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

Optimal. Leaf size=294 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 1.85241, antiderivative size = 294, 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 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 b^{8/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} b^{8/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 b^{8/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.442584, size = 290, normalized size = 0.99 \[ \frac{10 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+20 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )-20 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )+20 b^{2/3} (b c-a f) \log \left (a+b x^3\right )+60 b^{2/3} x (b d-a g)+30 b^{2/3} x^2 (b e-a h)+20 b^{5/3} f x^3+15 b^{5/3} g x^4+12 b^{5/3} h x^5}{60 b^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

(60*b^(2/3)*(b*d - a*g)*x + 30*b^(2/3)*(b*e - a*h)*x^2 + 20*b^(5/3)*f*x^3 + 15*b
^(5/3)*g*x^4 + 12*b^(5/3)*h*x^5 - 20*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e
+ a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 20*a^(
1/3)*(-(b^(4/3)*d) + a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3
)*x] + 10*a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3
) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*b^(2/3)*(b*c - a*f)*Log[a + b*x^3])/(6
0*b^(8/3))

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Maple [B]  time = 0.007, size = 483, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 95.2032, size = 789, normalized size = 2.68 \[ \operatorname{RootSum}{\left (27 t^{3} b^{8} + t^{2} \left (27 a b^{6} f - 27 b^{7} c\right ) + t \left (9 a^{3} b^{3} g h - 9 a^{2} b^{4} d h - 9 a^{2} b^{4} e g + 9 a^{2} b^{4} f^{2} - 18 a b^{5} c f + 9 a b^{5} d e + 9 b^{6} c^{2}\right ) + a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{4} b f g h - a^{4} b g^{3} - 3 a^{3} b^{2} c g h - 3 a^{3} b^{2} d f h + 3 a^{3} b^{2} d g^{2} + 3 a^{3} b^{2} e^{2} h - 3 a^{3} b^{2} e f g + a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c d h + 3 a^{2} b^{3} c e g - 3 a^{2} b^{3} c f^{2} - 3 a^{2} b^{3} d^{2} g + 3 a^{2} b^{3} d e f - a^{2} b^{3} e^{3} + 3 a b^{4} c^{2} f - 3 a b^{4} c d e + a b^{4} d^{3} - b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a b^{5} h - 9 t^{2} b^{6} e + 6 t a^{2} b^{3} f h + 3 t a^{2} b^{3} g^{2} - 6 t a b^{4} c h - 6 t a b^{4} d g - 6 t a b^{4} e f + 6 t b^{5} c e + 3 t b^{5} d^{2} + 2 a^{4} g h^{2} - 2 a^{3} b d h^{2} - 4 a^{3} b e g h + a^{3} b f^{2} h + a^{3} b f g^{2} - 2 a^{2} b^{2} c f h - a^{2} b^{2} c g^{2} + 4 a^{2} b^{2} d e h - 2 a^{2} b^{2} d f g + 2 a^{2} b^{2} e^{2} g - a^{2} b^{2} e f^{2} + a b^{3} c^{2} h + 2 a b^{3} c d g + 2 a b^{3} c e f + a b^{3} d^{2} f - 2 a b^{3} d e^{2} - b^{4} c^{2} e - b^{4} c d^{2}}{a^{4} h^{3} - 3 a^{3} b e h^{2} + a^{3} b g^{3} - 3 a^{2} b^{2} d g^{2} + 3 a^{2} b^{2} e^{2} h + 3 a b^{3} d^{2} g - a b^{3} e^{3} - b^{4} d^{3}} \right )} \right )\right )} + \frac{f x^{3}}{3 b} + \frac{g x^{4}}{4 b} + \frac{h x^{5}}{5 b} - \frac{x^{2} \left (a h - b e\right )}{2 b^{2}} - \frac{x \left (a g - b d\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(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**8 + _t**2*(27*a*b**6*f - 27*b**7*c) + _t*(9*a**3*b**3*g*h -
9*a**2*b**4*d*h - 9*a**2*b**4*e*g + 9*a**2*b**4*f**2 - 18*a*b**5*c*f + 9*a*b**5*
d*e + 9*b**6*c**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*b**5*h - 9*_t**2*b**6*e + 6*_t*a**2*b**3*f*h + 3*_t*a**2*b**3*g**2
- 6*_t*a*b**4*c*h - 6*_t*a*b**4*d*g - 6*_t*a*b**4*e*f + 6*_t*b**5*c*e + 3*_t*b**
5*d**2 + 2*a**4*g*h**2 - 2*a**3*b*d*h**2 - 4*a**3*b*e*g*h + a**3*b*f**2*h + a**3
*b*f*g**2 - 2*a**2*b**2*c*f*h - a**2*b**2*c*g**2 + 4*a**2*b**2*d*e*h - 2*a**2*b*
*2*d*f*g + 2*a**2*b**2*e**2*g - a**2*b**2*e*f**2 + a*b**3*c**2*h + 2*a*b**3*c*d*
g + 2*a*b**3*c*e*f + a*b**3*d**2*f - 2*a*b**3*d*e**2 - b**4*c**2*e - b**4*c*d**2
)/(a**4*h**3 - 3*a**3*b*e*h**2 + a**3*b*g**3 - 3*a**2*b**2*d*g**2 + 3*a**2*b**2*
e**2*h + 3*a*b**3*d**2*g - a*b**3*e**3 - b**4*d**3)))) + f*x**3/(3*b) + g*x**4/(
4*b) + h*x**5/(5*b) - x**2*(a*h - b*e)/(2*b**2) - x*(a*g - b*d)/b**2

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GIAC/XCAS [A]  time = 0.224048, size = 450, normalized size = 1.53 \[ \frac{{\left (b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} + \frac{12 \, b^{4} h x^{5} + 15 \, b^{4} g x^{4} + 20 \, b^{4} f x^{3} - 30 \, a b^{3} h x^{2} + 30 \, b^{4} x^{2} e + 60 \, b^{4} d x - 60 \, a b^{3} g x}{60 \, b^{5}} - \frac{{\left (a^{2} b^{9} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{10} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - a b^{10} d + a^{2} b^{9} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{11}} \]

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^2/(b*x^3 + a),x, algorithm="giac")

[Out]

1/3*(b*c - a*f)*ln(abs(b*x^3 + a))/b^2 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^2*d - (-a
*b^2)^(1/3)*a*b*g + (-a*b^2)^(2/3)*a*h - (-a*b^2)^(2/3)*b*e)*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*d - (-a*b^2)^(1
/3)*a*b*g - (-a*b^2)^(2/3)*a*h + (-a*b^2)^(2/3)*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (
-a/b)^(2/3))/b^4 + 1/60*(12*b^4*h*x^5 + 15*b^4*g*x^4 + 20*b^4*f*x^3 - 30*a*b^3*h
*x^2 + 30*b^4*x^2*e + 60*b^4*d*x - 60*a*b^3*g*x)/b^5 - 1/3*(a^2*b^9*h*(-a/b)^(1/
3) - a*b^10*(-a/b)^(1/3)*e - a*b^10*d + a^2*b^9*g)*(-a/b)^(1/3)*ln(abs(x - (-a/b
)^(1/3)))/(a*b^11)