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

Optimal. Leaf size=224 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{5/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{5/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b e-a f)}{b^2}-\frac{c}{2 a x^2}+\frac{f x^4}{4 b} \]

[Out]

-c/(2*a*x^2) + ((b*e - a*f)*x)/b^2 + (f*x^4)/(4*b) + ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*b^
(7/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/
3)*b^(7/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(6*a^(5/3)*b^(7/3))

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Rubi [A]  time = 0.379286, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{5/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{5/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b e-a f)}{b^2}-\frac{c}{2 a x^2}+\frac{f x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*a*x^2) + ((b*e - a*f)*x)/b^2 + (f*x^4)/(4*b) + ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*b^
(7/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/
3)*b^(7/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(6*a^(5/3)*b^(7/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \left (a f - b e\right ) \int \frac{1}{b^{2}}\, dx + \frac{f x^{4}}{4 b} - \frac{c}{2 a x^{2}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{7}{3}}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{7}{3}}} \]

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),x)

[Out]

-(a*f - b*e)*Integral(b**(-2), x) + f*x**4/(4*b) - c/(2*a*x**2) + (a**3*f - a**2
*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(5/3)*b**(7/3)) - (a*
*3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/
3)*x**2)/(6*a**(5/3)*b**(7/3)) - sqrt(3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)
*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(5/3)*b**(7/3))

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Mathematica [A]  time = 0.203563, size = 218, normalized size = 0.97 \[ \frac{1}{12} \left (\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{5/3} b^{7/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{5/3} b^{7/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{5/3} b^{7/3}}+\frac{12 x (b e-a f)}{b^2}-\frac{6 c}{a x^2}+\frac{3 f x^4}{b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-6*c)/(a*x^2) + (12*(b*e - a*f)*x)/b^2 + (3*f*x^4)/b + (4*Sqrt[3]*(b^3*c - a*b
^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(5/3)*b^
(7/3)) + (4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^
(5/3)*b^(7/3)) + (2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(a^(5/3)*b^(7/3)))/12

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Maple [B]  time = 0.007, size = 414, normalized size = 1.9 \[ \text{result too large to display} \]

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),x)

[Out]

1/4*f*x^4/b-1/b^2*a*f*x+e*x/b+1/3*a^2/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f-1/3*a/
b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e+1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-1/3/a/
(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/6*a^2/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/
b)^(2/3))*f+1/6*a/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e-1/6/b/(a/b
)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+1/6/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/
3)+(a/b)^(2/3))*c+1/3*a^2/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1
/3)*x-1))*f-1/3*a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)
)*e+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-1/3/a/(a
/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c-1/2*c/a/x^2

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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)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244861, size = 304, normalized size = 1.36 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (a b f x^{6} + 4 \,{\left (a b e - a^{2} f\right )} x^{3} - 2 \, b^{2} c\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{36 \, \left (a^{2} b\right )^{\frac{1}{3}} a b^{2} x^{2}} \]

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)*x^3),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(2*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2*log((a^2*b)^(2/3
)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 4*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)
*x^2*log((a^2*b)^(1/3)*x + a) - 12*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2*arcta
n(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) + 3*sqrt(3)*(a*b*f*x^6 + 4*(a*b
*e - a^2*f)*x^3 - 2*b^2*c)*(a^2*b)^(1/3))/((a^2*b)^(1/3)*a*b^2*x^2)

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Sympy [A]  time = 6.09596, size = 326, normalized size = 1.46 \[ \operatorname{RootSum}{\left (27 t^{3} a^{5} b^{7} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2} b^{2}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{4}}{4 b} - \frac{x \left (a f - b e\right )}{b^{2}} - \frac{c}{2 a x^{2}} \]

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),x)

[Out]

RootSum(27*_t**3*a**5*b**7 - a**9*f**3 + 3*a**8*b*e*f**2 - 3*a**7*b**2*d*f**2 -
3*a**7*b**2*e**2*f + 3*a**6*b**3*c*f**2 + 6*a**6*b**3*d*e*f + a**6*b**3*e**3 - 6
*a**5*b**4*c*e*f - 3*a**5*b**4*d**2*f - 3*a**5*b**4*d*e**2 + 6*a**4*b**5*c*d*f +
3*a**4*b**5*c*e**2 + 3*a**4*b**5*d**2*e - 3*a**3*b**6*c**2*f - 6*a**3*b**6*c*d*
e - a**3*b**6*d**3 + 3*a**2*b**7*c**2*e + 3*a**2*b**7*c*d**2 - 3*a*b**8*c**2*d +
b**9*c**3, Lambda(_t, _t*log(3*_t*a**2*b**2/(a**3*f - a**2*b*e + a*b**2*d - b**
3*c) + x))) + f*x**4/(4*b) - x*(a*f - b*e)/b**2 - c/(2*a*x**2)

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GIAC/XCAS [A]  time = 0.21652, size = 378, normalized size = 1.69 \[ \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2} b^{2}} - \frac{c}{2 \, a x^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b^{3}} + \frac{b^{3} f x^{4} - 4 \, a b^{2} f x + 4 \, b^{3} x e}{4 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{3}} \]

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)*x^3),x, algorithm="giac")

[Out]

1/3*(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(
a^2*b^2) - 1/2*c/(a*x^2) - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*
b^2*d - (-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^2*b^3) + 1/4*(b^3*f*x^4 - 4*a*b^2*f*x + 4*b^3*x*
e)/b^4 - 1/6*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d - (-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^2*b^3)