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

Optimal. Leaf size=274 \[ \frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4 + ((b^2*d - a*b*e + a^2*f)*x^4)/(4*b
^3) + ((b*e - a*f)*x^7)/(7*b^2) + (f*x^10)/(10*b) + (a^(1/3)*(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]*b^(
13/3)) - (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/
(3*b^(13/3)) + (a^(1/3)*(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*b^(13/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.592991, antiderivative size = 274, 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 \[ \frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4 + ((b^2*d - a*b*e + a^2*f)*x^4)/(4*b
^3) + ((b*e - a*f)*x^7)/(7*b^2) + (f*x^10)/(10*b) + (a^(1/3)*(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]*b^(
13/3)) - (a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/
(3*b^(13/3)) + (a^(1/3)*(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*b^(13/3))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [3]{a} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{13}{3}}} - \frac{\sqrt [3]{a} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{13}{3}}} - \frac{\sqrt{3} \sqrt [3]{a} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{13}{3}}} - \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int \frac{1}{b^{4}}\, dx + \frac{f x^{10}}{10 b} - \frac{x^{7} \left (a f - b e\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} f - a b e + b^{2} d\right )}{4 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*b
**(13/3)) - a**(1/3)*(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*b**(13/3)) - sqrt(3)*a**(1/3)*(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*b**(13/3)) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)*Integral(b**(-4), x) + f*
x**10/(10*b) - x**7*(a*f - b*e)/(7*b**2) + x**4*(a**2*f - a*b*e + b**2*d)/(4*b**
3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.170181, size = 264, normalized size = 0.96 \[ \frac{105 b^{4/3} x^4 \left (a^2 f-a b e+b^2 d\right )+420 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+60 b^{7/3} x^7 (b e-a f)+42 b^{10/3} f x^{10}}{420 b^{13/3}} \]

Antiderivative was successfully verified.

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

[Out]

(420*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x + 105*b^(4/3)*(b^2*d - a*b*e
+ a^2*f)*x^4 + 60*b^(7/3)*(b*e - a*f)*x^7 + 42*b^(10/3)*f*x^10 - 140*Sqrt[3]*a^(
1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]] + 140*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/
3)*x] - 70*a^(1/3)*(-(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])/(420*b^(13/3))

_______________________________________________________________________________________

Maple [B]  time = 0.005, size = 492, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*f*x^10/b-1/7/b^2*x^7*a*f+1/7/b*x^7*e+1/4/b^3*x^4*a^2*f-1/4/b^2*x^4*a*e+1/4/
b*x^4*d-1/b^4*a^3*f*x+1/b^3*a^2*e*x-1/b^2*a*d*x+c*x/b+1/3*a^4/b^5/(a/b)^(2/3)*ln
(x+(a/b)^(1/3))*f-1/3*a^3/b^4/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e+1/3*a^2/b^3/(a/b)^
(2/3)*ln(x+(a/b)^(1/3))*d-1/3*a/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/6*a^4/b^5/
(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/6*a^3/b^4/(a/b)^(2/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))*e-1/6*a^2/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))*d+1/6*a/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+1/3*a^4/b^5/(a
/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-1/3*a^3/b^4/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+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))*d-1/3*a/b^2/(a/b)^(2/3)*3^(1/2)*ar
ctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c

_______________________________________________________________________________________

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^3/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.2231, size = 355, normalized size = 1.3 \[ \frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 140 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (14 \, b^{3} f x^{10} + 20 \,{\left (b^{3} e - a b^{2} f\right )} x^{7} + 35 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{4} + 140 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x\right )}\right )}}{1260 \, b^{4}} \]

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, algorithm="fricas")

[Out]

1/1260*sqrt(3)*(70*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a/b)^(1/3)*log(x
^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 140*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3
*f)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 420*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(
a/b)^(1/3)*arctan(-1/3*(2*sqrt(3)*x - sqrt(3)*(a/b)^(1/3))/(a/b)^(1/3)) + 3*sqrt
(3)*(14*b^3*f*x^10 + 20*(b^3*e - a*b^2*f)*x^7 + 35*(b^3*d - a*b^2*e + a^2*b*f)*x
^4 + 140*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x))/b^4

_______________________________________________________________________________________

Sympy [A]  time = 3.44902, size = 371, normalized size = 1.35 \[ \operatorname{RootSum}{\left (27 t^{3} b^{13} - a^{10} f^{3} + 3 a^{9} b e f^{2} - 3 a^{8} b^{2} d f^{2} - 3 a^{8} b^{2} e^{2} f + 3 a^{7} b^{3} c f^{2} + 6 a^{7} b^{3} d e f + a^{7} b^{3} e^{3} - 6 a^{6} b^{4} c e f - 3 a^{6} b^{4} d^{2} f - 3 a^{6} b^{4} d e^{2} + 6 a^{5} b^{5} c d f + 3 a^{5} b^{5} c e^{2} + 3 a^{5} b^{5} d^{2} e - 3 a^{4} b^{6} c^{2} f - 6 a^{4} b^{6} c d e - a^{4} b^{6} d^{3} + 3 a^{3} b^{7} c^{2} e + 3 a^{3} b^{7} c d^{2} - 3 a^{2} b^{8} c^{2} d + a b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t b^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{10}}{10 b} - \frac{x^{7} \left (a f - b e\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} f - a b e + b^{2} d\right )}{4 b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*b**13 - a**10*f**3 + 3*a**9*b*e*f**2 - 3*a**8*b**2*d*f**2 - 3*a
**8*b**2*e**2*f + 3*a**7*b**3*c*f**2 + 6*a**7*b**3*d*e*f + a**7*b**3*e**3 - 6*a*
*6*b**4*c*e*f - 3*a**6*b**4*d**2*f - 3*a**6*b**4*d*e**2 + 6*a**5*b**5*c*d*f + 3*
a**5*b**5*c*e**2 + 3*a**5*b**5*d**2*e - 3*a**4*b**6*c**2*f - 6*a**4*b**6*c*d*e -
a**4*b**6*d**3 + 3*a**3*b**7*c**2*e + 3*a**3*b**7*c*d**2 - 3*a**2*b**8*c**2*d +
a*b**9*c**3, Lambda(_t, _t*log(3*_t*b**4/(a**3*f - a**2*b*e + a*b**2*d - b**3*c
) + x))) + f*x**10/(10*b) - x**7*(a*f - b*e)/(7*b**2) + x**4*(a**2*f - a*b*e + b
**2*d)/(4*b**3) - x*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/b**4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21903, size = 467, normalized size = 1.7 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{5}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{5}} + \frac{{\left (a b^{9} c - a^{2} b^{8} d - a^{4} b^{6} f + a^{3} b^{7} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac{14 \, b^{9} f x^{10} - 20 \, a b^{8} f x^{7} + 20 \, b^{9} x^{7} e + 35 \, b^{9} d x^{4} + 35 \, a^{2} b^{7} f x^{4} - 35 \, a b^{8} x^{4} e + 140 \, b^{9} c x - 140 \, a b^{8} d x - 140 \, a^{3} b^{6} f x + 140 \, a^{2} b^{7} x e}{140 \, b^{10}} \]

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, 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
))/b^5 - 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))/b^5 + 1/3*(
a*b^9*c - a^2*b^8*d - a^4*b^6*f + a^3*b^7*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3
)))/(a*b^10) + 1/140*(14*b^9*f*x^10 - 20*a*b^8*f*x^7 + 20*b^9*x^7*e + 35*b^9*d*x
^4 + 35*a^2*b^7*f*x^4 - 35*a*b^8*x^4*e + 140*b^9*c*x - 140*a*b^8*d*x - 140*a^3*b
^6*f*x + 140*a^2*b^7*x*e)/b^10