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

Optimal. Leaf size=311 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{18 a^{2/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} g+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 b^2 \left (a+b x^3\right )}+\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 1.29019, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{18 a^{2/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-4 a f)-\sqrt [3]{a} (2 b d-5 a g)\right )}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} g+2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{(b e-2 a h) \log \left (a+b x^3\right )}{3 b^3}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 b^2 \left (a+b x^3\right )}+\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(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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Mathematica [A]  time = 0.39672, size = 294, normalized size = 0.95 \[ \frac{-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d-4 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} g-2 \sqrt [3]{a} b d+4 a \sqrt [3]{b} f-b^{4/3} c\right )}{a^{2/3}}-\frac{6 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{a+b x^3}+6 (b e-2 a h) \log \left (a+b x^3\right )+18 b f x+9 b g x^2+6 b h x^3}{18 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 533, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*h*x^3/b^2+1/2*g*x^2/b^2+f*x/b^2+1/3/b^2/(b*x^3+a)*x^2*a*g-1/3*x^2*d/(b*x^3+a
)/b+1/3*a/b^2*x/(b*x^3+a)*f-1/3/b*x/(b*x^3+a)*c-1/3/b^3/(b*x^3+a)*a^2*h+1/3*a/b^
2/(b*x^3+a)*e-4/9*a/b^3*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+2/9*a/b^3*f/(a/b)^(2/3)*
ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9*a/b^3*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(a/b)^(1/3)*x-1))+1/9/b^2*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/18/b^2*c/(a/
b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/9/b^2*c/(a/b)^(2/3)*3^(1/2)*arctan(
1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+5/9/b^3*a*g/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-5/18/
b^3*a*g/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-5/9/b^3*a*g*3^(1/2)/(a/b)^
(1/3)*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))*d+1/9/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+2/9/b^2*3^(1/2)/(a
/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-2/3/b^3*ln(b*x^3+a)*a*h+1/3/
b^2*ln(b*x^3+a)*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^3/(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^3/(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**3*(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 = 0.22687, size = 475, normalized size = 1.53 \[ -\frac{{\left (2 \, a h - b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{4}} - \frac{a^{2} h +{\left (b^{2} d - a b g\right )} x^{2} - a b e +{\left (b^{2} c - a b f\right )} x}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{4}} - \frac{{\left (2 \, b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a b^{3} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + b^{4} c - 4 \, a b^{3} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac{2 \, b^{4} h x^{3} + 3 \, b^{4} g x^{2} + 6 \, b^{4} f x}{6 \, b^{6}} \]

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^3/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

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