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

Optimal. Leaf size=260 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{6 a^{5/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{5/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} a^{5/3} b^{2/3}}-\frac{(b e-a h) \log \left (a+b x^3\right )}{3 a b}-\frac{c}{2 a x^2}-\frac{d}{a x}+\frac{e \log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.813767, antiderivative size = 258, 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 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (b d-a g)}{\sqrt [3]{b}}-a f+b c\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b c-a f)-\sqrt [3]{a} (b d-a g)\right )}{3 a^{5/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{\sqrt{3} a^{5/3} b^{2/3}}-\frac{(b e-a h) \log \left (a+b x^3\right )}{3 a b}-\frac{c}{2 a x^2}-\frac{d}{a x}+\frac{e \log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 106.028, size = 228, normalized size = 0.88 \[ - \frac{c}{2 a x^{2}} - \frac{d}{a x} + \frac{e \log{\left (x \right )}}{a} + \frac{\left (a h - b e\right ) \log{\left (a + b x^{3} \right )}}{3 a b} - \frac{\left (\sqrt [3]{a} \left (a g - b d\right ) - \sqrt [3]{b} \left (a f - b c\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} + \frac{\left (\sqrt [3]{a} \left (a g - b d\right ) - \sqrt [3]{b} \left (a f - b c\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (a^{\frac{4}{3}} g - \sqrt [3]{a} b d + a \sqrt [3]{b} f - b^{\frac{4}{3}} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]

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

[Out]

-c/(2*a*x**2) - d/(a*x) + e*log(x)/a + (a*h - b*e)*log(a + b*x**3)/(3*a*b) - (a*
*(1/3)*(a*g - b*d) - b**(1/3)*(a*f - b*c))*log(a**(1/3) + b**(1/3)*x)/(3*a**(5/3
)*b**(2/3)) + (a**(1/3)*(a*g - b*d) - b**(1/3)*(a*f - b*c))*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)*g - a*
*(1/3)*b*d + a*b**(1/3)*f - b**(4/3)*c)*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 = 0.781863, size = 257, normalized size = 0.99 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} g-\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} (-g)+\sqrt [3]{a} b d-a \sqrt [3]{b} f+b^{4/3} c\right )}{b^{2/3}}+\frac{2 a^{2/3} (a h-b e) \log \left (a+b x^3\right )}{b}-\frac{3 a^{2/3} c}{x^2}-\frac{6 a^{2/3} d}{x}+6 a^{2/3} e \log (x)}{6 a^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 423, normalized size = 1.6 \[ -{\frac{d}{ax}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{c}{2\,a{x}^{2}}}+{\frac{f}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{f}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}f}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{g}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{g}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}g}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) h}{3\,b}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,a}} \]

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225554, size = 393, normalized size = 1.51 \[ \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a} + \frac{{\left (a h - b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b^{2}} + \frac{{\left (a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{2} c - a^{2} b f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3} b} - \frac{2 \, d x + c}{2 \, a x^{2}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}} \]

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^3),x, algorithm="giac")

[Out]

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