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

Optimal. Leaf size=289 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{h x}{b^2} \]

[Out]

(h*x)/b^2 - (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a*b^2*(
a + b*x^3)) - ((b^(5/3)*c + a^(2/3)*b*e + 2*a*b^(2/3)*f - 4*a^(5/3)*h)*ArcTan[(a
^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(4/3)*b^(7/3)) - ((b^(2/3
)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^
(7/3)) + ((b^(2/3)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(7/3)) + (g*Log[a + b*x^3])/(3*b^2)

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Rubi [A]  time = 1.04753, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{h x}{b^2} \]

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)^2,x]

[Out]

(h*x)/b^2 - (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a*b^2*(
a + b*x^3)) - ((b^(5/3)*c + a^(2/3)*b*e + 2*a*b^(2/3)*f - 4*a^(5/3)*h)*ArcTan[(a
^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(4/3)*b^(7/3)) - ((b^(2/3
)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^
(7/3)) + ((b^(2/3)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(7/3)) + (g*Log[a + b*x^3])/(3*b^2)

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Rubi in Sympy [A]  time = 160.671, size = 265, normalized size = 0.92 \[ \frac{g \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{h x}{b^{2}} + \frac{x \left (a \left (a h - b e\right ) - b x^{2} \left (a g - b d\right ) - b x \left (a f - b c\right )\right )}{3 a b^{2} \left (a + b x^{3}\right )} - \frac{\left (a^{\frac{2}{3}} \left (4 a h - b e\right ) + b^{\frac{2}{3}} \left (2 a f + b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{7}{3}}} + \frac{\left (a^{\frac{2}{3}} \left (4 a h - b e\right ) + b^{\frac{2}{3}} \left (2 a f + b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{7}{3}}} + \frac{\sqrt{3} \left (4 a^{\frac{5}{3}} h - a^{\frac{2}{3}} b e - 2 a b^{\frac{2}{3}} f - b^{\frac{5}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{7}{3}}} \]

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)**2,x)

[Out]

g*log(a + b*x**3)/(3*b**2) + h*x/b**2 + x*(a*(a*h - b*e) - b*x**2*(a*g - b*d) -
b*x*(a*f - b*c))/(3*a*b**2*(a + b*x**3)) - (a**(2/3)*(4*a*h - b*e) + b**(2/3)*(2
*a*f + b*c))*log(a**(1/3) + b**(1/3)*x)/(9*a**(4/3)*b**(7/3)) + (a**(2/3)*(4*a*h
- b*e) + b**(2/3)*(2*a*f + b*c))*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*
x**2)/(18*a**(4/3)*b**(7/3)) + sqrt(3)*(4*a**(5/3)*h - a**(2/3)*b*e - 2*a*b**(2/
3)*f - b**(5/3)*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(4
/3)*b**(7/3))

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Mathematica [A]  time = 0.385777, size = 285, normalized size = 0.99 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{2/3} b^{4/3} e-4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}+\frac{6 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{a \left (a+b x^3\right )}+6 b^{2/3} g \log \left (a+b x^3\right )+18 b^{2/3} h x}{18 b^{8/3}} \]

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)^2,x]

[Out]

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

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Maple [B]  time = 0.014, size = 504, normalized size = 1.7 \[ \text{result too large to display} \]

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)^2,x)

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227123, size = 451, normalized size = 1.56 \[ \frac{h x}{b^{2}} + \frac{g{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} - \frac{a b d - a^{2} g -{\left (b^{2} c - a b f\right )} x^{2} -{\left (a^{2} h - a b e\right )} x}{3 \,{\left (b x^{3} + a\right )} a b^{2}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} - \frac{{\left (a b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{2} b^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{3} b^{3} h + a^{2} b^{4} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3} b^{5}} \]

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, algorithm="giac")

[Out]

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