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

Optimal. Leaf size=245 \[ \frac{x^2 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 \sqrt [3]{a} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 \sqrt [3]{a} b^{11/3}}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b} \]

[Out]

((b^2*d - a*b*e + a^2*f)*x^2)/(2*b^3) + ((b*e - a*f)*x^5)/(5*b^2) + (f*x^8)/(8*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^(1/3)*b^(11/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Lo
g[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*b^(11/3)) + ((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^(1/3)*b^(11/3))

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Rubi [A]  time = 0.480416, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x^2 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 \sqrt [3]{a} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 \sqrt [3]{a} b^{11/3}}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[(x*(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^2)/(2*b^3) + ((b*e - a*f)*x^5)/(5*b^2) + (f*x^8)/(8*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^(1/3)*b^(11/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Lo
g[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*b^(11/3)) + ((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^(1/3)*b^(11/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{f x^{8}}{8 b} - \frac{x^{5} \left (a f - b e\right )}{5 b^{2}} + \frac{\left (a^{2} f - a b e + b^{2} d\right ) \int x\, dx}{b^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{a} b^{\frac{11}{3}}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{a} b^{\frac{11}{3}}} + \frac{\sqrt{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} b^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x**8/(8*b) - x**5*(a*f - b*e)/(5*b**2) + (a**2*f - a*b*e + b**2*d)*Integral(x,
x)/b**3 + (a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3
*a**(1/3)*b**(11/3)) - (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**(1/3)*b**(11/3)) + 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**(1/3)*b**(11/3))

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Mathematica [A]  time = 0.2958, size = 231, normalized size = 0.94 \[ \frac{60 b^{2/3} x^2 \left (a^2 f-a b e+b^2 d\right )+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt [3]{a}}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt [3]{a}}+24 b^{5/3} x^5 (b e-a f)+15 b^{8/3} f x^8}{120 b^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

(60*b^(2/3)*(b^2*d - a*b*e + a^2*f)*x^2 + 24*b^(5/3)*(b*e - a*f)*x^5 + 15*b^(8/3
)*f*x^8 + (40*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^(1/3) + (40*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*L
og[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(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^(1/3))/(120*b^(11/3))

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Maple [B]  time = 0.005, size = 450, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*f*x^8/b-1/5/b^2*x^5*a*f+1/5/b*x^5*e+1/2/b^3*x^2*a^2*f-1/2/b^2*x^2*a*e+1/2*d*
x^2/b+1/3/b^4/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*a^3*f-1/3/b^3/(a/b)^(1/3)*ln(x+(a/b)
^(1/3))*a^2*e+1/3/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*a*d-1/3/b/(a/b)^(1/3)*ln(x+(
a/b)^(1/3))*c-1/6/b^4/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^3*f+1/6/b^
3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^2*e-1/6/b^2/(a/b)^(1/3)*ln(x^2
-x*(a/b)^(1/3)+(a/b)^(2/3))*a*d+1/6/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/
3))*c-1/3/b^4*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a^3*f+
1/3/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a^2*e-1/3/b^
2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a*d+1/3/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)*x/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222158, size = 313, normalized size = 1.28 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 40 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 120 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (5 \, b^{2} f x^{8} + 8 \,{\left (b^{2} e - a b f\right )} x^{5} + 20 \,{\left (b^{2} d - a b e + a^{2} f\right )} x^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{360 \, \left (a b^{2}\right )^{\frac{1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/360*sqrt(3)*(20*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log((a*b^2)^(1/3)*
b*x^2 + a*b - (a*b^2)^(2/3)*x) - 40*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*
log(a*b + (a*b^2)^(2/3)*x) + 120*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*arctan(-1/3
*(sqrt(3)*a*b - 2*sqrt(3)*(a*b^2)^(2/3)*x)/(a*b)) + 3*sqrt(3)*(5*b^2*f*x^8 + 8*(
b^2*e - a*b*f)*x^5 + 20*(b^2*d - a*b*e + a^2*f)*x^2)*(a*b^2)^(1/3))/((a*b^2)^(1/
3)*b^3)

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Sympy [A]  time = 2.89422, size = 422, normalized size = 1.72 \[ \operatorname{RootSum}{\left (27 t^{3} a b^{11} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a b^{7}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{8}}{8 b} - \frac{x^{5} \left (a f - b e\right )}{5 b^{2}} + \frac{x^{2} \left (a^{2} f - a b e + b^{2} d\right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a*b**11 - 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(9*_t**2*a*b**7/(a**6*f**2 - 2*a**5*b*e*f + 2*a**4*b*
*2*d*f + a**4*b**2*e**2 - 2*a**3*b**3*c*f - 2*a**3*b**3*d*e + 2*a**2*b**4*c*e +
a**2*b**4*d**2 - 2*a*b**5*c*d + b**6*c**2) + x))) + f*x**8/(8*b) - x**5*(a*f - b
*e)/(5*b**2) + x**2*(a**2*f - a*b*e + b**2*d)/(2*b**3)

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GIAC/XCAS [A]  time = 0.219204, size = 466, normalized size = 1.9 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{5}} - \frac{{\left (b^{8} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{7} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{5} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{6} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{8}} + \frac{5 \, b^{7} f x^{8} - 8 \, a b^{6} f x^{5} + 8 \, b^{7} x^{5} e + 20 \, b^{7} d x^{2} + 20 \, a^{2} b^{5} f x^{2} - 20 \, a b^{6} x^{2} e}{40 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-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*b^5) + 1/6*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-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*b^5)
- 1/3*(b^8*c*(-a/b)^(1/3) - a*b^7*d*(-a/b)^(1/3) - a^3*b^5*f*(-a/b)^(1/3) + a^2
*b^6*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^8) + 1/40*(5*b^
7*f*x^8 - 8*a*b^6*f*x^5 + 8*b^7*x^5*e + 20*b^7*d*x^2 + 20*a^2*b^5*f*x^2 - 20*a*b
^6*x^2*e)/b^8