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

Optimal. Leaf size=290 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{18 a^{2/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} h+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.994918, antiderivative size = 288, 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 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}-4 a g+b d\right )}{18 a^{2/3} b^{7/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} h+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{f \log \left (a+b x^3\right )}{3 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*g*x)/(3*b^2) + (5*h*x^2)/(6*b^2) - (c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/
(3*b*(a + b*x^3)) - ((b^(4/3)*d + 2*a^(1/3)*b*e - 4*a*b^(1/3)*g - 5*a^(4/3)*h)*A
rcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(8/3)) +
((b^(1/3)*(b*d - 4*a*g) - a^(1/3)*(2*b*e - 5*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*
a^(2/3)*b^(8/3)) - ((b*d - 4*a*g - (a^(1/3)*(2*b*e - 5*a*h))/b^(1/3))*Log[a^(2/3
) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(7/3)) + (f*Log[a + b*x^3])/
(3*b^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{c + d x + e x^{2} + f x^{3} + g x^{4} + h x^{5}}{3 b \left (a + b x^{3}\right )} + \frac{f \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{4 g x}{3 b^{2}} + \frac{5 h \int x\, dx}{3 b^{2}} + \frac{\left (\sqrt [3]{a} \left (5 a h - 2 b e\right ) - \sqrt [3]{b} \left (4 a g - b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{8}{3}}} - \frac{\left (\sqrt [3]{a} \left (5 a h - 2 b e\right ) - \sqrt [3]{b} \left (4 a g - b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{8}{3}}} + \frac{\sqrt{3} \left (5 a^{\frac{4}{3}} h - 2 \sqrt [3]{a} b e + 4 a \sqrt [3]{b} g - b^{\frac{4}{3}} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x + e*x**2 + f*x**3 + g*x**4 + h*x**5)/(3*b*(a + b*x**3)) + f*log(a + b*
x**3)/(3*b**2) + 4*g*x/(3*b**2) + 5*h*Integral(x, x)/(3*b**2) + (a**(1/3)*(5*a*h
- 2*b*e) - b**(1/3)*(4*a*g - b*d))*log(a**(1/3) + b**(1/3)*x)/(9*a**(2/3)*b**(8
/3)) - (a**(1/3)*(5*a*h - 2*b*e) - b**(1/3)*(4*a*g - b*d))*log(a**(2/3) - a**(1/
3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(2/3)*b**(8/3)) + sqrt(3)*(5*a**(4/3)*h -
2*a**(1/3)*b*e + 4*a*b**(1/3)*g - b**(4/3)*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1
/3)*x/3)/a**(1/3))/(9*a**(2/3)*b**(8/3))

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Mathematica [A]  time = 0.368352, size = 280, normalized size = 0.97 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} h-2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{4/3} h-2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} h-2 \sqrt [3]{a} b e+4 a \sqrt [3]{b} g-b^{4/3} d\right )}{a^{2/3}}-\frac{6 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{a+b x^3}+6 b^{2/3} f \log \left (a+b x^3\right )+18 b^{2/3} g x+9 b^{2/3} h x^2}{18 b^{8/3}} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.014, size = 506, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*h*x^2/b^2+g*x/b^2+1/3/b^2/(b*x^3+a)*x^2*a*h-1/3/b*x^2/(b*x^3+a)*e+1/3/b^2/(b
*x^3+a)*x*a*g-1/3/b*x/(b*x^3+a)*d+1/3*a/b^2/(b*x^3+a)*f-1/3/b/(b*x^3+a)*c-4/9/b^
3*a*g/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+2/9/b^3*a*g/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)
+(a/b)^(2/3))-4/9/b^3*a*g/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*
x-1))+1/9/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-1/18/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))*d+1/9/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(
1/3)*x-1))*d+5/9/b^3*a*h/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-5/18/b^3*a*h/(a/b)^(1/3)*
ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-5/9/b^3*a*h*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))-2/9/b^2*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/9/b^2*e/(a/b
)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/9/b^2*e*3^(1/2)/(a/b)^(1/3)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*f*ln(b*x^3+a)/b^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)*x^2/(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^2/(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**2*(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 = 0.227333, size = 444, normalized size = 1.53 \[ \frac{f{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{{\left (a h - b e\right )} x^{2} - b c + a f -{\left (b d - a g\right )} x}{3 \,{\left (b x^{3} + a\right )} b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{4}} + \frac{b^{2} h x^{2} + 2 \, b^{2} g x}{2 \, b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{4}} + \frac{{\left (5 \, a b^{3} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, b^{4} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - b^{4} d + 4 \, a b^{3} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{5}} \]

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^2/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

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