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

Optimal. Leaf size=297 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left (x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.901279, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left (x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2} \]

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)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 114.252, size = 279, normalized size = 0.94 \[ - \frac{c + d x + e x^{2} + f x^{3} + g x^{4} + h x^{5}}{6 b \left (a + b x^{3}\right )^{2}} - \frac{x \left (4 a g - b d - 3 b f x^{2} + x \left (5 a h - 2 b e\right )\right )}{18 a b^{2} \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (5 a h + b e\right ) - \sqrt [3]{b} \left (2 a g + b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{5}{3}} b^{\frac{8}{3}}} + \frac{\left (\sqrt [3]{a} \left (5 a h + b e\right ) - \sqrt [3]{b} \left (2 a g + b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{5}{3}} b^{\frac{8}{3}}} - \frac{\sqrt{3} \left (5 a^{\frac{4}{3}} h + \sqrt [3]{a} b e + 2 a \sqrt [3]{b} g + b^{\frac{4}{3}} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{5}{3}} b^{\frac{8}{3}}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.461016, size = 287, normalized size = 0.97 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e-2 a \sqrt [3]{b} g-b^{4/3} d\right )}{a^{5/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^{4/3} h-\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{5/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{a^{5/3}}-\frac{9 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{\left (a+b x^3\right )^2}+\frac{3 b^{2/3} (b x (d+2 e x)-a (6 f+x (7 g+8 h x)))}{a \left (a+b x^3\right )}}{54 b^{8/3}} \]

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)^3,x]

[Out]

((-9*b^(2/3)*(b*(c + x*(d + e*x)) - a*(f + x*(g + h*x))))/(a + b*x^3)^2 + (3*b^(
2/3)*(b*x*(d + 2*e*x) - a*(6*f + x*(7*g + 8*h*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]
*(b^(4/3)*d + a^(1/3)*b*e + 2*a*b^(1/3)*g + 5*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*
x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^(4/3)*d - a^(1/3)*b*e + 2*a*b^(1/3)*g - 5*
a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + ((-(b^(4/3)*d) + a^(1/3)*b*e - 2*
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^(5/
3))/(54*b^(8/3))

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Maple [A]  time = 0.016, size = 490, 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)^3,x)

[Out]

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

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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)^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^2/(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**2*(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 = 0.227394, size = 454, normalized size = 1.53 \[ -\frac{{\left (5 \, a h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + b d + 2 \, a g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{4}} - \frac{8 \, a b h x^{5} - 2 \, b^{2} x^{5} e - b^{2} d x^{4} + 7 \, a b g x^{4} + 6 \, a b f x^{3} + 5 \, a^{2} h x^{2} + a b x^{2} e + 2 \, a b d x + 4 \, a^{2} g x + 3 \, a b c + 3 \, a^{2} f}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{4}} \]

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)^3,x, algorithm="giac")

[Out]

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