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

Optimal. Leaf size=331 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 b^{10/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} b^{10/3}}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac{a x (b e-a h)}{b^3}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}+\frac{x^4 (b e-a h)}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b} \]

[Out]

-((a*(b*e - a*h)*x)/b^3) + ((b*c - a*f)*x^2)/(2*b^2) + ((b*d - a*g)*x^3)/(3*b^2)
+ ((b*e - a*h)*x^4)/(4*b^2) + (f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) + (
a^(2/3)*(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]*b^(10/3)) + (a^(2/3)*(b^(2/3)*(b*c - a*f
) + a^(2/3)*(b*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/3)) - (a^(2/3)*(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*b^(10/3)) - (a*(b*d - a*g)*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 2.02567, antiderivative size = 331, 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 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 b^{10/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} b^{10/3}}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac{a x (b e-a h)}{b^3}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}+\frac{x^4 (b e-a h)}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b} \]

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),x]

[Out]

-((a*(b*e - a*h)*x)/b^3) + ((b*c - a*f)*x^2)/(2*b^2) + ((b*d - a*g)*x^3)/(3*b^2)
+ ((b*e - a*h)*x^4)/(4*b^2) + (f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) + (
a^(2/3)*(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]*b^(10/3)) + (a^(2/3)*(b^(2/3)*(b*c - a*f
) + a^(2/3)*(b*e - a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/3)) - (a^(2/3)*(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*b^(10/3)) - (a*(b*d - a*g)*Log[a + b*x^3])/(3*b^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{3} a^{\frac{2}{3}} \left (a^{\frac{2}{3}} \left (a h - b e\right ) - b^{\frac{2}{3}} \left (a f - b c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{10}{3}}} - \frac{a^{\frac{2}{3}} \left (a^{\frac{2}{3}} \left (a h - b e\right ) + b^{\frac{2}{3}} \left (a f - b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{10}{3}}} + \frac{a^{\frac{2}{3}} \left (a^{\frac{2}{3}} \left (a h - b e\right ) + b^{\frac{2}{3}} \left (a f - b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{10}{3}}} + \frac{a \left (a g - b d\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{f x^{5}}{5 b} + \frac{g x^{6}}{6 b} + \frac{h x^{7}}{7 b} - \frac{x^{4} \left (a h - b e\right )}{4 b^{2}} - \frac{x^{3} \left (a g - b d\right )}{3 b^{2}} - \frac{\left (a f - b c\right ) \int x\, dx}{b^{2}} + \frac{\left (a h - b e\right ) \int a\, dx}{b^{3}} \]

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),x)

[Out]

sqrt(3)*a**(2/3)*(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*b**(10/3)) - a**(2/3)*(a**(2/3)*(a*h - b*
e) + b**(2/3)*(a*f - b*c))*log(a**(1/3) + b**(1/3)*x)/(3*b**(10/3)) + a**(2/3)*(
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*b**(10/3)) + a*(a*g - b*d)*log(a + b*x**3)/(3*b**3) + f*x**5
/(5*b) + g*x**6/(6*b) + h*x**7/(7*b) - x**4*(a*h - b*e)/(4*b**2) - x**3*(a*g - b
*d)/(3*b**2) - (a*f - b*c)*Integral(x, x)/b**2 + (a*h - b*e)*Integral(a, x)/b**3

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Mathematica [A]  time = 1.23964, size = 334, normalized size = 1.01 \[ \frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{6 b^{10/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} b^{10/3}}+\frac{a (a g-b d) \log \left (a+b x^3\right )}{3 b^3}+\frac{a x (a h-b e)}{b^3}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}+\frac{x^4 (b e-a h)}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b} \]

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),x]

[Out]

(a*(-(b*e) + a*h)*x)/b^3 + ((b*c - a*f)*x^2)/(2*b^2) + ((b*d - a*g)*x^3)/(3*b^2)
+ ((b*e - a*h)*x^4)/(4*b^2) + (f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) + (
a^(2/3)*(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]])/(Sqrt[3]*b^(10/3)) + (a^(2/3)*(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])/(3*b^(10/3)) + (a^(2/3)*(-(
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])/(6*b^(10/3)) + (a*(-(b*d) + a*g)*Log[a + b*x^3])/(3*b^3)

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Maple [B]  time = 0.009, size = 533, normalized size = 1.6 \[ \text{result too large to display} \]

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),x)

[Out]

-1/b^2*a*e*x+1/6*g*x^6/b+1/7*h*x^7/b-1/2/b^2*x^2*a*f+1/3*a/b^2/(a/b)^(1/3)*ln(x+
(a/b)^(1/3))*c+1/3/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^2*e-1/6*a/b^2/(a/b)^(1/3)
*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+1/4/b*x^4*e-1/3*a^2/b^3/(a/b)^(1/3)*ln(x+(a
/b)^(1/3))*f-1/4/b^2*x^4*a*h-1/3/b^2*x^3*a*g+1/3/b^3*a^2*ln(b*x^3+a)*g-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))*c+1/2/b*x^2*c+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*e+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))*f+1/6*a^2/b^3/(a/b)
^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/3/b^2*ln(b*x^3+a)*a*d+1/3/b*d*x^3-1
/3/b^4*a^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*h-1/6/b^3
/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^2*e-1/3/b^4*a^3/(a/b)^(2/3)*ln(
x+(a/b)^(1/3))*h+1/5*f*x^5/b+1/b^3*x*a^2*h+1/6/b^4*a^3/(a/b)^(2/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))*h

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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),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),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 50.4846, size = 874, normalized size = 2.64 \[ \operatorname{RootSum}{\left (27 t^{3} b^{10} + t^{2} \left (- 27 a^{2} b^{7} g + 27 a b^{8} d\right ) + t \left (- 9 a^{4} b^{4} f h + 9 a^{4} b^{4} g^{2} + 9 a^{3} b^{5} c h - 18 a^{3} b^{5} d g + 9 a^{3} b^{5} e f - 9 a^{2} b^{6} c e + 9 a^{2} b^{6} d^{2}\right ) + a^{7} h^{3} - 3 a^{6} b e h^{2} + 3 a^{6} b f g h - a^{6} b g^{3} - 3 a^{5} b^{2} c g h - 3 a^{5} b^{2} d f h + 3 a^{5} b^{2} d g^{2} + 3 a^{5} b^{2} e^{2} h - 3 a^{5} b^{2} e f g + a^{5} b^{2} f^{3} + 3 a^{4} b^{3} c d h + 3 a^{4} b^{3} c e g - 3 a^{4} b^{3} c f^{2} - 3 a^{4} b^{3} d^{2} g + 3 a^{4} b^{3} d e f - a^{4} b^{3} e^{3} + 3 a^{3} b^{4} c^{2} f - 3 a^{3} b^{4} c d e + a^{3} b^{4} d^{3} - a^{2} b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t^{2} a b^{7} f + 9 t^{2} b^{8} c - 3 t a^{4} b^{3} h^{2} + 6 t a^{3} b^{4} e h + 6 t a^{3} b^{4} f g - 6 t a^{2} b^{5} c g - 6 t a^{2} b^{5} d f - 3 t a^{2} b^{5} e^{2} + 6 t a b^{6} c d + a^{6} g h^{2} - a^{5} b d h^{2} - 2 a^{5} b e g h + 2 a^{5} b f^{2} h - a^{5} b f g^{2} - 4 a^{4} b^{2} c f h + a^{4} b^{2} c g^{2} + 2 a^{4} b^{2} d e h + 2 a^{4} b^{2} d f g + a^{4} b^{2} e^{2} g - 2 a^{4} b^{2} e f^{2} + 2 a^{3} b^{3} c^{2} h - 2 a^{3} b^{3} c d g + 4 a^{3} b^{3} c e f - a^{3} b^{3} d^{2} f - a^{3} b^{3} d e^{2} - 2 a^{2} b^{4} c^{2} e + a^{2} b^{4} c d^{2}}{a^{6} h^{3} - 3 a^{5} b e h^{2} + 3 a^{4} b^{2} e^{2} h - a^{4} b^{2} f^{3} + 3 a^{3} b^{3} c f^{2} - a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + a b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{5}}{5 b} + \frac{g x^{6}}{6 b} + \frac{h x^{7}}{7 b} - \frac{x^{4} \left (a h - b e\right )}{4 b^{2}} - \frac{x^{3} \left (a g - b d\right )}{3 b^{2}} - \frac{x^{2} \left (a f - b c\right )}{2 b^{2}} + \frac{x \left (a^{2} h - a b e\right )}{b^{3}} \]

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),x)

[Out]

RootSum(27*_t**3*b**10 + _t**2*(-27*a**2*b**7*g + 27*a*b**8*d) + _t*(-9*a**4*b**
4*f*h + 9*a**4*b**4*g**2 + 9*a**3*b**5*c*h - 18*a**3*b**5*d*g + 9*a**3*b**5*e*f
- 9*a**2*b**6*c*e + 9*a**2*b**6*d**2) + a**7*h**3 - 3*a**6*b*e*h**2 + 3*a**6*b*f
*g*h - a**6*b*g**3 - 3*a**5*b**2*c*g*h - 3*a**5*b**2*d*f*h + 3*a**5*b**2*d*g**2
+ 3*a**5*b**2*e**2*h - 3*a**5*b**2*e*f*g + a**5*b**2*f**3 + 3*a**4*b**3*c*d*h +
3*a**4*b**3*c*e*g - 3*a**4*b**3*c*f**2 - 3*a**4*b**3*d**2*g + 3*a**4*b**3*d*e*f
- a**4*b**3*e**3 + 3*a**3*b**4*c**2*f - 3*a**3*b**4*c*d*e + a**3*b**4*d**3 - a**
2*b**5*c**3, Lambda(_t, _t*log(x + (-9*_t**2*a*b**7*f + 9*_t**2*b**8*c - 3*_t*a*
*4*b**3*h**2 + 6*_t*a**3*b**4*e*h + 6*_t*a**3*b**4*f*g - 6*_t*a**2*b**5*c*g - 6*
_t*a**2*b**5*d*f - 3*_t*a**2*b**5*e**2 + 6*_t*a*b**6*c*d + a**6*g*h**2 - a**5*b*
d*h**2 - 2*a**5*b*e*g*h + 2*a**5*b*f**2*h - a**5*b*f*g**2 - 4*a**4*b**2*c*f*h +
a**4*b**2*c*g**2 + 2*a**4*b**2*d*e*h + 2*a**4*b**2*d*f*g + a**4*b**2*e**2*g - 2*
a**4*b**2*e*f**2 + 2*a**3*b**3*c**2*h - 2*a**3*b**3*c*d*g + 4*a**3*b**3*c*e*f -
a**3*b**3*d**2*f - a**3*b**3*d*e**2 - 2*a**2*b**4*c**2*e + a**2*b**4*c*d**2)/(a*
*6*h**3 - 3*a**5*b*e*h**2 + 3*a**4*b**2*e**2*h - a**4*b**2*f**3 + 3*a**3*b**3*c*
f**2 - a**3*b**3*e**3 - 3*a**2*b**4*c**2*f + a*b**5*c**3)))) + f*x**5/(5*b) + g*
x**6/(6*b) + h*x**7/(7*b) - x**4*(a*h - b*e)/(4*b**2) - x**3*(a*g - b*d)/(3*b**2
) - x**2*(a*f - b*c)/(2*b**2) + x*(a**2*h - a*b*e)/b**3

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GIAC/XCAS [A]  time = 0.225604, size = 513, normalized size = 1.55 \[ -\frac{{\left (a b d - a^{2} g\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} + \frac{60 \, b^{6} h x^{7} + 70 \, b^{6} g x^{6} + 84 \, b^{6} f x^{5} - 105 \, a b^{5} h x^{4} + 105 \, b^{6} x^{4} e + 140 \, b^{6} d x^{3} - 140 \, a b^{5} g x^{3} + 210 \, b^{6} c x^{2} - 210 \, a b^{5} f x^{2} + 420 \, a^{2} b^{4} h x - 420 \, a b^{5} x e}{420 \, b^{7}} + \frac{{\left (a b^{14} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{13} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{12} h - a^{2} b^{13} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{15}} \]

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),x, algorithm="giac")

[Out]

-1/3*(a*b*d - a^2*g)*ln(abs(b*x^3 + a))/b^3 - 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*sqr
t(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 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))/b^4 + 1/420*(60*b^6*h*x^7 + 70*b^6*g*x^6 + 84*b^6*f*x^5 - 105*
a*b^5*h*x^4 + 105*b^6*x^4*e + 140*b^6*d*x^3 - 140*a*b^5*g*x^3 + 210*b^6*c*x^2 -
210*a*b^5*f*x^2 + 420*a^2*b^4*h*x - 420*a*b^5*x*e)/b^7 + 1/3*(a*b^14*c*(-a/b)^(1
/3) - a^2*b^13*f*(-a/b)^(1/3) + a^3*b^12*h - a^2*b^13*e)*(-a/b)^(1/3)*ln(abs(x -
(-a/b)^(1/3)))/(a*b^15)