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

Optimal. Leaf size=313 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 b^{8/3}}-\frac{a (b e-a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{x (b c-a f)}{b^2}+\frac{x^2 (b d-a g)}{2 b^2}+\frac{x^3 (b e-a h)}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b} \]

[Out]

((b*c - a*f)*x)/b^2 + ((b*d - a*g)*x^2)/(2*b^2) + ((b*e - a*h)*x^3)/(3*b^2) + (f
*x^4)/(4*b) + (g*x^5)/(5*b) + (h*x^6)/(6*b) + (a^(1/3)*(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))])/(S
qrt[3]*b^(8/3)) - (a^(1/3)*(b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*Log[a^(1/
3) + b^(1/3)*x])/(3*b^(8/3)) + (a^(1/3)*(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*b^(8/3)) - (a*(b*e - a*h)
*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 1.97049, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 b^{8/3}}-\frac{a (b e-a h) \log \left (a+b x^3\right )}{3 b^3}+\frac{x (b c-a f)}{b^2}+\frac{x^2 (b d-a g)}{2 b^2}+\frac{x^3 (b e-a h)}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b} \]

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),x]

[Out]

((b*c - a*f)*x)/b^2 + ((b*d - a*g)*x^2)/(2*b^2) + ((b*e - a*h)*x^3)/(3*b^2) + (f
*x^4)/(4*b) + (g*x^5)/(5*b) + (h*x^6)/(6*b) + (a^(1/3)*(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))])/(S
qrt[3]*b^(8/3)) - (a^(1/3)*(b^(1/3)*(b*c - a*f) - a^(1/3)*(b*d - a*g))*Log[a^(1/
3) + b^(1/3)*x])/(3*b^(8/3)) + (a^(1/3)*(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*b^(8/3)) - (a*(b*e - a*h)
*Log[a + b*x^3])/(3*b^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt [3]{a} \left (\sqrt [3]{a} \left (a g - b d\right ) - \sqrt [3]{b} \left (a f - b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{8}{3}}} + \frac{\sqrt [3]{a} \left (\sqrt [3]{a} \left (a g - b d\right ) - \sqrt [3]{b} \left (a f - b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{8}{3}}} - \frac{\sqrt{3} \sqrt [3]{a} \left (a^{\frac{4}{3}} g - \sqrt [3]{a} b d + a \sqrt [3]{b} f - b^{\frac{4}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{8}{3}}} + \frac{a \left (a h - b e\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} - \left (a f - b c\right ) \int \frac{1}{b^{2}}\, dx + \frac{f x^{4}}{4 b} + \frac{g x^{5}}{5 b} + \frac{h x^{6}}{6 b} - \frac{x^{3} \left (a h - b e\right )}{3 b^{2}} - \frac{\left (a g - b d\right ) \int x\, dx}{b^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.514915, size = 299, normalized size = 0.96 \[ \frac{10 \sqrt [3]{a} \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )-20 \sqrt [3]{a} \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )-20 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g-\sqrt [3]{a} b d+a \sqrt [3]{b} f-b^{4/3} c\right )+60 b x (b c-a f)+30 b x^2 (b d-a g)+20 b x^3 (b e-a h)+20 a (a h-b e) \log \left (a+b x^3\right )+15 b^2 f x^4+12 b^2 g x^5+10 b^2 h x^6}{60 b^3} \]

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),x]

[Out]

(60*b*(b*c - a*f)*x + 30*b*(b*d - a*g)*x^2 + 20*b*(b*e - a*h)*x^3 + 15*b^2*f*x^4
+ 12*b^2*g*x^5 + 10*b^2*h*x^6 - 20*Sqrt[3]*a^(1/3)*b^(1/3)*(-(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]] -
20*a^(1/3)*b^(1/3)*(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] + 10*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a
^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*a*(-(b*e) + a*h)*L
og[a + b*x^3])/(60*b^3)

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Maple [B]  time = 0.007, size = 505, normalized size = 1.6 \[ \text{result too large to display} \]

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),x)

[Out]

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

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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),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),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 47.2096, size = 842, normalized size = 2.69 \[ \operatorname{RootSum}{\left (27 t^{3} b^{9} + t^{2} \left (- 27 a^{2} b^{6} h + 27 a b^{7} e\right ) + t \left (9 a^{4} b^{3} h^{2} - 18 a^{3} b^{4} e h + 9 a^{3} b^{4} f g - 9 a^{2} b^{5} c g - 9 a^{2} b^{5} d f + 9 a^{2} b^{5} e^{2} + 9 a b^{6} c d\right ) - a^{6} h^{3} + 3 a^{5} b e h^{2} - 3 a^{5} b f g h + a^{5} b g^{3} + 3 a^{4} b^{2} c g h + 3 a^{4} b^{2} d f h - 3 a^{4} b^{2} d g^{2} - 3 a^{4} b^{2} e^{2} h + 3 a^{4} b^{2} e f g - a^{4} b^{2} f^{3} - 3 a^{3} b^{3} c d h - 3 a^{3} b^{3} c e g + 3 a^{3} b^{3} c f^{2} + 3 a^{3} b^{3} d^{2} g - 3 a^{3} b^{3} d e f + a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + 3 a^{2} b^{4} c d e - a^{2} b^{4} d^{3} + a b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a b^{6} g - 9 t^{2} b^{7} d - 6 t a^{3} b^{3} g h + 6 t a^{2} b^{4} d h + 6 t a^{2} b^{4} e g + 3 t a^{2} b^{4} f^{2} - 6 t a b^{5} c f - 6 t a b^{5} d e + 3 t b^{6} c^{2} + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h - a^{4} b f^{2} h + 2 a^{4} b f g^{2} + 2 a^{3} b^{2} c f h - 2 a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h - 4 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g + a^{3} b^{2} e f^{2} - a^{2} b^{3} c^{2} h + 4 a^{2} b^{3} c d g - 2 a^{2} b^{3} c e f + 2 a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} + a b^{4} c^{2} e - 2 a b^{4} c d^{2}}{a^{4} b g^{3} - 3 a^{3} b^{2} d g^{2} + a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g + 3 a b^{4} c^{2} f - a b^{4} d^{3} - b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{4}}{4 b} + \frac{g x^{5}}{5 b} + \frac{h x^{6}}{6 b} - \frac{x^{3} \left (a h - b e\right )}{3 b^{2}} - \frac{x^{2} \left (a g - b d\right )}{2 b^{2}} - \frac{x \left (a f - b c\right )}{b^{2}} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223959, size = 477, normalized size = 1.52 \[ \frac{{\left (a^{2} h - a b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} + \frac{10 \, b^{5} h x^{6} + 12 \, b^{5} g x^{5} + 15 \, b^{5} f x^{4} - 20 \, a b^{4} h x^{3} + 20 \, b^{5} x^{3} e + 30 \, b^{5} d x^{2} - 30 \, a b^{4} g x^{2} + 60 \, b^{5} c x - 60 \, a b^{4} f x}{60 \, b^{6}} + \frac{{\left (a b^{12} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{11} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{12} c - a^{2} b^{11} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{13}} \]

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),x, algorithm="giac")

[Out]

1/3*(a^2*h - a*b*e)*ln(abs(b*x^3 + a))/b^3 - 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))/b^4 - 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))/b^4 + 1/60*(10*b^5*h*x^6 + 12*b^5*g*x^5 + 15*b^5*f*x^4 - 20*a*b
^4*h*x^3 + 20*b^5*x^3*e + 30*b^5*d*x^2 - 30*a*b^4*g*x^2 + 60*b^5*c*x - 60*a*b^4*
f*x)/b^6 + 1/3*(a*b^12*d*(-a/b)^(1/3) - a^2*b^11*g*(-a/b)^(1/3) + a*b^12*c - a^2
*b^11*f)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^13)