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

Optimal. Leaf size=259 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{6 a^{2/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} a^{2/3} b^{5/3}}+\frac{(b e-a h) \log \left (a+b x^3\right )}{3 b^2}+\frac{f x}{b}+\frac{g x^2}{2 b}+\frac{h x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.801398, antiderivative size = 257, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}-a f+b c\right )}{6 a^{2/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} a^{2/3} b^{5/3}}+\frac{(b e-a h) \log \left (a+b x^3\right )}{3 b^2}+\frac{f x}{b}+\frac{g x^2}{2 b}+\frac{h x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{g \int x\, dx}{b} + \frac{h x^{3}}{3 b} + \frac{\int f\, dx}{b} - \frac{\left (a h - b e\right ) \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{\left (\sqrt [3]{a} \left (a g - b d\right ) - \sqrt [3]{b} \left (a f - b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{5}{3}}} - \frac{\left (\sqrt [3]{a} \left (a g - b d\right ) - \sqrt [3]{b} \left (a f - b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{5}{3}}} + \frac{\sqrt{3} \left (a^{\frac{4}{3}} g - \sqrt [3]{a} b d + a \sqrt [3]{b} f - b^{\frac{4}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

g*Integral(x, x)/b + h*x**3/(3*b) + Integral(f, x)/b - (a*h - b*e)*log(a + b*x**
3)/(3*b**2) + (a**(1/3)*(a*g - b*d) - b**(1/3)*(a*f - b*c))*log(a**(1/3) + b**(1
/3)*x)/(3*a**(2/3)*b**(5/3)) - (a**(1/3)*(a*g - b*d) - b**(1/3)*(a*f - b*c))*log
(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(2/3)*b**(5/3)) + sqrt(3)
*(a**(4/3)*g - a**(1/3)*b*d + a*b**(1/3)*f - b**(4/3)*c)*atan(sqrt(3)*(a**(1/3)/
3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(2/3)*b**(5/3))

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Mathematica [A]  time = 0.740207, size = 254, normalized size = 0.98 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g-\sqrt [3]{a} b d+a \sqrt [3]{b} f-b^{4/3} c\right )}{a^{2/3}}+\frac{2 (b e-a h) \log \left (a+b x^3\right )}{\sqrt [3]{b}}+6 b^{2/3} f x+3 b^{2/3} g x^2+2 b^{2/3} h x^3}{6 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 429, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*h*x^3/b+1/2*g*x^2/b+f*x/b-1/3/b^2*a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f+1/3*c/b/
(a/b)^(2/3)*ln(x+(a/b)^(1/3))+1/6/b^2*a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))*f-1/6*c/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3/b^2*a/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+1/3*c/b/(a/b)^(2/3)*3^(1/2
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*
a*g-1/3*d/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/6/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/
3)+(a/b)^(2/3))*a*g+1/6*d/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3/b^
2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a*g+1/3*d*3^(1/2)/
b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3/b^2*ln(b*x^3+a)*a*h+1/
3*e*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)/(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)/(b*x^3 + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 49.168, size = 804, normalized size = 3.1 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{6} + t^{2} \left (27 a^{3} b^{4} h - 27 a^{2} b^{5} e\right ) + t \left (9 a^{4} b^{2} h^{2} - 18 a^{3} b^{3} e h + 9 a^{3} b^{3} f g - 9 a^{2} b^{4} c g - 9 a^{2} b^{4} d f + 9 a^{2} b^{4} e^{2} + 9 a b^{5} c d\right ) + a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{4} b f g h - a^{4} b g^{3} - 3 a^{3} b^{2} c g h - 3 a^{3} b^{2} d f h + 3 a^{3} b^{2} d g^{2} + 3 a^{3} b^{2} e^{2} h - 3 a^{3} b^{2} e f g + a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c d h + 3 a^{2} b^{3} c e g - 3 a^{2} b^{3} c f^{2} - 3 a^{2} b^{3} d^{2} g + 3 a^{2} b^{3} d e f - a^{2} b^{3} e^{3} + 3 a b^{4} c^{2} f - 3 a b^{4} c d e + a b^{4} d^{3} - b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a^{3} b^{4} g - 9 t^{2} a^{2} b^{5} d + 6 t a^{4} b^{2} g h - 6 t a^{3} b^{3} d h - 6 t a^{3} b^{3} e g - 3 t a^{3} b^{3} f^{2} + 6 t a^{2} b^{4} c f + 6 t a^{2} b^{4} d e - 3 t a b^{5} c^{2} + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h - a^{4} b f^{2} h + 2 a^{4} b f g^{2} + 2 a^{3} b^{2} c f h - 2 a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h - 4 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g + a^{3} b^{2} e f^{2} - a^{2} b^{3} c^{2} h + 4 a^{2} b^{3} c d g - 2 a^{2} b^{3} c e f + 2 a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} + a b^{4} c^{2} e - 2 a b^{4} c d^{2}}{a^{4} b g^{3} - 3 a^{3} b^{2} d g^{2} + a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g + 3 a b^{4} c^{2} f - a b^{4} d^{3} - b^{5} c^{3}} \right )} \right )\right )} + \frac{f x}{b} + \frac{g x^{2}}{2 b} + \frac{h x^{3}}{3 b} \]

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)/(b*x**3+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223974, size = 397, normalized size = 1.53 \[ -\frac{{\left (a h - b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} + \frac{2 \, b^{2} h x^{3} + 3 \, b^{2} g x^{2} + 6 \, b^{2} f x}{6 \, b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} - \frac{{\left (b^{7} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{6} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + b^{7} c - a b^{6} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} \]

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)/(b*x^3 + a),x, algorithm="giac")

[Out]

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