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

Optimal. Leaf size=276 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 a^{5/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{5/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} a^{5/3} b^{2/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac{\log (x) (b c-a f)}{a^2}-\frac{c}{3 a x^3}-\frac{d}{2 a x^2}-\frac{e}{a x} \]

[Out]

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

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Rubi [A]  time = 0.959213, antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{5/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} a^{5/3} b^{2/3}}+\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac{\log (x) (b c-a f)}{a^2}-\frac{c}{3 a x^3}-\frac{d}{2 a x^2}-\frac{e}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 121.797, size = 243, normalized size = 0.88 \[ - \frac{c}{3 a x^{3}} - \frac{d}{2 a x^{2}} - \frac{e}{a x} + \frac{\left (a f - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a f - b c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{2}} - \frac{\left (\sqrt [3]{a} \left (a h - b e\right ) - \sqrt [3]{b} \left (a g - b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} + \frac{\left (\sqrt [3]{a} \left (a h - b e\right ) - \sqrt [3]{b} \left (a g - b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (a^{\frac{4}{3}} h - \sqrt [3]{a} b e + a \sqrt [3]{b} g - b^{\frac{4}{3}} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]

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)/x**4/(b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 1.08363, size = 264, normalized size = 0.96 \[ -\frac{-\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{b^{2/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )}{b^{2/3}}-2 (b c-a f) \log \left (a+b x^3\right )+6 \log (x) (b c-a f)+\frac{2 a c}{x^3}+\frac{3 a d}{x^2}+\frac{6 a e}{x}}{6 a^2} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a*c)/x^3 + (3*a*d)/x^2 + (6*a*e)/x + (2*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]])
/b^(2/3) + 6*(b*c - a*f)*Log[x] + (2*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])/b^(2/3) - (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]
)/b^(2/3) - 2*(b*c - a*f)*Log[a + b*x^3])/(6*a^2)

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Maple [B]  time = 0.01, size = 442, normalized size = 1.6 \[ \text{result too large to display} \]

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)/x^4/(b*x^3+a),x)

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226677, size = 423, normalized size = 1.53 \[ \frac{{\left (b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac{{\left (b c - a f\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b^{2}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}} - \frac{{\left (a^{4} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - a^{3} b^{2} d + a^{4} b g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{5} b} - \frac{6 \, a x^{2} e + 3 \, a d x + 2 \, a c}{6 \, a^{2} x^{3}} \]

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^4),x, algorithm="giac")

[Out]

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