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

Optimal. Leaf size=240 \[ \frac{x \left (a^2 f-a b e+b^2 d\right )}{b^3}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{2/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{2/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{x^4 (b e-a f)}{4 b^2}+\frac{f x^7}{7 b} \]

[Out]

((b^2*d - a*b*e + a^2*f)*x)/b^3 + ((b*e - a*f)*x^4)/(4*b^2) + (f*x^7)/(7*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^(2/3)*b^(10/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1
/3) + b^(1/3)*x])/(3*a^(2/3)*b^(10/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(10/3))

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Rubi [A]  time = 0.342271, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{x \left (a^2 f-a b e+b^2 d\right )}{b^3}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{2/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{2/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{x^4 (b e-a f)}{4 b^2}+\frac{f x^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*d - a*b*e + a^2*f)*x)/b^3 + ((b*e - a*f)*x^4)/(4*b^2) + (f*x^7)/(7*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^(2/3)*b^(10/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1
/3) + b^(1/3)*x])/(3*a^(2/3)*b^(10/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(10/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \left (a^{2} f - a b e + b^{2} d\right ) \int \frac{1}{b^{3}}\, dx + \frac{f x^{7}}{7 b} - \frac{x^{4} \left (a f - b e\right )}{4 b^{2}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{10}{3}}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{10}{3}}} + \frac{\sqrt{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a**2*f - a*b*e + b**2*d)*Integral(b**(-3), x) + f*x**7/(7*b) - x**4*(a*f - b*e)
/(4*b**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**(2/3)*b**(10/3)) + (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**(2/3)*b**(10/3)) + 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**(2/3)*b**(10/3))

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Mathematica [A]  time = 0.297413, size = 229, normalized size = 0.95 \[ \frac{84 \sqrt [3]{b} x \left (a^2 f-a b e+b^2 d\right )+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{2/3}}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{2/3}}+21 b^{4/3} x^4 (b e-a f)+12 b^{7/3} f x^7}{84 b^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

(84*b^(1/3)*(b^2*d - a*b*e + a^2*f)*x + 21*b^(4/3)*(b*e - a*f)*x^4 + 12*b^(7/3)*
f*x^7 + (28*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^(2/3) + (28*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^
(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(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^(2/3))/(84*b^(10/3))

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Maple [B]  time = 0.004, size = 442, 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)/(b*x^3+a),x)

[Out]

1/7*f*x^7/b-1/4/b^2*x^4*a*f+1/4/b*x^4*e+1/b^3*a^2*f*x-1/b^2*a*e*x+d*x/b-1/3/b^4/
(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^3*f+1/3/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^2*e-
1/3/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a*d+1/3*c/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+
1/6/b^4/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^3*f-1/6/b^3/(a/b)^(2/3)*
ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^2*e+1/6/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)
+(a/b)^(2/3))*a*d-1/6*c/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3/b^4/
(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a^3*f+1/3/b^3/(a/b)^
(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a^2*e-1/3/b^2/(a/b)^(2/3)*
3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a*d+1/3*c/b/(a/b)^(2/3)*3^(1/2)*
arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219365, size = 312, normalized size = 1.3 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (4 \, b^{2} f x^{7} + 7 \,{\left (b^{2} e - a b f\right )} x^{4} + 28 \,{\left (b^{2} d - a b e + a^{2} f\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{252 \, \left (-a^{2} b\right )^{\frac{1}{3}} b^{3}} \]

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, algorithm="fricas")

[Out]

1/252*sqrt(3)*(14*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log((-a^2*b)^(2/3)
*x^2 + (-a^2*b)^(1/3)*a*x + a^2) - 28*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*log((-a^2*b)^(1/3)*x - a) + 84*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*arctan(1/3*
(2*sqrt(3)*(-a^2*b)^(1/3)*x + sqrt(3)*a)/a) + 3*sqrt(3)*(4*b^2*f*x^7 + 7*(b^2*e
- a*b*f)*x^4 + 28*(b^2*d - a*b*e + a^2*f)*x)*(-a^2*b)^(1/3))/((-a^2*b)^(1/3)*b^3
)

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Sympy [A]  time = 3.60088, size = 340, normalized size = 1.42 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{10} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t a b^{3}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{7}}{7 b} - \frac{x^{4} \left (a f - b e\right )}{4 b^{2}} + \frac{x \left (a^{2} f - a b e + b^{2} d\right )}{b^{3}} \]

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)

[Out]

RootSum(27*_t**3*a**2*b**10 + 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*b**3/(a**3*f - a**2*b*e + a*b**2*d - b**3
*c) + x))) + f*x**7/(7*b) - x**4*(a*f - b*e)/(4*b**2) + x*(a**2*f - a*b*e + b**2
*d)/b**3

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GIAC/XCAS [A]  time = 0.216909, size = 414, normalized size = 1.72 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{4}} - \frac{{\left (b^{7} c - a b^{6} d - a^{3} b^{4} f + a^{2} b^{5} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac{4 \, b^{6} f x^{7} - 7 \, a b^{5} f x^{4} + 7 \, b^{6} x^{4} e + 28 \, b^{6} d x + 28 \, a^{2} b^{4} f x - 28 \, a b^{5} x e}{28 \, b^{7}} \]

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, algorithm="giac")

[Out]

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*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*b^4)
- 1/3*(b^7*c - a*b^6*d - a^3*b^4*f + a^2*b^5*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(
1/3)))/(a*b^7) + 1/28*(4*b^6*f*x^7 - 7*a*b^5*f*x^4 + 7*b^6*x^4*e + 28*b^6*d*x +
28*a^2*b^4*f*x - 28*a*b^5*x*e)/b^7