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

Optimal. Leaf size=323 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{54 a^{7/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{27 a^{7/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{9 \sqrt{3} a^{7/3} b^{7/3}}+\frac{x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2} \]

[Out]

-(x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*a*b^2*(a + b*x^3)^
2) + (x*(a*(b*e - 7*a*h) + 2*b*(2*b*c + a*f)*x + 3*b*(b*d + a*g)*x^2))/(18*a^2*b
^2*(a + b*x^3)) - ((2*b^(5/3)*c + a^(2/3)*b*e + a*b^(2/3)*f + 2*a^(5/3)*h)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(7/3)) - ((b^
(2/3)*(2*b*c + a*f) - a^(2/3)*(b*e + 2*a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/
3)*b^(7/3)) + ((b^(2/3)*(2*b*c + a*f) - a^(2/3)*(b*e + 2*a*h))*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(7/3))

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Rubi [A]  time = 1.01775, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{54 a^{7/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{27 a^{7/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{9 \sqrt{3} a^{7/3} b^{7/3}}+\frac{x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*a*b^2*(a + b*x^3)^
2) + (x*(a*(b*e - 7*a*h) + 2*b*(2*b*c + a*f)*x + 3*b*(b*d + a*g)*x^2))/(18*a^2*b
^2*(a + b*x^3)) - ((2*b^(5/3)*c + a^(2/3)*b*e + a*b^(2/3)*f + 2*a^(5/3)*h)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(7/3)) - ((b^
(2/3)*(2*b*c + a*f) - a^(2/3)*(b*e + 2*a*h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/
3)*b^(7/3)) + ((b^(2/3)*(2*b*c + a*f) - a^(2/3)*(b*e + 2*a*h))*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(7/3))

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Rubi in Sympy [A]  time = 144.401, size = 299, normalized size = 0.93 \[ \frac{x \left (a \left (a h - b e\right ) - b x^{2} \left (a g - b d\right ) - b x \left (a f - b c\right )\right )}{6 a b^{2} \left (a + b x^{3}\right )^{2}} - \frac{x \left (a \left (7 a h - b e\right ) - 3 b x^{2} \left (a g + b d\right ) - 2 b x \left (a f + 2 b c\right )\right )}{18 a^{2} b^{2} \left (a + b x^{3}\right )} + \frac{\left (a^{\frac{2}{3}} \left (2 a h + b e\right ) - b^{\frac{2}{3}} \left (a f + 2 b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{7}{3}}} - \frac{\left (a^{\frac{2}{3}} \left (2 a h + b e\right ) - b^{\frac{2}{3}} \left (a f + 2 b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{7}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (2 a^{\frac{5}{3}} h + a^{\frac{2}{3}} b e + a b^{\frac{2}{3}} f + 2 b^{\frac{5}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*(a*h - b*e) - b*x**2*(a*g - b*d) - b*x*(a*f - b*c))/(6*a*b**2*(a + b*x**3)*
*2) - x*(a*(7*a*h - b*e) - 3*b*x**2*(a*g + b*d) - 2*b*x*(a*f + 2*b*c))/(18*a**2*
b**2*(a + b*x**3)) + (a**(2/3)*(2*a*h + b*e) - b**(2/3)*(a*f + 2*b*c))*log(a**(1
/3) + b**(1/3)*x)/(27*a**(7/3)*b**(7/3)) - (a**(2/3)*(2*a*h + b*e) - b**(2/3)*(a
*f + 2*b*c))*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(7/3)*b*
*(7/3)) - sqrt(3)*(2*a**(5/3)*h + a**(2/3)*b*e + a*b**(2/3)*f + 2*b**(5/3)*c)*at
an(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(7/3)*b**(7/3))

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Mathematica [A]  time = 0.576882, size = 297, normalized size = 0.92 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b e-2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} b e+2 a^{5/3} h-a b^{2/3} f-2 b^{5/3} c\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )-\frac{3 \sqrt [3]{a} \sqrt [3]{b} \left (a^2 (6 g+7 h x)-a b x (e+2 f x)-4 b^2 c x^2\right )}{a+b x^3}+\frac{9 a^{4/3} \sqrt [3]{b} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{\left (a+b x^3\right )^2}}{54 a^{7/3} b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-3*a^(1/3)*b^(1/3)*(-4*b^2*c*x^2 - a*b*x*(e + 2*f*x) + a^2*(6*g + 7*h*x)))/(a
+ b*x^3) + (9*a^(4/3)*b^(1/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))
))/(a + b*x^3)^2 - 2*Sqrt[3]*(2*b^(5/3)*c + a^(2/3)*b*e + a*b^(2/3)*f + 2*a^(5/3
)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-2*b^(5/3)*c + a^(2/3)*b*e
- a*b^(2/3)*f + 2*a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x] + (2*b^(5/3)*c - a^(2/3)*
b*e + a*b^(2/3)*f - 2*a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/(54*a^(7/3)*b^(7/3))

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Maple [A]  time = 0.015, size = 498, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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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)*x/(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)*x/(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(x*(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 = 0.22797, size = 481, normalized size = 1.49 \[ -\frac{{\left (2 \, b^{2} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{2} h + a b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} + \frac{4 \, b^{3} c x^{5} + 2 \, a b^{2} f x^{5} - 7 \, a^{2} b h x^{4} + a b^{2} x^{4} e - 6 \, a^{2} b g x^{3} + 7 \, a b^{2} c x^{2} - a^{2} b f x^{2} - 4 \, a^{3} h x - 2 \, a^{2} b x e - 3 \, a^{2} b d - 3 \, a^{3} g}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \]

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)*x/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

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