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

Optimal. Leaf size=337 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{18 \sqrt [3]{a} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{9 \sqrt [3]{a} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{10/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 b^3 \left (a+b x^3\right )}+\frac{(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{x (b e-2 a h)}{b^3}+\frac{f x^2}{2 b^2}+\frac{g x^3}{3 b^2}+\frac{h x^4}{4 b^2} \]

[Out]

((b*e - 2*a*h)*x)/b^3 + (f*x^2)/(2*b^2) + (g*x^3)/(3*b^2) + (h*x^4)/(4*b^2) + (x
*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*b^3*(a + b*x^3)) - ((
2*b^(5/3)*c - 4*a^(2/3)*b*e - 5*a*b^(2/3)*f + 7*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(10/3)) - ((b^(2/3)*(2*b*c -
5*a*f) + a^(2/3)*(4*b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(10/3))
+ ((b^(2/3)*(2*b*c - 5*a*f) + a^(2/3)*(4*b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(18*a^(1/3)*b^(10/3)) + ((b*d - 2*a*g)*Log[a + b*x^3])/(
3*b^3)

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Rubi [A]  time = 1.47564, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{18 \sqrt [3]{a} b^{10/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{9 \sqrt [3]{a} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right )}{3 \sqrt{3} \sqrt [3]{a} b^{10/3}}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 b^3 \left (a+b x^3\right )}+\frac{(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{x (b e-2 a h)}{b^3}+\frac{f x^2}{2 b^2}+\frac{g x^3}{3 b^2}+\frac{h x^4}{4 b^2} \]

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)^2,x]

[Out]

((b*e - 2*a*h)*x)/b^3 + (f*x^2)/(2*b^2) + (g*x^3)/(3*b^2) + (h*x^4)/(4*b^2) + (x
*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*b^3*(a + b*x^3)) - ((
2*b^(5/3)*c - 4*a^(2/3)*b*e - 5*a*b^(2/3)*f + 7*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(10/3)) - ((b^(2/3)*(2*b*c -
5*a*f) + a^(2/3)*(4*b*e - 7*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(10/3))
+ ((b^(2/3)*(2*b*c - 5*a*f) + a^(2/3)*(4*b*e - 7*a*h))*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(18*a^(1/3)*b^(10/3)) + ((b*d - 2*a*g)*Log[a + b*x^3])/(
3*b^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.27878, size = 334, normalized size = 0.99 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^{2/3} b^{4/3} e-7 a^{5/3} \sqrt [3]{b} h-5 a b f+2 b^2 c\right )}{\sqrt [3]{a}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^{2/3} b^{4/3} e+7 a^{5/3} \sqrt [3]{b} h+5 a b f-2 b^2 c\right )}{\sqrt [3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-4 a^{2/3} b^{4/3} e+7 a^{5/3} \sqrt [3]{b} h-5 a b f+2 b^2 c\right )}{\sqrt [3]{a}}-\frac{12 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{a+b x^3}+12 b^{2/3} (b d-2 a g) \log \left (a+b x^3\right )+36 b^{2/3} x (b e-2 a h)+18 b^{5/3} f x^2+12 b^{5/3} g x^3+9 b^{5/3} h x^4}{36 b^{11/3}} \]

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)^2,x]

[Out]

(36*b^(2/3)*(b*e - 2*a*h)*x + 18*b^(5/3)*f*x^2 + 12*b^(5/3)*g*x^3 + 9*b^(5/3)*h*
x^4 - (12*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^
3) - (4*Sqrt[3]*(2*b^2*c - 4*a^(2/3)*b^(4/3)*e - 5*a*b*f + 7*a^(5/3)*b^(1/3)*h)*
ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-2*b^2*c - 4*a^(2/3)*
b^(4/3)*e + 5*a*b*f + 7*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (
2*(2*b^2*c + 4*a^(2/3)*b^(4/3)*e - 5*a*b*f - 7*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + 12*b^(2/3)*(b*d - 2*a*g)*Log[a + b*x
^3])/(36*b^(11/3))

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Maple [B]  time = 0.015, size = 562, normalized size = 1.7 \[ \text{result too large to display} \]

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)^2,x)

[Out]

-4/9/b^3*a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e-5/9*a/b
^3*f*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+7/9/b^4*a^2*h/(
a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/2*f*x^2/b^2+1/b^2*e
*x+1/3/b^2/(b*x^3+a)*a*d-1/3/b*x^2/(b*x^3+a)*c-2/9/b^2*c/(a/b)^(1/3)*ln(x+(a/b)^
(1/3))+1/9/b^2*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/9*a/b^3*f/(a/b)
^(1/3)*ln(x+(a/b)^(1/3))-5/18*a/b^3*f/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/
3))+2/9/b^2*c*3^(1/2)/(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)*d-2/b^3*x*a*h-1/3/b^3/(b*x^3+a)*a^2*g-2/3/b^3*ln(b*x^3+a)*a*g+1/3/
b^2*x*a/(b*x^3+a)*e+7/9/b^4*a^2*h/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/18/b^4*a^2*h/(
a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*a/b^2*x^2/(b*x^3+a)*f+1/3*g*x^3
/b^2+1/4*h*x^4/b^2-1/3/b^3/(b*x^3+a)*x*a^2*h-4/9/b^3*a/(a/b)^(2/3)*ln(x+(a/b)^(1
/3))*e+2/9/b^3*a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e

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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)^2,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)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230377, size = 512, normalized size = 1.52 \[ \frac{{\left (b d - 2 \, a g\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{4}} + \frac{a b d - a^{2} g -{\left (b^{2} c - a b f\right )} x^{2} -{\left (a^{2} h - a b e\right )} x}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{4}} - \frac{{\left (2 \, b^{6} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 7 \, a^{2} b^{4} h - 4 \, a b^{5} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{7}} + \frac{3 \, b^{6} h x^{4} + 4 \, b^{6} g x^{3} + 6 \, b^{6} f x^{2} - 24 \, a b^{5} h x + 12 \, b^{6} x e}{12 \, b^{8}} \]

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)^2,x, algorithm="giac")

[Out]

1/3*(b*d - 2*a*g)*ln(abs(b*x^3 + a))/b^3 + 1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*a^2*h -
4*(-a*b^2)^(1/3)*a*b*e - 2*(-a*b^2)^(2/3)*b*c + 5*(-a*b^2)^(2/3)*a*f)*arctan(1/
3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) + 1/3*(a*b*d - a^2*g - (b^2
*c - a*b*f)*x^2 - (a^2*h - a*b*e)*x)/((b*x^3 + a)*b^3) + 1/18*(7*(-a*b^2)^(1/3)*
a^2*h - 4*(-a*b^2)^(1/3)*a*b*e + 2*(-a*b^2)^(2/3)*b*c - 5*(-a*b^2)^(2/3)*a*f)*ln
(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^4) - 1/9*(2*b^6*c*(-a/b)^(1/3) - 5*a*
b^5*f*(-a/b)^(1/3) + 7*a^2*b^4*h - 4*a*b^5*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/
3)))/(a*b^7) + 1/12*(3*b^6*h*x^4 + 4*b^6*g*x^3 + 6*b^6*f*x^2 - 24*a*b^5*h*x + 12
*b^6*x*e)/b^8