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

Optimal. Leaf size=301 \[ -\frac{c}{2 a^3 x^2}-\frac{x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{54 a^{11/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{27 a^{11/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} a^{11/3} b^{7/3}} \]

[Out]

-c/(2*a^3*x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^2*b^2*(a + b*x^3)^
2) - ((11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3*f)*x)/(18*a^3*b^2*(a + b*x^3)) + (
(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(7/3)) - ((20*b^3*c - 5*a*b^2*d - a^2*b*e -
2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(7/3)) + ((20*b^3*c - 5*a*b^2*
d - a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11
/3)*b^(7/3))

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Rubi [A]  time = 0.786445, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{c}{2 a^3 x^2}-\frac{x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{18 a^3 b^2 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{54 a^{11/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{27 a^{11/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{9 \sqrt{3} a^{11/3} b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*a^3*x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^2*b^2*(a + b*x^3)^
2) - ((11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3*f)*x)/(18*a^3*b^2*(a + b*x^3)) + (
(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(7/3)) - ((20*b^3*c - 5*a*b^2*d - a^2*b*e -
2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(7/3)) + ((20*b^3*c - 5*a*b^2*
d - a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11
/3)*b^(7/3))

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Rubi in Sympy [A]  time = 148.259, size = 287, normalized size = 0.95 \[ - \frac{x \left (\frac{a^{3} f}{x^{3}} - \frac{a^{2} b e}{x^{3}} + \frac{a b^{2} d}{x^{3}} - \frac{b^{3} c}{x^{3}}\right )}{6 a b^{3} \left (a + b x^{3}\right )^{2}} - \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{3 a^{2} b^{2} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{2 a^{2} b^{3} x^{2}} - \frac{\left (6 a^{2} f - 7 a b e + 5 b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{8}{3}} b^{\frac{7}{3}}} + \frac{\left (6 a^{2} f - 7 a b e + 5 b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{8}{3}} b^{\frac{7}{3}}} + \frac{\sqrt{3} \left (6 a^{2} f - 7 a b e + 5 b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{8}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a**3*f/x**3 - a**2*b*e/x**3 + a*b**2*d/x**3 - b**3*c/x**3)/(6*a*b**3*(a + b*
x**3)**2) - x*(3*a**2*f - 2*a*b*e + b**2*d)/(3*a**2*b**2*(a + b*x**3)) - (a**2*f
- a*b*e + b**2*d)/(2*a**2*b**3*x**2) - (6*a**2*f - 7*a*b*e + 5*b**2*d)*log(a**(
1/3) + b**(1/3)*x)/(9*a**(8/3)*b**(7/3)) + (6*a**2*f - 7*a*b*e + 5*b**2*d)*log(a
**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(8/3)*b**(7/3)) + sqrt(3)*
(6*a**2*f - 7*a*b*e + 5*b**2*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1
/3))/(9*a**(8/3)*b**(7/3))

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Mathematica [A]  time = 0.409312, size = 283, normalized size = 0.94 \[ \frac{-\frac{27 a^{2/3} c}{x^2}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e+5 a b^2 d-20 b^3 c\right )}{b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right )}{b^{7/3}}+\frac{9 a^{5/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )^2}-\frac{3 a^{2/3} x \left (7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right )}{b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e+5 a b^2 d-20 b^3 c\right )}{b^{7/3}}}{54 a^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-27*a^(2/3)*c)/x^2 + (9*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(b^2
*(a + b*x^3)^2) - (3*a^(2/3)*(11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3*f)*x)/(b^2*
(a + b*x^3)) + (2*Sqrt[3]*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*ArcTan[(1 -
(2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(7/3) + (2*(-20*b^3*c + 5*a*b^2*d + a^2*b*e
+ 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(7/3) - ((-20*b^3*c + 5*a*b^2*d + a^2*b*e
+ 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(7/3))/(54*a^(11/3
))

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Maple [B]  time = 0.02, size = 539, 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^3/(b*x^3+a)^3,x)

[Out]

-1/2*c/a^3/x^2-7/18/(b*x^3+a)^2/b*x^4*f+1/18/a/(b*x^3+a)^2*x^4*e+5/18/a^2/(b*x^3
+a)^2*x^4*b*d-11/18/a^3/(b*x^3+a)^2*x^4*b^2*c-2/9*a/(b*x^3+a)^2/b^2*x*f-1/9/(b*x
^3+a)^2/b*x*e+4/9/a/(b*x^3+a)^2*x*d-7/9/a^2/(b*x^3+a)^2*x*b*c+2/27/b^3/(a/b)^(2/
3)*ln(x+(a/b)^(1/3))*f+1/27/a/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e+5/27/a^2/b/(a/
b)^(2/3)*ln(x+(a/b)^(1/3))*d-20/27/a^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/27/b^3/
(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/54/a/b^2/(a/b)^(2/3)*ln(x^2-x*
(a/b)^(1/3)+(a/b)^(2/3))*e-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/
3))*d+10/27/a^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+2/27/b^3/(a/b)^(
2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+1/27/a/b^2/(a/b)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+5/27/a^2/b/(a/b)^(2/3)*3^(1/2)*a
rctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-20/27/a^3/(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)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266457, size = 737, normalized size = 2.45 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (20 \, b^{5} c - 5 \, a b^{4} d - a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{8} + 2 \,{\left (20 \, a b^{4} c - 5 \, a^{2} b^{3} d - a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{5} +{\left (20 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d - a^{4} b e - 2 \, a^{5} f\right )} x^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (20 \, b^{5} c - 5 \, a b^{4} d - a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{8} + 2 \,{\left (20 \, a b^{4} c - 5 \, a^{2} b^{3} d - a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{5} +{\left (20 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d - a^{4} b e - 2 \, a^{5} f\right )} x^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left ({\left (20 \, b^{5} c - 5 \, a b^{4} d - a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{8} + 2 \,{\left (20 \, a b^{4} c - 5 \, a^{2} b^{3} d - a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{5} +{\left (20 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d - a^{4} b e - 2 \, a^{5} f\right )} x^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left ({\left (20 \, b^{4} c - 5 \, a b^{3} d - a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{6} + 9 \, a^{2} b^{2} c + 2 \,{\left (16 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{3} b^{4} x^{8} + 2 \, a^{4} b^{3} x^{5} + a^{5} b^{2} x^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

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)^3*x^3),x, algorithm="fricas")

[Out]

1/162*sqrt(3)*(sqrt(3)*((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*b^2*f)*x^8 + 2
*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*
b^2*d - a^4*b*e - 2*a^5*f)*x^2)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2)
- 2*sqrt(3)*((20*b^5*c - 5*a*b^4*d - a^2*b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4
*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^
4*b*e - 2*a^5*f)*x^2)*log((a^2*b)^(1/3)*x + a) - 6*((20*b^5*c - 5*a*b^4*d - a^2*
b^3*e - 2*a^3*b^2*f)*x^8 + 2*(20*a*b^4*c - 5*a^2*b^3*d - a^3*b^2*e - 2*a^4*b*f)*
x^5 + (20*a^2*b^3*c - 5*a^3*b^2*d - a^4*b*e - 2*a^5*f)*x^2)*arctan(1/3*(2*sqrt(3
)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*((20*b^4*c - 5*a*b^3*d - a^2*b^2*e
+ 7*a^3*b*f)*x^6 + 9*a^2*b^2*c + 2*(16*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*
f)*x^3)*(a^2*b)^(1/3))/((a^3*b^4*x^8 + 2*a^4*b^3*x^5 + a^5*b^2*x^2)*(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**3/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218071, size = 486, normalized size = 1.61 \[ \frac{{\left (20 \, b^{3} c - 5 \, a b^{2} d - 2 \, a^{3} f - a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{4} b^{2}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{3}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{3}} - \frac{20 \, b^{4} c x^{6} - 5 \, a b^{3} d x^{6} + 7 \, a^{3} b f x^{6} - a^{2} b^{2} x^{6} e + 32 \, a b^{3} c x^{3} - 8 \, a^{2} b^{2} d x^{3} + 4 \, a^{4} f x^{3} + 2 \, a^{3} b x^{3} e + 9 \, a^{2} b^{2} c}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3} b^{2}} \]

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)^3*x^3),x, algorithm="giac")

[Out]

1/27*(20*b^3*c - 5*a*b^2*d - 2*a^3*f - a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(
1/3)))/(a^4*b^2) - 1/27*sqrt(3)*(20*(-a*b^2)^(1/3)*b^3*c - 5*(-a*b^2)^(1/3)*a*b^
2*d - 2*(-a*b^2)^(1/3)*a^3*f - (-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^3) - 1/54*(20*(-a*b^2)^(1/3)*b^3*c - 5*(-a*b
^2)^(1/3)*a*b^2*d - 2*(-a*b^2)^(1/3)*a^3*f - (-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*
(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^3) - 1/18*(20*b^4*c*x^6 - 5*a*b^3*d*x^6 + 7*
a^3*b*f*x^6 - a^2*b^2*x^6*e + 32*a*b^3*c*x^3 - 8*a^2*b^2*d*x^3 + 4*a^4*f*x^3 + 2
*a^3*b*x^3*e + 9*a^2*b^2*c)/((b*x^4 + a*x)^2*a^3*b^2)