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

Optimal. Leaf size=347 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{54 a^{8/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]

[Out]

(x*(a*(b*d - a*g) + a*(b*e - a*h)*x - b*(b*c - a*f)*x^2))/(6*a^2*b*(a + b*x^3)^2
) + (x*(a*(5*b*d + a*g) + 2*a*(2*b*e + a*h)*x - 3*b*(3*b*c - a*f)*x^2))/(18*a^3*
b*(a + b*x^3)) - ((5*b^(4/3)*d + 2*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))])/(9*Sqrt[3]*a^(8/3)*b^(5/3)) + (c*Lo
g[x])/a^3 + ((b^(1/3)*(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/
3)*x])/(27*a^(8/3)*b^(5/3)) - ((b^(1/3)*(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(5/3)) - (c*Log[a +
b*x^3])/(3*a^3)

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Rubi [A]  time = 1.40924, antiderivative size = 345, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (a h+2 b e)}{\sqrt [3]{b}}+a g+5 b d\right )}{54 a^{8/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left (-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]

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)^3),x]

[Out]

(x*(a*(b*d - a*g) + a*(b*e - a*h)*x - b*(b*c - a*f)*x^2))/(6*a^2*b*(a + b*x^3)^2
) + (x*(a*(5*b*d + a*g) + 2*a*(2*b*e + a*h)*x - 3*b*(3*b*c - a*f)*x^2))/(18*a^3*
b*(a + b*x^3)) - ((5*b^(4/3)*d + 2*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))])/(9*Sqrt[3]*a^(8/3)*b^(5/3)) + (c*Lo
g[x])/a^3 + ((b^(1/3)*(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/
3)*x])/(27*a^(8/3)*b^(5/3)) - ((5*b*d + a*g - (a^(1/3)*(2*b*e + a*h))/b^(1/3))*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(4/3)) - (c*Log[a +
b*x^3])/(3*a^3)

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Rubi in Sympy [A]  time = 136.863, size = 284, normalized size = 0.82 \[ - \frac{x \left (\frac{a f}{x} + a g + a h x - \frac{b c}{x} - b d - b e x\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x \left (\frac{6 a f}{x} + a g + 2 a h x + 5 b d + 4 b e x\right )}{18 a^{2} b \left (a + b x^{3}\right )} - \frac{\left (\sqrt [3]{a} \left (a h + 2 b e\right ) - \sqrt [3]{b} \left (a g + 5 b d\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} + \frac{\left (\sqrt [3]{a} \left (a h + 2 b e\right ) - \sqrt [3]{b} \left (a g + 5 b d\right )\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (a^{\frac{4}{3}} h + 2 \sqrt [3]{a} b e + a \sqrt [3]{b} g + 5 b^{\frac{4}{3}} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{5}{3}}} \]

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)**3,x)

[Out]

-x*(a*f/x + a*g + a*h*x - b*c/x - b*d - b*e*x)/(6*a*b*(a + b*x**3)**2) + x*(6*a*
f/x + a*g + 2*a*h*x + 5*b*d + 4*b*e*x)/(18*a**2*b*(a + b*x**3)) - (a**(1/3)*(a*h
+ 2*b*e) - b**(1/3)*(a*g + 5*b*d))*log(a**(1/3) + b**(1/3)*x)/(27*a**(8/3)*b**(
5/3)) + (a**(1/3)*(a*h + 2*b*e) - b**(1/3)*(a*g + 5*b*d))*log(a**(2/3) - a**(1/3
)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(8/3)*b**(5/3)) - sqrt(3)*(a**(4/3)*h + 2*a
**(1/3)*b*e + a*b**(1/3)*g + 5*b**(4/3)*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)
*x/3)/a**(1/3))/(27*a**(8/3)*b**(5/3))

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Mathematica [A]  time = 0.50999, size = 311, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{5/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} (-h)-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{5/3}}-\frac{9 a^2 (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left (a+b x^3\right )^2}+\frac{3 a (a x (g+2 h x)+6 b c+b x (5 d+4 e x))}{b \left (a+b x^3\right )}-18 c \log \left (a+b x^3\right )+54 c \log (x)}{54 a^3} \]

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)^3),x]

[Out]

((3*a*(6*b*c + b*x*(5*d + 4*e*x) + a*x*(g + 2*h*x)))/(b*(a + b*x^3)) - (9*a^2*(-
(b*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(b*(a + b*x^3)^2) - (2*Sqrt[3]*a^(
1/3)*(5*b^(4/3)*d + 2*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^(5/3) + 54*c*Log[x] + (2*a^(1/3)*(5*b^(4/3)*d - 2*a^
(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (a^(1/3
)*(-5*b^(4/3)*d + 2*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^(5/3) - 18*c*Log[a + b*x^3])/(54*a^3)

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Maple [B]  time = 0.029, size = 620, normalized size = 1.8 \[ \text{result too large to display} \]

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)^3,x)

[Out]

1/2/a/(b*x^3+a)^2*c+5/27/a^2/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(
1/3)*x-1))*d+2/27/a^2/b*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-
1))*e+1/27/a^2/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+4/9/a/(b*x^3+a)
^2*x*d+7/18/a/(b*x^3+a)^2*x^2*e-2/27/a^2/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e+5/27/
a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3
)+(a/b)^(2/3))*d+c*ln(x)/a^3+5/18/a^2/(b*x^3+a)^2*x^4*b*d-1/3/a^3*c*ln(b*(b*x^3+
a))+1/27/a*g/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/54/a*g/b^2/(a/b)^(2/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))-1/27/a*h/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/54/a*h/b^
2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/9/a^2/(b*x^3+a)^2*x^5*b*e+1/9/
a/(b*x^3+a)^2*x^5*h+1/18/a/(b*x^3+a)^2*x^4*g-1/18/(b*x^3+a)^2/b*x^2*h-1/9/(b*x^3
+a)^2/b*x*g-1/6/b/(b*x^3+a)^2*f+1/3/a^2/(b*x^3+a)^2*x^3*c*b+1/27/a*g/b^2/(a/b)^(
2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/27/a*h*3^(1/2)/b^2/(a/b)^
(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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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)^3*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)^3*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232004, size = 529, normalized size = 1.52 \[ -\frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} + \frac{6 \, a b^{2} c x^{3} + 2 \,{\left (a^{2} b h + 2 \, a b^{2} e\right )} x^{5} +{\left (5 \, a b^{2} d + a^{2} b g\right )} x^{4} + 9 \, a^{2} b c - 3 \, a^{3} f -{\left (a^{3} h - 7 \, a^{2} b e\right )} x^{2} + 2 \,{\left (4 \, a^{2} b d - a^{3} g\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b} - \frac{{\left (a^{5} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{4} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, a^{4} b^{3} d + a^{5} b^{2} g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b^{3}} \]

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)^3*x),x, algorithm="giac")

[Out]

-1/3*c*ln(abs(b*x^3 + a))/a^3 + c*ln(abs(x))/a^3 + 1/27*sqrt(3)*(5*(-a*b^2)^(1/3
)*b^2*d + (-a*b^2)^(1/3)*a*b*g - (-a*b^2)^(2/3)*a*h - 2*(-a*b^2)^(2/3)*b*e)*arct
an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^3) + 1/54*(5*(-a*b^2)^(
1/3)*b^2*d + (-a*b^2)^(1/3)*a*b*g + (-a*b^2)^(2/3)*a*h + 2*(-a*b^2)^(2/3)*b*e)*l
n(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^3) + 1/18*(6*a*b^2*c*x^3 + 2*(a^2*
b*h + 2*a*b^2*e)*x^5 + (5*a*b^2*d + a^2*b*g)*x^4 + 9*a^2*b*c - 3*a^3*f - (a^3*h
- 7*a^2*b*e)*x^2 + 2*(4*a^2*b*d - a^3*g)*x)/((b*x^3 + a)^2*a^3*b) - 1/27*(a^5*b^
2*h*(-a/b)^(1/3) + 2*a^4*b^3*(-a/b)^(1/3)*e + 5*a^4*b^3*d + a^5*b^2*g)*(-a/b)^(1
/3)*ln(abs(x - (-a/b)^(1/3)))/(a^7*b^3)