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

Optimal. Leaf size=345 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{54 a^{4/3} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{27 a^{4/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right )}{9 \sqrt{3} a^{4/3} b^{10/3}}-\frac{x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{h x}{b^3} \]

[Out]

(h*x)/b^3 + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*b^3*(a
+ b*x^3)^2) - (x*(a*(7*b*e - 13*a*h) - 2*b*(b*c - 4*a*f)*x - 3*b*(b*d - 3*a*g)*x
^2))/(18*a*b^3*(a + b*x^3)) - ((b^(5/3)*c + 2*a^(2/3)*b*e + 5*a*b^(2/3)*f - 14*a
^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*
b^(10/3)) - ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(1/3) + b^(
1/3)*x])/(27*a^(4/3)*b^(10/3)) + ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*
h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(10/3)) + (g*L
og[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 1.71711, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{54 a^{4/3} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right )}{27 a^{4/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right )}{9 \sqrt{3} a^{4/3} b^{10/3}}-\frac{x \left (-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right )}{18 a b^3 \left (a+b x^3\right )}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 b^3 \left (a+b x^3\right )^2}+\frac{g \log \left (a+b x^3\right )}{3 b^3}+\frac{h x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(h*x)/b^3 + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*b^3*(a
+ b*x^3)^2) - (x*(a*(7*b*e - 13*a*h) - 2*b*(b*c - 4*a*f)*x - 3*b*(b*d - 3*a*g)*x
^2))/(18*a*b^3*(a + b*x^3)) - ((b^(5/3)*c + 2*a^(2/3)*b*e + 5*a*b^(2/3)*f - 14*a
^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*
b^(10/3)) - ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*h))*Log[a^(1/3) + b^(
1/3)*x])/(27*a^(4/3)*b^(10/3)) + ((b^(2/3)*(b*c + 5*a*f) - 2*a^(2/3)*(b*e - 7*a*
h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(10/3)) + (g*L
og[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**4*(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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Mathematica [A]  time = 0.594297, size = 342, normalized size = 0.99 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^{2/3} b^{4/3} e+14 a^{5/3} \sqrt [3]{b} h+5 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^{2/3} b^{4/3} e+14 a^{5/3} \sqrt [3]{b} h+5 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{2/3} b^{4/3} e-14 a^{5/3} \sqrt [3]{b} h+5 a b f+b^2 c\right )}{a^{4/3}}-\frac{9 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{\left (a+b x^3\right )^2}+\frac{3 b^{2/3} \left (a^2 (12 g+13 h x)-a b (6 d+x (7 e+8 f x))+2 b^2 c x^2\right )}{a \left (a+b x^3\right )}+18 b^{2/3} g \log \left (a+b x^3\right )+54 b^{2/3} h x}{54 b^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

(54*b^(2/3)*h*x - (9*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x)))
)/(a + b*x^3)^2 + (3*b^(2/3)*(2*b^2*c*x^2 + a^2*(12*g + 13*h*x) - a*b*(6*d + x*(
7*e + 8*f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c + 2*a^(2/3)*b^(4/3)*e + 5*a*
b*f - 14*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 - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(1/3)
+ b^(1/3)*x])/a^(4/3) + ((b^2*c - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14*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) + 18*b^(2/3)*g*
Log[a + b*x^3])/(54*b^(11/3))

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Maple [B]  time = 0.019, size = 621, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27/b^2/a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c-5/18*a/
(b*x^3+a)^2/b^2*x^2*f+1/9/(b*x^3+a)^2/a*x^5*c-1/18/b/(b*x^3+a)^2*x^2*c-4/9/(b*x^
3+a)^2/b*x^5*f-5/27/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f+5/54/b^3/(a/b)^(1/3)*ln(
x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/3/b^3*g*ln(a*(b*x^3+a))-1/27/b^2/a/(a/b)^(1/3
)*ln(x+(a/b)^(1/3))*c+h*x/b^3+2/3/b^2/(b*x^3+a)^2*x^3*a*g-1/3/b/(b*x^3+a)^2*d*x^
3+1/2/b^3/(b*x^3+a)^2*a^2*g-1/6/b^2/(b*x^3+a)^2*a*d+1/54/b^2/a/(a/b)^(1/3)*ln(x^
2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-2/9/b^2/(b*x^3+a)^2*a*e*x+5/9/b^3/(b*x^3+a)^2*x*a
^2*h-14/27/b^4*a*h/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+2
/27/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+7/27/b^4*a
*h/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-14/27/b^4*a*h/(a/b)^(2/3)*ln(x+
(a/b)^(1/3))+13/18/b^2/(b*x^3+a)^2*x^4*a*h+5/27/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-7/18/b/(b*x^3+a)^2*x^4*e+2/27/b^3/(a/b)^(2/3)*
ln(x+(a/b)^(1/3))*e-1/27/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*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^4/(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^4/(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**4*(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 = 0.228512, size = 541, normalized size = 1.57 \[ \frac{h x}{b^{3}} + \frac{g{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{4}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{4}} + \frac{2 \,{\left (b^{3} c - 4 \, a b^{2} f\right )} x^{5} +{\left (13 \, a^{2} b h - 7 \, a b^{2} e\right )} x^{4} - 3 \, a^{2} b d + 9 \, a^{3} g - 6 \,{\left (a b^{2} d - 2 \, a^{2} b g\right )} x^{3} -{\left (a b^{2} c + 5 \, a^{2} b f\right )} x^{2} + 2 \,{\left (5 \, a^{3} h - 2 \, a^{2} b e\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{3}} - \frac{{\left (a b^{6} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{3} b^{4} h + 2 \, a^{2} b^{5} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{7}} \]

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^4/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

h*x/b^3 + 1/3*g*ln(abs(b*x^3 + a))/b^3 - 1/27*sqrt(3)*(14*(-a*b^2)^(1/3)*a^2*h -
2*(-a*b^2)^(1/3)*a*b*e + (-a*b^2)^(2/3)*b*c + 5*(-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^4) - 1/54*(14*(-a*b^2)^(1/3)*a
^2*h - 2*(-a*b^2)^(1/3)*a*b*e - (-a*b^2)^(2/3)*b*c - 5*(-a*b^2)^(2/3)*a*f)*ln(x^
2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^4) + 1/18*(2*(b^3*c - 4*a*b^2*f)*x^5 +
(13*a^2*b*h - 7*a*b^2*e)*x^4 - 3*a^2*b*d + 9*a^3*g - 6*(a*b^2*d - 2*a^2*b*g)*x^
3 - (a*b^2*c + 5*a^2*b*f)*x^2 + 2*(5*a^3*h - 2*a^2*b*e)*x)/((b*x^3 + a)^2*a*b^3)
- 1/27*(a*b^6*c*(-a/b)^(1/3) + 5*a^2*b^5*f*(-a/b)^(1/3) - 14*a^3*b^4*h + 2*a^2*
b^5*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^3*b^7)