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

Optimal. Leaf size=258 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 a^{2/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} a^{2/3} b^{5/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{c \log (x)}{a}+\frac{g x}{b}+\frac{h x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.967689, antiderivative size = 256, 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 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{2/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} a^{2/3} b^{5/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{c \log (x)}{a}+\frac{g x}{b}+\frac{h x^2}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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

[Out]

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

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Mathematica [A]  time = 0.443865, size = 258, normalized size = 1. \[ \frac{-\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )-2 b^{2/3} (b c-a f) \log \left (a+b x^3\right )+6 a b^{2/3} g x+3 a b^{2/3} h x^2+6 b^{5/3} c \log (x)}{6 a b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

(6*a*b^(2/3)*g*x + 3*a*b^(2/3)*h*x^2 + 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]] + 6*
b^(5/3)*c*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] - 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] - 2*b^(2/3)*(b*c - a*f)*L
og[a + b*x^3])/(6*a*b^(5/3))

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Maple [B]  time = 0.01, size = 426, normalized size = 1.7 \[ \text{result too large to display} \]

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225447, size = 409, normalized size = 1.59 \[ \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{{\left (b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} + \frac{b h x^{2} + 2 \, b g x}{2 \, b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} + \frac{{\left (a^{3} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - a^{2} b^{3} d + a^{3} b^{2} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3} b^{3}} \]

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),x, algorithm="giac")

[Out]

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