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

Optimal. Leaf size=276 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{18 a^{5/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{h \log \left (a+b x^3\right )}{3 b^2}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.807343, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{18 a^{5/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{h \log \left (a+b x^3\right )}{3 b^2}+\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )} \]

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)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 118.282, size = 252, normalized size = 0.91 \[ \frac{h \log{\left (a + b x^{3} \right )}}{3 b^{2}} - \frac{x \left (a f - b c + x^{2} \left (a h - b e\right ) + x \left (a g - b d\right )\right )}{3 a b \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (2 a g + b d\right ) - \sqrt [3]{b} \left (a f + 2 b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} \left (2 a g + b d\right ) - \sqrt [3]{b} \left (a f + 2 b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (2 a^{\frac{4}{3}} g + \sqrt [3]{a} b d + a \sqrt [3]{b} f + 2 b^{\frac{4}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{5}{3}}} \]

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)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.339598, size = 268, normalized size = 0.97 \[ \frac{\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} g+\sqrt [3]{a} b d-a \sqrt [3]{b} f-2 b^{4/3} c\right )}{a^{5/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^{4/3} g-\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{a^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{a^{5/3}}+\frac{6 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{a \left (a+b x^3\right )}+6 h \log \left (a+b x^3\right )}{18 b^2} \]

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)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 465, 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)/(b*x^3+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224916, size = 429, normalized size = 1.55 \[ \frac{h{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{{\left (b d - a g\right )} x^{2} +{\left (b c - a f\right )} x + \frac{a^{2} h - a b e}{b}}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \left (-a b^{2}\right )^{\frac{2}{3}} b d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \left (-a b^{2}\right )^{\frac{2}{3}} b d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} - \frac{{\left (a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{2} b^{2} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a b^{3} c + a^{2} b^{2} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3} b^{3}} \]

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, algorithm="giac")

[Out]

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