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

Optimal. Leaf size=275 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 1.82005, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \left (a h - b e\right ) \int \frac{1}{b^{2}}\, dx + \frac{f \int x\, dx}{b} + \frac{g x^{3}}{3 b} + \frac{h x^{4}}{4 b} - \frac{\left (a g - b d\right ) \log{\left (a + b x^{3} \right )}}{3 b^{2}} - \frac{\sqrt{3} \left (a^{\frac{2}{3}} \left (a h - b e\right ) - b^{\frac{2}{3}} \left (a f - b c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} b^{\frac{7}{3}}} + \frac{\left (a^{\frac{2}{3}} \left (a h - b e\right ) + b^{\frac{2}{3}} \left (a f - b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{a} b^{\frac{7}{3}}} - \frac{\left (a^{\frac{2}{3}} \left (a h - b e\right ) + b^{\frac{2}{3}} \left (a f - b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{a} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*h - b*e)*Integral(b**(-2), x) + f*Integral(x, x)/b + g*x**3/(3*b) + h*x**4/(
4*b) - (a*g - b*d)*log(a + b*x**3)/(3*b**2) - sqrt(3)*(a**(2/3)*(a*h - b*e) - b*
*(2/3)*(a*f - b*c))*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(
1/3)*b**(7/3)) + (a**(2/3)*(a*h - b*e) + b**(2/3)*(a*f - b*c))*log(a**(1/3) + b*
*(1/3)*x)/(3*a**(1/3)*b**(7/3)) - (a**(2/3)*(a*h - b*e) + b**(2/3)*(a*f - b*c))*
log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(1/3)*b**(7/3))

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Mathematica [A]  time = 1.26289, size = 272, normalized size = 0.99 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{\sqrt [3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+4 \sqrt [3]{b} (b d-a g) \log \left (a+b x^3\right )+12 \sqrt [3]{b} x (b e-a h)+6 b^{4/3} f x^2+4 b^{4/3} g x^3+3 b^{4/3} h x^4}{12 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(12*b^(1/3)*(b*e - a*h)*x + 6*b^(4/3)*f*x^2 + 4*b^(4/3)*g*x^3 + 3*b^(4/3)*h*x^4
- (4*Sqrt[3]*(b^(5/3)*c - a^(2/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*
b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-(b^(5/3)*c) - a^(2/3)*b*e + a*b^(2/
3)*f + a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (2*(b^(5/3)*c + a^(2/3)*b*
e - a*b^(2/3)*f - a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(
1/3) + 4*b^(1/3)*(b*d - a*g)*Log[a + b*x^3])/(12*b^(7/3))

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Maple [B]  time = 0.006, size = 455, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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/(b*x^3 + a),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/(b*x^3 + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 56.4232, size = 811, normalized size = 2.95 \[ \operatorname{RootSum}{\left (27 t^{3} a b^{7} + t^{2} \left (27 a^{2} b^{5} g - 27 a b^{6} d\right ) + t \left (- 9 a^{3} b^{3} f h + 9 a^{3} b^{3} g^{2} + 9 a^{2} b^{4} c h - 18 a^{2} b^{4} d g + 9 a^{2} b^{4} e f - 9 a b^{5} c e + 9 a b^{5} d^{2}\right ) - a^{5} h^{3} + 3 a^{4} b e h^{2} - 3 a^{4} b f g h + a^{4} b g^{3} + 3 a^{3} b^{2} c g h + 3 a^{3} b^{2} d f h - 3 a^{3} b^{2} d g^{2} - 3 a^{3} b^{2} e^{2} h + 3 a^{3} b^{2} e f g - a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c d h - 3 a^{2} b^{3} c e g + 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g - 3 a^{2} b^{3} d e f + a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + 3 a b^{4} c d e - a b^{4} d^{3} + b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t^{2} a^{2} b^{5} f + 9 t^{2} a b^{6} c + 3 t a^{4} b^{2} h^{2} - 6 t a^{3} b^{3} e h - 6 t a^{3} b^{3} f g + 6 t a^{2} b^{4} c g + 6 t a^{2} b^{4} d f + 3 t a^{2} b^{4} e^{2} - 6 t a b^{5} c d + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h + 2 a^{4} b f^{2} h - a^{4} b f g^{2} - 4 a^{3} b^{2} c f h + a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h + 2 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g - 2 a^{3} b^{2} e f^{2} + 2 a^{2} b^{3} c^{2} h - 2 a^{2} b^{3} c d g + 4 a^{2} b^{3} c e f - a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} - 2 a b^{4} c^{2} e + a b^{4} c d^{2}}{a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{3} b^{2} e^{2} h - a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c f^{2} - a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{2}}{2 b} + \frac{g x^{3}}{3 b} + \frac{h x^{4}}{4 b} - \frac{x \left (a h - b e\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a*b**7 + _t**2*(27*a**2*b**5*g - 27*a*b**6*d) + _t*(-9*a**3*b**
3*f*h + 9*a**3*b**3*g**2 + 9*a**2*b**4*c*h - 18*a**2*b**4*d*g + 9*a**2*b**4*e*f
- 9*a*b**5*c*e + 9*a*b**5*d**2) - a**5*h**3 + 3*a**4*b*e*h**2 - 3*a**4*b*f*g*h +
a**4*b*g**3 + 3*a**3*b**2*c*g*h + 3*a**3*b**2*d*f*h - 3*a**3*b**2*d*g**2 - 3*a*
*3*b**2*e**2*h + 3*a**3*b**2*e*f*g - a**3*b**2*f**3 - 3*a**2*b**3*c*d*h - 3*a**2
*b**3*c*e*g + 3*a**2*b**3*c*f**2 + 3*a**2*b**3*d**2*g - 3*a**2*b**3*d*e*f + a**2
*b**3*e**3 - 3*a*b**4*c**2*f + 3*a*b**4*c*d*e - a*b**4*d**3 + b**5*c**3, Lambda(
_t, _t*log(x + (-9*_t**2*a**2*b**5*f + 9*_t**2*a*b**6*c + 3*_t*a**4*b**2*h**2 -
6*_t*a**3*b**3*e*h - 6*_t*a**3*b**3*f*g + 6*_t*a**2*b**4*c*g + 6*_t*a**2*b**4*d*
f + 3*_t*a**2*b**4*e**2 - 6*_t*a*b**5*c*d + a**5*g*h**2 - a**4*b*d*h**2 - 2*a**4
*b*e*g*h + 2*a**4*b*f**2*h - a**4*b*f*g**2 - 4*a**3*b**2*c*f*h + a**3*b**2*c*g**
2 + 2*a**3*b**2*d*e*h + 2*a**3*b**2*d*f*g + a**3*b**2*e**2*g - 2*a**3*b**2*e*f**
2 + 2*a**2*b**3*c**2*h - 2*a**2*b**3*c*d*g + 4*a**2*b**3*c*e*f - a**2*b**3*d**2*
f - a**2*b**3*d*e**2 - 2*a*b**4*c**2*e + a*b**4*c*d**2)/(a**5*h**3 - 3*a**4*b*e*
h**2 + 3*a**3*b**2*e**2*h - a**3*b**2*f**3 + 3*a**2*b**3*c*f**2 - a**2*b**3*e**3
- 3*a*b**4*c**2*f + b**5*c**3)))) + f*x**2/(2*b) + g*x**3/(3*b) + h*x**4/(4*b)
- x*(a*h - b*e)/b**2

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GIAC/XCAS [A]  time = 0.227269, size = 428, normalized size = 1.56 \[ \frac{{\left (b d - a g\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} + \frac{3 \, b^{3} h x^{4} + 4 \, b^{3} g x^{3} + 6 \, b^{3} f x^{2} - 12 \, a b^{2} h x + 12 \, b^{3} x e}{12 \, b^{4}} - \frac{{\left (b^{9} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{8} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{7} h - a b^{8} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{9}} \]

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/(b*x^3 + a),x, algorithm="giac")

[Out]

1/3*(b*d - a*g)*ln(abs(b*x^3 + a))/b^2 + 1/3*sqrt(3)*((-a*b^2)^(1/3)*a^2*h - (-a
*b^2)^(1/3)*a*b*e - (-a*b^2)^(2/3)*b*c + (-a*b^2)^(2/3)*a*f)*arctan(1/3*sqrt(3)*
(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3) + 1/6*((-a*b^2)^(1/3)*a^2*h - (-a*b^2
)^(1/3)*a*b*e + (-a*b^2)^(2/3)*b*c - (-a*b^2)^(2/3)*a*f)*ln(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a*b^3) + 1/12*(3*b^3*h*x^4 + 4*b^3*g*x^3 + 6*b^3*f*x^2 - 12*a*
b^2*h*x + 12*b^3*x*e)/b^4 - 1/3*(b^9*c*(-a/b)^(1/3) - a*b^8*f*(-a/b)^(1/3) + a^2
*b^7*h - a*b^8*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^9)