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

Optimal. Leaf size=316 \[ \frac{3 b c-a d}{2 a^4 x^2}-\frac{c}{5 a^3 x^5}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{54 a^{14/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{27 a^{14/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{9 \sqrt{3} a^{14/3} b^{4/3}}+\frac{x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{18 a^4 b \left (a+b x^3\right )} \]

[Out]

-c/(5*a^3*x^5) + (3*b*c - a*d)/(2*a^4*x^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x)/(6*a^3*b*(a + b*x^3)^2) + ((17*b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(
18*a^4*b*(a + b*x^3)) - ((44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*ArcTan[(a^(
1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(14/3)*b^(4/3)) + ((44*b^3*
c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(14/3)*b^(4/
3)) - ((44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(54*a^(14/3)*b^(4/3))

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Rubi [A]  time = 0.879611, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{3 b c-a d}{2 a^4 x^2}-\frac{c}{5 a^3 x^5}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{54 a^{14/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{27 a^{14/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{9 \sqrt{3} a^{14/3} b^{4/3}}+\frac{x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{18 a^4 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-c/(5*a^3*x^5) + (3*b*c - a*d)/(2*a^4*x^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x)/(6*a^3*b*(a + b*x^3)^2) + ((17*b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(
18*a^4*b*(a + b*x^3)) - ((44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*ArcTan[(a^(
1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(14/3)*b^(4/3)) + ((44*b^3*
c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(14/3)*b^(4/
3)) - ((44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(54*a^(14/3)*b^(4/3))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.426866, size = 299, normalized size = 0.95 \[ \frac{-\frac{135 a^{2/3} (a d-3 b c)}{x^2}-\frac{54 a^{5/3} c}{x^5}+\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{b^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{b^{4/3}}-\frac{45 a^{5/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )^2}+\frac{15 a^{2/3} x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{b \left (a+b x^3\right )}-\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{b^{4/3}}}{270 a^{14/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-54*a^(5/3)*c)/x^5 - (135*a^(2/3)*(-3*b*c + a*d))/x^2 - (45*a^(5/3)*(-(b^3*c)
+ a*b^2*d - a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)^2) + (15*a^(2/3)*(17*b^3*c - 11*a
*b^2*d + 5*a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)) - (10*Sqrt[3]*(44*b^3*c - 20*a*b^
2*d + 5*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) +
(10*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3
) - (5*(44*b^3*c - 20*a*b^2*d + 5*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))/(270*a^(14/3))

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Maple [B]  time = 0.023, size = 566, 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)^3,x)

[Out]

-1/5*c/a^3/x^5-1/2*d/a^3/x^2+3/2/a^4/x^2*b*c+1/18/a/(b*x^3+a)^2*x^4*f+5/18/a^2/(
b*x^3+a)^2*x^4*b*e-11/18/a^3/(b*x^3+a)^2*x^4*b^2*d+17/18/a^4/(b*x^3+a)^2*x^4*b^3
*c-1/9/(b*x^3+a)^2*x/b*f+4/9/a/(b*x^3+a)^2*x*e-7/9/a^2/(b*x^3+a)^2*x*b*d+10/9/a^
3/(b*x^3+a)^2*x*b^2*c+1/27/a/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f+5/27/a^2/b/(a/b
)^(2/3)*ln(x+(a/b)^(1/3))*e-20/27/a^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d+44/27/a^4*
b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/54/a/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a
/b)^(2/3))*f-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+10/27/a^
3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d-22/27/a^4*b/(a/b)^(2/3)*ln(x^2
-x*(a/b)^(1/3)+(a/b)^(2/3))*c+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))*f+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-20/27/a^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*
x-1))*d+44/27/a^4*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)^3*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22673, size = 752, normalized size = 2.38 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (44 \, b^{5} c - 20 \, a b^{4} d + 5 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{11} + 2 \,{\left (44 \, a b^{4} c - 20 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e + a^{4} b f\right )} x^{8} +{\left (44 \, a^{2} b^{3} c - 20 \, a^{3} b^{2} d + 5 \, a^{4} b e + a^{5} f\right )} x^{5}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 10 \, \sqrt{3}{\left ({\left (44 \, b^{5} c - 20 \, a b^{4} d + 5 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{11} + 2 \,{\left (44 \, a b^{4} c - 20 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e + a^{4} b f\right )} x^{8} +{\left (44 \, a^{2} b^{3} c - 20 \, a^{3} b^{2} d + 5 \, a^{4} b e + a^{5} f\right )} x^{5}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 30 \,{\left ({\left (44 \, b^{5} c - 20 \, a b^{4} d + 5 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{11} + 2 \,{\left (44 \, a b^{4} c - 20 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e + a^{4} b f\right )} x^{8} +{\left (44 \, a^{2} b^{3} c - 20 \, a^{3} b^{2} d + 5 \, a^{4} b e + a^{5} f\right )} x^{5}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (5 \,{\left (44 \, b^{4} c - 20 \, a b^{3} d + 5 \, a^{2} b^{2} e + a^{3} b f\right )} x^{9} + 2 \,{\left (176 \, a b^{3} c - 80 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{6} - 18 \, a^{3} b c + 9 \,{\left (11 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{810 \,{\left (a^{4} b^{3} x^{11} + 2 \, a^{5} b^{2} x^{8} + a^{6} b x^{5}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

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)^3*x^6),x, algorithm="fricas")

[Out]

-1/810*sqrt(3)*(5*sqrt(3)*((44*b^5*c - 20*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^1
1 + 2*(44*a*b^4*c - 20*a^2*b^3*d + 5*a^3*b^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c -
20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x
+ a^2) - 10*sqrt(3)*((44*b^5*c - 20*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2
*(44*a*b^4*c - 20*a^2*b^3*d + 5*a^3*b^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^
3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*log((a^2*b)^(1/3)*x + a) - 30*((44*b^5*c - 20*
a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c - 20*a^2*b^3*d + 5*a^3*b
^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*arc
tan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(5*(44*b^4*c - 20
*a*b^3*d + 5*a^2*b^2*e + a^3*b*f)*x^9 + 2*(176*a*b^3*c - 80*a^2*b^2*d + 20*a^3*b
*e - 5*a^4*f)*x^6 - 18*a^3*b*c + 9*(11*a^2*b^2*c - 5*a^3*b*d)*x^3)*(a^2*b)^(1/3)
)/((a^4*b^3*x^11 + 2*a^5*b^2*x^8 + a^6*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218416, size = 491, normalized size = 1.55 \[ -\frac{{\left (44 \, b^{3} c - 20 \, a b^{2} d + a^{3} f + 5 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{5} b} + \frac{\sqrt{3}{\left (44 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b^{2}} + \frac{{\left (44 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{5} b^{2}} + \frac{17 \, b^{4} c x^{4} - 11 \, a b^{3} d x^{4} + a^{3} b f x^{4} + 5 \, a^{2} b^{2} x^{4} e + 20 \, a b^{3} c x - 14 \, a^{2} b^{2} d x - 2 \, a^{4} f x + 8 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} b} + \frac{15 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{4} x^{5}} \]

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)^3*x^6),x, algorithm="giac")

[Out]

-1/27*(44*b^3*c - 20*a*b^2*d + a^3*f + 5*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)
^(1/3)))/(a^5*b) + 1/27*sqrt(3)*(44*(-a*b^2)^(1/3)*b^3*c - 20*(-a*b^2)^(1/3)*a*b
^2*d + (-a*b^2)^(1/3)*a^3*f + 5*(-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^5*b^2) + 1/54*(44*(-a*b^2)^(1/3)*b^3*c - 20*(-a
*b^2)^(1/3)*a*b^2*d + (-a*b^2)^(1/3)*a^3*f + 5*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 +
x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b^2) + 1/18*(17*b^4*c*x^4 - 11*a*b^3*d*x^4 +
a^3*b*f*x^4 + 5*a^2*b^2*x^4*e + 20*a*b^3*c*x - 14*a^2*b^2*d*x - 2*a^4*f*x + 8*a
^3*b*x*e)/((b*x^3 + a)^2*a^4*b) + 1/10*(15*b*c*x^3 - 5*a*d*x^3 - 2*a*c)/(a^4*x^5
)