g + 2 b d\right ) - \sqrt [3]{b} \left (a f + 5 b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (a^{\frac{4}{3}} g + 2 \sqrt [3]{a} b d + a \sqrt [3]{b} f + 5 b^{\frac{4}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} \]

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

[Out]

-x*(a*f - b*c + x**2*(a*h - b*e) + x*(a*g - b*d))/(6*a*b*(a + b*x**3)**2) - (3*a
*(a*h + b*e) - b*x*(a*f + 5*b*c + x*(2*a*g + 4*b*d)))/(18*a**2*b**2*(a + b*x**3)
) - (a**(1/3)*(a*g + 2*b*d) - b**(1/3)*(a*f + 5*b*c))*log(a**(1/3) + b**(1/3)*x)
/(27*a**(8/3)*b**(5/3)) + (a**(1/3)*(a*g + 2*b*d) - b**(1/3)*(a*f + 5*b*c))*log(
a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(8/3)*b**(5/3)) - sqrt(3)
*(a**(4/3)*g + 2*a**(1/3)*b*d + a*b**(1/3)*f + 5*b**(4/3)*c)*atan(sqrt(3)*(a**(1
/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(8/3)*b**(5/3))

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Mathematica [A]  time = 0.45834, size = 295, normalized size = 0.94 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d-a \sqrt [3]{b} f-5 b^{4/3} c\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-g)-2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )-2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )+\frac{9 a^{5/3} \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{\left (a+b x^3\right )^2}+\frac{3 a^{2/3} \left (-6 a^2 h+a b x (f+2 g x)+b^2 x (5 c+4 d x)\right )}{a+b x^3}}{54 a^{8/3} b^2} \]

Antiderivative was successfully verified.

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

[Out]

((3*a^(2/3)*(-6*a^2*h + b^2*x*(5*c + 4*d*x) + a*b*x*(f + 2*g*x)))/(a + b*x^3) +
(9*a^(5/3)*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3)^2 - 2*
Sqrt[3]*b^(1/3)*(5*b^(4/3)*c + 2*a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(
1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(5*b^(4/3)*c - 2*a^(1/3)*b*d + a
*b^(1/3)*f - a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(-5*b^(4/3)*c + 2*a^(
1/3)*b*d - 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])/(54*a^(8/3)*b^2)

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Maple [A]  time = 0.014, size = 506, normalized size = 1.6 \[ \text{result too large to display} \]

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

[Out]

(1/9*(a*g+2*b*d)/a^2*x^5+1/18*(a*f+5*b*c)/a^2*x^4-1/3*h*x^3/b-1/18*(a*g-7*b*d)/a
/b*x^2-1/9*(a*f-4*b*c)/a/b*x-1/6*(a*h+b*e)/b^2)/(b*x^3+a)^2+1/27/a/b^2/(a/b)^(2/
3)*ln(x+(a/b)^(1/3))*f-1/54/a/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*
f+1/27/a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+5/27*
c/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-5/54*c/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1
/3)+(a/b)^(2/3))+5/27*c/a^2/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1
/3)*x-1))-1/27/a*g/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/54/a*g/b^2/(a/b)^(1/3)*ln
(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/27/a*g*3^(1/2)/b^2/(a/b)^(1/3)*arctan(1/3*3^(1
/2)*(2/(a/b)^(1/3)*x-1))-2/27*d/a^2/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/27*d/a^2/b
/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27*d/a^2*3^(1/2)/b/(a/b)^(1/3)*
arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225924, size = 467, normalized size = 1.49 \[ -\frac{{\left (2 \, b d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, b c + a f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} + \frac{4 \, b^{3} d x^{5} + 2 \, a b^{2} g x^{5} + 5 \, b^{3} c x^{4} + a b^{2} f x^{4} - 6 \, a^{2} b h x^{3} + 7 \, a b^{2} d x^{2} - a^{2} b g x^{2} + 8 \, a b^{2} c x - 2 \, a^{2} b f x - 3 \, a^{3} h - 3 \, a^{2} b e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-1/27*(2*b*d*(-a/b)^(1/3) + a*g*(-a/b)^(1/3) + 5*b*c + a*f)*(-a/b)^(1/3)*ln(abs(
x - (-a/b)^(1/3)))/(a^3*b) + 1/27*sqrt(3)*(5*(-a*b^2)^(1/3)*b^2*c + (-a*b^2)^(1/
3)*a*b*f - 2*(-a*b^2)^(2/3)*b*d - (-a*b^2)^(2/3)*a*g)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^3) + 1/54*(5*(-a*b^2)^(1/3)*b^2*c + (-a*b^2)^
(1/3)*a*b*f + 2*(-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^3) + 1/18*(4*b^3*d*x^5 + 2*a*b^2*g*x^5 + 5*b^3*c*x^4 + a
*b^2*f*x^4 - 6*a^2*b*h*x^3 + 7*a*b^2*d*x^2 - a^2*b*g*x^2 + 8*a*b^2*c*x - 2*a^2*b
*f*x - 3*a^3*h - 3*a^2*b*e)/((b*x^3 + a)^2*a^2*b^2)