3.269 \(\int \frac{c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=269 \[ \frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}} \]

[Out]

-c/(4*a^2*x^4) + (2*b*c - a*d)/(a^3*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^
2)/(3*a^3*b*(a + b*x^3)) - ((7*b^3*c - 4*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))])/(3*Sqrt[3]*a^(10/3)*b^(5/3)) - ((7*b^3*
c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(10/3)*b^(5/3)
) + ((7*b^3*c - 4*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])/(18*a^(10/3)*b^(5/3))

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Rubi [A]  time = 0.656291, antiderivative size = 269, 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 \[ \frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]

[Out]

-c/(4*a^2*x^4) + (2*b*c - a*d)/(a^3*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^
2)/(3*a^3*b*(a + b*x^3)) - ((7*b^3*c - 4*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))])/(3*Sqrt[3]*a^(10/3)*b^(5/3)) - ((7*b^3*
c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(10/3)*b^(5/3)
) + ((7*b^3*c - 4*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])/(18*a^(10/3)*b^(5/3))

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Rubi in Sympy [A]  time = 139.282, size = 270, normalized size = 1. \[ - \frac{x \left (\frac{a^{3} f}{x^{5}} - \frac{a^{2} b e}{x^{5}} + \frac{a b^{2} d}{x^{5}} - \frac{b^{3} c}{x^{5}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{4 a b^{3} x^{4}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{a^{2} b^{2} x} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{7}{3}} b^{\frac{5}{3}}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{7}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**5/(b*x**3+a)**2,x)

[Out]

-x*(a**3*f/x**5 - a**2*b*e/x**5 + a*b**2*d/x**5 - b**3*c/x**5)/(3*a*b**3*(a + b*
x**3)) - (a**2*f - a*b*e + b**2*d)/(4*a*b**3*x**4) + (2*a**2*f - 2*a*b*e + b**2*
d)/(a**2*b**2*x) - (3*a**2*f - 2*a*b*e + b**2*d)*log(a**(1/3) + b**(1/3)*x)/(3*a
**(7/3)*b**(5/3)) + (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**(7/3)*b**(5/3)) - 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**(7/3)*b**(5/3))

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Mathematica [A]  time = 0.301137, size = 255, normalized size = 0.95 \[ \frac{-\frac{9 a^{4/3} c}{x^4}-\frac{12 \sqrt [3]{a} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{36 \sqrt [3]{a} (a d-2 b c)}{x}}{36 a^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]

[Out]

((-9*a^(4/3)*c)/x^4 - (36*a^(1/3)*(-2*b*c + a*d))/x - (12*a^(1/3)*(-(b^3*c) + a*
b^2*d - a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)) - (4*Sqrt[3]*(7*b^3*c - 4*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^(5/3) - (4*(
7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*
(7*b^3*c - 4*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^(5/3))/(36*a^(10/3))

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Maple [B]  time = 0.022, size = 486, 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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23711, size = 532, normalized size = 1.98 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 4 \, \sqrt{3}{\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 12 \,{\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (4 \,{\left (7 \, b^{3} c - 4 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 3 \, a^{2} b c + 3 \,{\left (7 \, a b^{2} c - 4 \, a^{2} b d\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{108 \,{\left (a^{3} b^{2} x^{7} + a^{4} b x^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

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)^2*x^5),x, algorithm="fricas")

[Out]

-1/108*sqrt(3)*(2*sqrt(3)*((7*b^4*c - 4*a*b^3*d + a^2*b^2*e + 2*a^3*b*f)*x^7 + (
7*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*f)*x^4)*log((-a*b^2)^(1/3)*b*x^2 - a*b
+ (-a*b^2)^(2/3)*x) - 4*sqrt(3)*((7*b^4*c - 4*a*b^3*d + a^2*b^2*e + 2*a^3*b*f)*
x^7 + (7*a*b^3*c - 4*a^2*b^2*d + a^3*b*e + 2*a^4*f)*x^4)*log(a*b + (-a*b^2)^(2/3
)*x) + 12*((7*b^4*c - 4*a*b^3*d + a^2*b^2*e + 2*a^3*b*f)*x^7 + (7*a*b^3*c - 4*a^
2*b^2*d + a^3*b*e + 2*a^4*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*(7*b^3*c - 4*a*b^2*d + a^2*b*e - a^3*f)*x^6 - 3*a
^2*b*c + 3*(7*a*b^2*c - 4*a^2*b*d)*x^3)*(-a*b^2)^(1/3))/((a^3*b^2*x^7 + a^4*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220432, size = 483, normalized size = 1.8 \[ -\frac{{\left (7 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4} b} + \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{3} b} - \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4} b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{4} b^{3}} + \frac{8 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{3} x^{4}} \]

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)^2*x^5),x, algorithm="giac")

[Out]

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