3.297 \(\int \frac{c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=317 \[ \frac{3 b c-a d}{a^4 x}-\frac{c}{4 a^3 x^4}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{54 a^{13/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{27 a^{13/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} a^{13/3} b^{5/3}}+\frac{x^2 \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{9 a^4 b \left (a+b x^3\right )} \]

[Out]

-c/(4*a^3*x^4) + (3*b*c - a*d)/(a^4*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^
2)/(6*a^3*b*(a + b*x^3)^2) + ((8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*x^2)/(9*
a^4*b*(a + b*x^3)) - ((35*b^3*c - 14*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))])/(9*Sqrt[3]*a^(13/3)*b^(5/3)) - ((35*b^3*c -
14*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(13/3)*b^(5/3))
+ ((35*b^3*c - 14*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])/(54*a^(13/3)*b^(5/3))

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Rubi [A]  time = 0.899201, antiderivative size = 317, 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 \[ \frac{3 b c-a d}{a^4 x}-\frac{c}{4 a^3 x^4}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{54 a^{13/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{27 a^{13/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} a^{13/3} b^{5/3}}+\frac{x^2 \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{9 a^4 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^3),x]

[Out]

-c/(4*a^3*x^4) + (3*b*c - a*d)/(a^4*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^
2)/(6*a^3*b*(a + b*x^3)^2) + ((8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*x^2)/(9*
a^4*b*(a + b*x^3)) - ((35*b^3*c - 14*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))])/(9*Sqrt[3]*a^(13/3)*b^(5/3)) - ((35*b^3*c -
14*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(13/3)*b^(5/3))
+ ((35*b^3*c - 14*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])/(54*a^(13/3)*b^(5/3))

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

[Out]

Timed out

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Mathematica [A]  time = 0.43246, size = 303, normalized size = 0.96 \[ \frac{-\frac{27 a^{4/3} c}{x^4}+\frac{12 \sqrt [3]{a} x^2 \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b \left (a+b x^3\right )}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{b^{5/3}}-\frac{18 a^{4/3} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )^2}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right )}{b^{5/3}}-\frac{108 \sqrt [3]{a} (a d-3 b c)}{x}}{108 a^{13/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^3),x]

[Out]

((-27*a^(4/3)*c)/x^4 - (108*a^(1/3)*(-3*b*c + a*d))/x - (18*a^(4/3)*(-(b^3*c) +
a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)^2) + (12*a^(1/3)*(8*b^3*c - 5*a*b
^2*d + 2*a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)) - (4*Sqrt[3]*(35*b^3*c - 14*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^(5/3) - (
4*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3)
+ (2*(35*b^3*c - 14*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^(5/3))/(108*a^(13/3))

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Maple [B]  time = 0.023, size = 574, 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^5/(b*x^3+a)^3,x)

[Out]

-1/4*c/a^3/x^4-d/a^3/x+3/a^4/x*b*c+1/9/a/(b*x^3+a)^2*x^5*f+2/9/a^2/(b*x^3+a)^2*x
^5*b*e-5/9/a^3/(b*x^3+a)^2*x^5*b^2*d+8/9/a^4/(b*x^3+a)^2*x^5*b^3*c-1/18/(b*x^3+a
)^2*x^2/b*f+7/18/a/(b*x^3+a)^2*x^2*e-13/18/a^2/(b*x^3+a)^2*x^2*b*d+19/18/a^3/(b*
x^3+a)^2*c*x^2*b^2-1/27/a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-2/27/a^2/b/(a/b)^(
1/3)*ln(x+(a/b)^(1/3))*e+14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d-35/27/a^4*b/(
a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+1/54/a/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)
^(2/3))*f+1/27/a^2/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e-7/27/a^3/(a
/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+35/54/a^4*b/(a/b)^(1/3)*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))*c+1/27/a/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(
a/b)^(1/3)*x-1))*f+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-14/27/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)
)*d+35/27/a^4*b*3^(1/2)/(a/b)^(1/3)*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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240055, size = 765, normalized size = 2.41 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (35 \, b^{5} c - 14 \, a b^{4} d + 2 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{10} + 2 \,{\left (35 \, a b^{4} c - 14 \, a^{2} b^{3} d + 2 \, a^{3} b^{2} e + a^{4} b f\right )} x^{7} +{\left (35 \, a^{2} b^{3} c - 14 \, a^{3} b^{2} d + 2 \, a^{4} b e + a^{5} f\right )} x^{4}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 4 \, \sqrt{3}{\left ({\left (35 \, b^{5} c - 14 \, a b^{4} d + 2 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{10} + 2 \,{\left (35 \, a b^{4} c - 14 \, a^{2} b^{3} d + 2 \, a^{3} b^{2} e + a^{4} b f\right )} x^{7} +{\left (35 \, a^{2} b^{3} c - 14 \, a^{3} b^{2} d + 2 \, a^{4} b e + a^{5} f\right )} x^{4}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 12 \,{\left ({\left (35 \, b^{5} c - 14 \, a b^{4} d + 2 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{10} + 2 \,{\left (35 \, a b^{4} c - 14 \, a^{2} b^{3} d + 2 \, a^{3} b^{2} e + a^{4} b f\right )} x^{7} +{\left (35 \, a^{2} b^{3} c - 14 \, a^{3} b^{2} d + 2 \, a^{4} b e + a^{5} f\right )} x^{4}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (4 \,{\left (35 \, b^{4} c - 14 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{9} +{\left (245 \, a b^{3} c - 98 \, a^{2} b^{2} d + 14 \, a^{3} b e - 2 \, a^{4} f\right )} x^{6} - 9 \, a^{3} b c + 18 \,{\left (5 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{324 \,{\left (a^{4} b^{3} x^{10} + 2 \, a^{5} b^{2} x^{7} + a^{6} b x^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

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^5),x, algorithm="fricas")

[Out]

-1/324*sqrt(3)*(2*sqrt(3)*((35*b^5*c - 14*a*b^4*d + 2*a^2*b^3*e + a^3*b^2*f)*x^1
0 + 2*(35*a*b^4*c - 14*a^2*b^3*d + 2*a^3*b^2*e + a^4*b*f)*x^7 + (35*a^2*b^3*c -
14*a^3*b^2*d + 2*a^4*b*e + a^5*f)*x^4)*log((-a*b^2)^(1/3)*b*x^2 - a*b + (-a*b^2)
^(2/3)*x) - 4*sqrt(3)*((35*b^5*c - 14*a*b^4*d + 2*a^2*b^3*e + a^3*b^2*f)*x^10 +
2*(35*a*b^4*c - 14*a^2*b^3*d + 2*a^3*b^2*e + a^4*b*f)*x^7 + (35*a^2*b^3*c - 14*a
^3*b^2*d + 2*a^4*b*e + a^5*f)*x^4)*log(a*b + (-a*b^2)^(2/3)*x) + 12*((35*b^5*c -
14*a*b^4*d + 2*a^2*b^3*e + a^3*b^2*f)*x^10 + 2*(35*a*b^4*c - 14*a^2*b^3*d + 2*a
^3*b^2*e + a^4*b*f)*x^7 + (35*a^2*b^3*c - 14*a^3*b^2*d + 2*a^4*b*e + a^5*f)*x^4)
*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x)/(a*b)) - 3*sqrt(3)*(4*(3
5*b^4*c - 14*a*b^3*d + 2*a^2*b^2*e + a^3*b*f)*x^9 + (245*a*b^3*c - 98*a^2*b^2*d
+ 14*a^3*b*e - 2*a^4*f)*x^6 - 9*a^3*b*c + 18*(5*a^2*b^2*c - 2*a^3*b*d)*x^3)*(-a*
b^2)^(1/3))/((a^4*b^3*x^10 + 2*a^5*b^2*x^7 + a^6*b*x^4)*(-a*b^2)^(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**5/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221169, size = 547, normalized size = 1.73 \[ -\frac{{\left (35 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{5} b} - \frac{\sqrt{3}{\left (35 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b^{3}} + \frac{16 \, b^{4} c x^{5} - 10 \, a b^{3} d x^{5} + 2 \, a^{3} b f x^{5} + 4 \, a^{2} b^{2} x^{5} e + 19 \, a b^{3} c x^{2} - 13 \, a^{2} b^{2} d x^{2} - a^{4} f x^{2} + 7 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} b} + \frac{{\left (35 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{5} b^{3}} + \frac{12 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{4} x^{4}} \]

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^5),x, algorithm="giac")

[Out]

-1/27*(35*b^3*c*(-a/b)^(1/3) - 14*a*b^2*d*(-a/b)^(1/3) + a^3*f*(-a/b)^(1/3) + 2*
a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^5*b) - 1/27*sqrt
(3)*(35*(-a*b^2)^(2/3)*b^3*c - 14*(-a*b^2)^(2/3)*a*b^2*d + (-a*b^2)^(2/3)*a^3*f
+ 2*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3)
)/(a^5*b^3) + 1/18*(16*b^4*c*x^5 - 10*a*b^3*d*x^5 + 2*a^3*b*f*x^5 + 4*a^2*b^2*x^
5*e + 19*a*b^3*c*x^2 - 13*a^2*b^2*d*x^2 - a^4*f*x^2 + 7*a^3*b*x^2*e)/((b*x^3 + a
)^2*a^4*b) + 1/54*(35*(-a*b^2)^(2/3)*b^3*c - 14*(-a*b^2)^(2/3)*a*b^2*d + (-a*b^2
)^(2/3)*a^3*f + 2*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)
)/(a^5*b^3) + 1/4*(12*b*c*x^3 - 4*a*d*x^3 - a*c)/(a^4*x^4)