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

Optimal. Leaf size=362 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{54 a^{10/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{27 a^{10/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{2/3} b e+a^{5/3} (-h)-2 a b^{2/3} f+14 b^{5/3} c\right )}{9 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x \left (-2 b x (5 b c-2 a f)-3 b x^2 (3 b d-a g)+a (a h+5 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]

[Out]

-(c/(a^3*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*a^2*
b*(a + b*x^3)^2) + (x*(a*(5*b*e + a*h) - 2*b*(5*b*c - 2*a*f)*x - 3*b*(3*b*d - a*
g)*x^2))/(18*a^3*b*(a + b*x^3)) + ((14*b^(5/3)*c - 5*a^(2/3)*b*e - 2*a*b^(2/3)*f
- a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1
0/3)*b^(4/3)) + (d*Log[x])/a^3 + ((2*b^(2/3)*(7*b*c - a*f) + a^(2/3)*(5*b*e + a*
h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(4/3)) - ((2*b^(2/3)*(7*b*c - a*f)
+ a^(2/3)*(5*b*e + a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(
10/3)*b^(4/3)) - (d*Log[a + b*x^3])/(3*a^3)

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Rubi [A]  time = 1.63075, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{54 a^{10/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{27 a^{10/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{2/3} b e+a^{5/3} (-h)-2 a b^{2/3} f+14 b^{5/3} c\right )}{9 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x \left (-2 b x (5 b c-2 a f)-3 b x^2 (3 b d-a g)+a (a h+5 b e)\right )}{18 a^3 b \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^3*x)) + (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(6*a^2*
b*(a + b*x^3)^2) + (x*(a*(5*b*e + a*h) - 2*b*(5*b*c - 2*a*f)*x - 3*b*(3*b*d - a*
g)*x^2))/(18*a^3*b*(a + b*x^3)) + ((14*b^(5/3)*c - 5*a^(2/3)*b*e - 2*a*b^(2/3)*f
- a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1
0/3)*b^(4/3)) + (d*Log[x])/a^3 + ((2*b^(2/3)*(7*b*c - a*f) + a^(2/3)*(5*b*e + a*
h))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(4/3)) - ((2*b^(2/3)*(7*b*c - a*f)
+ a^(2/3)*(5*b*e + a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(
10/3)*b^(4/3)) - (d*Log[a + b*x^3])/(3*a^3)

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Rubi in Sympy [A]  time = 96.0268, size = 219, normalized size = 0.6 \[ - \frac{x \left (\frac{a f}{x^{2}} + \frac{a g}{x} + a h - \frac{b c}{x^{2}} - \frac{b d}{x} - b e\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x \left (\frac{6 a f}{x^{2}} + \frac{6 a g}{x} + a h + 5 b e\right )}{18 a^{2} b \left (a + b x^{3}\right )} + \frac{\left (a h + 5 b e\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{4}{3}}} - \frac{\left (a h + 5 b e\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a h + 5 b e\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{4}{3}}} \]

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

[Out]

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

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Mathematica [A]  time = 1.15837, size = 336, normalized size = 0.93 \[ -\frac{\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{2/3} b e+a^{5/3} h-2 a b^{2/3} f+14 b^{5/3} c\right )}{b^{4/3}}-\frac{2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{2/3} b e+a^{5/3} h-2 a b^{2/3} f+14 b^{5/3} c\right )}{b^{4/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{2/3} b e+a^{5/3} h+2 a b^{2/3} f-14 b^{5/3} c\right )}{b^{4/3}}+\frac{9 a^2 \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{b \left (a+b x^3\right )^2}-\frac{3 a \left (a^2 h x+a b (6 d+x (5 e+4 f x))-10 b^2 c x^2\right )}{b \left (a+b x^3\right )}+18 a d \log \left (a+b x^3\right )+\frac{54 a c}{x}-54 a d \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

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

[Out]

-((54*a*c)/x + (9*a^2*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(b*(a
+ b*x^3)^2) - (3*a*(a^2*h*x - 10*b^2*c*x^2 + a*b*(6*d + x*(5*e + 4*f*x))))/(b*(
a + b*x^3)) + (2*Sqrt[3]*a^(2/3)*(-14*b^(5/3)*c + 5*a^(2/3)*b*e + 2*a*b^(2/3)*f
+ a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) - 54*a*d*Log[x
] - (2*a^(2/3)*(14*b^(5/3)*c + 5*a^(2/3)*b*e - 2*a*b^(2/3)*f + a^(5/3)*h)*Log[a^
(1/3) + b^(1/3)*x])/b^(4/3) + (a^(2/3)*(14*b^(5/3)*c + 5*a^(2/3)*b*e - 2*a*b^(2/
3)*f + a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3) + 18*a
*d*Log[a + b*x^3])/(54*a^4)

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Maple [B]  time = 0.025, size = 624, 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^2/(b*x^3+a)^3,x)

[Out]

5/27/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)
^(1/3)+(a/b)^(2/3))*e+1/27/a*h/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+7/18/a/(b*x^3+a
)^2*x^2*f-13/18/a^2/(b*x^3+a)^2*x^2*b*c+1/3/a^2/(b*x^3+a)^2*x^3*b*d+1/27/a^2*f/b
/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^
(1/3))*c-7/27/a^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-c/a^3/x+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))*e+2/27/a^2*f*3^
(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+4/9/a/(b*x^3+a)^2*x*
e+1/2/a/(b*x^3+a)^2*d-5/9/a^3/(b*x^3+a)^2*b^2*x^5*c+1/18/a/(b*x^3+a)^2*x^4*h-1/9
/(b*x^3+a)^2/b*x*h-2/27/a^2*f/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+2/9/a^2/(b*x^3+a)^
2*x^5*f*b-1/54/a*h/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/6/(b*x^3+
a)^2/b*g-1/3/a^3*d*ln(b*(b*x^3+a))-14/27/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1
/2)*(2/(a/b)^(1/3)*x-1))*c+1/27/a*h/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(
2/(a/b)^(1/3)*x-1))+5/18/a^2/(b*x^3+a)^2*x^4*b*e+d*ln(x)/a^3

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230017, size = 556, normalized size = 1.54 \[ -\frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{2}} + \frac{6 \, a b^{2} d x^{4} - 4 \,{\left (7 \, b^{3} c - a b^{2} f\right )} x^{6} +{\left (a^{2} b h + 5 \, a b^{2} e\right )} x^{5} - 18 \, a^{2} b c - 7 \,{\left (7 \, a b^{2} c - a^{2} b f\right )} x^{3} - 2 \,{\left (a^{3} h - 4 \, a^{2} b e\right )} x^{2} + 3 \,{\left (3 \, a^{2} b d - a^{3} g\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b x} + \frac{{\left (14 \, a^{3} b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{4} b^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{5} b^{2} h - 5 \, a^{4} b^{3} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b^{3}} \]

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^2),x, algorithm="giac")

[Out]

-1/3*d*ln(abs(b*x^3 + a))/a^3 + d*ln(abs(x))/a^3 + 1/27*sqrt(3)*((-a*b^2)^(1/3)*
a^2*h + 5*(-a*b^2)^(1/3)*a*b*e + 14*(-a*b^2)^(2/3)*b*c - 2*(-a*b^2)^(2/3)*a*f)*a
rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/54*((-a*b^2)^
(1/3)*a^2*h + 5*(-a*b^2)^(1/3)*a*b*e - 14*(-a*b^2)^(2/3)*b*c + 2*(-a*b^2)^(2/3)*
a*f)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^2) + 1/18*(6*a*b^2*d*x^4 - 4
*(7*b^3*c - a*b^2*f)*x^6 + (a^2*b*h + 5*a*b^2*e)*x^5 - 18*a^2*b*c - 7*(7*a*b^2*c
- a^2*b*f)*x^3 - 2*(a^3*h - 4*a^2*b*e)*x^2 + 3*(3*a^2*b*d - a^3*g)*x)/((b*x^3 +
a)^2*a^3*b*x) + 1/27*(14*a^3*b^4*c*(-a/b)^(1/3) - 2*a^4*b^3*f*(-a/b)^(1/3) - a^
5*b^2*h - 5*a^4*b^3*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^7*b^3)