3.270 \(\int \frac{c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=270 \[ \frac{2 b c-a d}{2 a^3 x^2}-\frac{c}{5 a^2 x^5}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{18 a^{11/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{9 a^{11/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{3 \sqrt{3} a^{11/3} b^{4/3}} \]

[Out]

-c/(5*a^2*x^5) + (2*b*c - a*d)/(2*a^3*x^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x)/(3*a^3*b*(a + b*x^3)) - ((8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(11/3)*b^(4/3)) + ((8*b^
3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(11/3)*b^(4/
3)) - ((8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(18*a^(11/3)*b^(4/3))

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Rubi [A]  time = 0.646128, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 b c-a d}{2 a^3 x^2}-\frac{c}{5 a^2 x^5}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{18 a^{11/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{9 a^{11/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{3 \sqrt{3} a^{11/3} b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^2),x]

[Out]

-c/(5*a^2*x^5) + (2*b*c - a*d)/(2*a^3*x^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x)/(3*a^3*b*(a + b*x^3)) - ((8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(11/3)*b^(4/3)) + ((8*b^
3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(11/3)*b^(4/
3)) - ((8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(18*a^(11/3)*b^(4/3))

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Rubi in Sympy [A]  time = 141.426, size = 274, normalized size = 1.01 \[ - \frac{x \left (\frac{a^{3} f}{x^{6}} - \frac{a^{2} b e}{x^{6}} + \frac{a b^{2} d}{x^{6}} - \frac{b^{3} c}{x^{6}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{5 a b^{3} x^{5}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{2 a^{2} b^{2} x^{2}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{8}{3}} b^{\frac{4}{3}}} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{8}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**6/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.293454, size = 253, normalized size = 0.94 \[ \frac{-\frac{45 a^{2/3} (a d-2 b c)}{x^2}-\frac{18 a^{5/3} c}{x^5}+\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}-\frac{30 a^{2/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}-\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}}{90 a^{11/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^2),x]

[Out]

((-18*a^(5/3)*c)/x^5 - (45*a^(2/3)*(-2*b*c + a*d))/x^2 - (30*a^(2/3)*(-(b^3*c) +
a*b^2*d - a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)) - (10*Sqrt[3]*(8*b^3*c - 5*a*b^2*
d + 2*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) + (1
0*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) -
(5*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/b^(4/3))/(90*a^(11/3))

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Maple [B]  time = 0.019, size = 477, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^6/(b*x^3+a)^2,x)

[Out]

-1/5*c/a^2/x^5-1/2*d/a^2/x^2+1/a^3/x^2*b*c-1/3*x/(b*x^3+a)/b*f+1/3/a*x/(b*x^3+a)
*e-1/3/a^2*x/(b*x^3+a)*b*d+1/3/a^3*x/(b*x^3+a)*b^2*c+1/9/b^2/(a/b)^(2/3)*ln(x+(a
/b)^(1/3))*f+2/9/a/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-5/9/a^2/(a/b)^(2/3)*ln(x+(a
/b)^(1/3))*d+8/9/a^3*b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/18/b^2/(a/b)^(2/3)*ln(x
^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/9/a/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))*e+5/18/a^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d-4/9/a^3*b/(a/b)
^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+1/9/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/
3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+2/9/a/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*
(2/(a/b)^(1/3)*x-1))*e-5/9/a^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(
1/3)*x-1))*d+8/9/a^3*b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))*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((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246023, size = 518, normalized size = 1.92 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} +{\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 10 \, \sqrt{3}{\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} +{\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 30 \,{\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} +{\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (5 \,{\left (8 \, b^{3} c - 5 \, a b^{2} d + 2 \, a^{2} b e - 2 \, a^{3} f\right )} x^{6} - 6 \, a^{2} b c + 3 \,{\left (8 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{270 \,{\left (a^{3} b^{2} x^{8} + a^{4} b x^{5}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^6),x, algorithm="fricas")

[Out]

-1/270*sqrt(3)*(5*sqrt(3)*((8*b^4*c - 5*a*b^3*d + 2*a^2*b^2*e + a^3*b*f)*x^8 + (
8*a*b^3*c - 5*a^2*b^2*d + 2*a^3*b*e + a^4*f)*x^5)*log((a^2*b)^(2/3)*x^2 - (a^2*b
)^(1/3)*a*x + a^2) - 10*sqrt(3)*((8*b^4*c - 5*a*b^3*d + 2*a^2*b^2*e + a^3*b*f)*x
^8 + (8*a*b^3*c - 5*a^2*b^2*d + 2*a^3*b*e + a^4*f)*x^5)*log((a^2*b)^(1/3)*x + a)
- 30*((8*b^4*c - 5*a*b^3*d + 2*a^2*b^2*e + a^3*b*f)*x^8 + (8*a*b^3*c - 5*a^2*b^
2*d + 2*a^3*b*e + a^4*f)*x^5)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)
/a) - 3*sqrt(3)*(5*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e - 2*a^3*f)*x^6 - 6*a^2*b*c +
3*(8*a*b^2*c - 5*a^2*b*d)*x^3)*(a^2*b)^(1/3))/((a^3*b^2*x^8 + a^4*b*x^5)*(a^2*b
)^(1/3))

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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((f*x**9+e*x**6+d*x**3+c)/x**6/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219709, size = 429, normalized size = 1.59 \[ -\frac{{\left (8 \, b^{3} c - 5 \, a b^{2} d + a^{3} f + 2 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4} b} + \frac{\sqrt{3}{\left (8 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4} b^{2}} + \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{3} b} + \frac{{\left (8 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{4} b^{2}} + \frac{10 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^6),x, algorithm="giac")

[Out]

-1/9*(8*b^3*c - 5*a*b^2*d + a^3*f + 2*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1
/3)))/(a^4*b) + 1/9*sqrt(3)*(8*(-a*b^2)^(1/3)*b^3*c - 5*(-a*b^2)^(1/3)*a*b^2*d +
(-a*b^2)^(1/3)*a^3*f + 2*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/
b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/3*(b^3*c*x - a*b^2*d*x - a^3*f*x + a^2*b*x
*e)/((b*x^3 + a)*a^3*b) + 1/18*(8*(-a*b^2)^(1/3)*b^3*c - 5*(-a*b^2)^(1/3)*a*b^2*
d + (-a*b^2)^(1/3)*a^3*f + 2*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (
-a/b)^(2/3))/(a^4*b^2) + 1/10*(10*b*c*x^3 - 5*a*d*x^3 - 2*a*c)/(a^3*x^5)