3.236 \(\int \frac{x^4 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=279 \[ \frac{x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^4}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{14/3}}+\frac{x^8 (b e-a f)}{8 b^2}+\frac{f x^{11}}{11 b} \]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(2*b^4) + ((b^2*d - a*b*e + a^2*f)*x^5
)/(5*b^3) + ((b*e - a*f)*x^8)/(8*b^2) + (f*x^11)/(11*b) + (a^(2/3)*(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^(14/3)) + (a^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3
)*x])/(3*b^(14/3)) - (a^(2/3)*(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^(14/3))

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Rubi [A]  time = 0.61526, antiderivative size = 279, 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 \[ \frac{x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^4}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{14/3}}+\frac{x^8 (b e-a f)}{8 b^2}+\frac{f x^{11}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(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^2)/(2*b^4) + ((b^2*d - a*b*e + a^2*f)*x^5
)/(5*b^3) + ((b*e - a*f)*x^8)/(8*b^2) + (f*x^11)/(11*b) + (a^(2/3)*(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^(14/3)) + (a^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3
)*x])/(3*b^(14/3)) - (a^(2/3)*(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^(14/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{\frac{2}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{14}{3}}} + \frac{a^{\frac{2}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{14}{3}}} - \frac{\sqrt{3} a^{\frac{2}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{14}{3}}} + \frac{f x^{11}}{11 b} - \frac{x^{8} \left (a f - b e\right )}{8 b^{2}} + \frac{x^{5} \left (a^{2} f - a b e + b^{2} d\right )}{5 b^{3}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int x\, dx}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

-a**(2/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*
b**(14/3)) + a**(2/3)*(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**(14/3)) - sqrt(3)*a**(2/3)*(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**(14/3)) + f*x**11/(11*b) - x**8*(a*f - b*e)/(8*b**2) + x**5*(a**2*f - a*b*
e + b**2*d)/(5*b**3) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)*Integral(x, x)/b*
*4

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Mathematica [A]  time = 0.165219, size = 266, normalized size = 0.95 \[ \frac{264 b^{5/3} x^5 \left (a^2 f-a b e+b^2 d\right )+660 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-440 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+165 b^{8/3} x^8 (b e-a f)+120 b^{11/3} f x^{11}}{1320 b^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(660*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2 + 264*b^(5/3)*(b^2*d - a*b*
e + a^2*f)*x^5 + 165*b^(8/3)*(b*e - a*f)*x^8 + 120*b^(11/3)*f*x^11 - 440*Sqrt[3]
*a^(2/3)*(-(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]] - 440*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b
^(1/3)*x] + 220*a^(2/3)*(-(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])/(1320*b^(14/3))

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Maple [B]  time = 0.006, size = 502, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

1/11*f*x^11/b-1/8/b^2*x^8*a*f+1/8/b*x^8*e+1/5/b^3*x^5*a^2*f-1/5/b^2*x^5*a*e+1/5/
b*x^5*d-1/2/b^4*x^2*a^3*f+1/2/b^3*x^2*a^2*e-1/2/b^2*x^2*a*d+1/2/b*x^2*c-1/3*a^4/
b^5/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f+1/3*a^3/b^4/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-
1/3*a^2/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d+1/3*a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/
3))*c+1/6*a^4/b^5/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/6*a^3/b^4/(a
/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+1/6*a^2/b^3/(a/b)^(1/3)*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))*d-1/6*a/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)*c+1/3*a^4/b^5*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-1/
3*a^3/b^4*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+1/3*a^2/
b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-1/3*a/b^2*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^4/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220689, size = 410, normalized size = 1.47 \[ \frac{\sqrt{3}{\left (220 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 440 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 1320 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (40 \, b^{3} f x^{11} + 55 \,{\left (b^{3} e - a b^{2} f\right )} x^{8} + 88 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{5} + 220 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2}\right )}\right )}}{3960 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/3960*sqrt(3)*(220*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2/b^2)^(1/3)
*log(a*x^2 - b*x*(-a^2/b^2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 440*sqrt(3)*(b^3*c - a
*b^2*d + a^2*b*e - a^3*f)*(-a^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/3)) - 1320*
(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2/b^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*a*x
- sqrt(3)*b*(-a^2/b^2)^(2/3))/(b*(-a^2/b^2)^(2/3))) + 3*sqrt(3)*(40*b^3*f*x^11 +
55*(b^3*e - a*b^2*f)*x^8 + 88*(b^3*d - a*b^2*e + a^2*b*f)*x^5 + 220*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x^2))/b^4

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Sympy [A]  time = 2.84197, size = 459, normalized size = 1.65 \[ \operatorname{RootSum}{\left (27 t^{3} b^{14} + a^{11} f^{3} - 3 a^{10} b e f^{2} + 3 a^{9} b^{2} d f^{2} + 3 a^{9} b^{2} e^{2} f - 3 a^{8} b^{3} c f^{2} - 6 a^{8} b^{3} d e f - a^{8} b^{3} e^{3} + 6 a^{7} b^{4} c e f + 3 a^{7} b^{4} d^{2} f + 3 a^{7} b^{4} d e^{2} - 6 a^{6} b^{5} c d f - 3 a^{6} b^{5} c e^{2} - 3 a^{6} b^{5} d^{2} e + 3 a^{5} b^{6} c^{2} f + 6 a^{5} b^{6} c d e + a^{5} b^{6} d^{3} - 3 a^{4} b^{7} c^{2} e - 3 a^{4} b^{7} c d^{2} + 3 a^{3} b^{8} c^{2} d - a^{2} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{9}}{a^{7} f^{2} - 2 a^{6} b e f + 2 a^{5} b^{2} d f + a^{5} b^{2} e^{2} - 2 a^{4} b^{3} c f - 2 a^{4} b^{3} d e + 2 a^{3} b^{4} c e + a^{3} b^{4} d^{2} - 2 a^{2} b^{5} c d + a b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{11}}{11 b} - \frac{x^{8} \left (a f - b e\right )}{8 b^{2}} + \frac{x^{5} \left (a^{2} f - a b e + b^{2} d\right )}{5 b^{3}} - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**14 + a**11*f**3 - 3*a**10*b*e*f**2 + 3*a**9*b**2*d*f**2 + 3*
a**9*b**2*e**2*f - 3*a**8*b**3*c*f**2 - 6*a**8*b**3*d*e*f - a**8*b**3*e**3 + 6*a
**7*b**4*c*e*f + 3*a**7*b**4*d**2*f + 3*a**7*b**4*d*e**2 - 6*a**6*b**5*c*d*f - 3
*a**6*b**5*c*e**2 - 3*a**6*b**5*d**2*e + 3*a**5*b**6*c**2*f + 6*a**5*b**6*c*d*e
+ a**5*b**6*d**3 - 3*a**4*b**7*c**2*e - 3*a**4*b**7*c*d**2 + 3*a**3*b**8*c**2*d
- a**2*b**9*c**3, Lambda(_t, _t*log(9*_t**2*b**9/(a**7*f**2 - 2*a**6*b*e*f + 2*a
**5*b**2*d*f + a**5*b**2*e**2 - 2*a**4*b**3*c*f - 2*a**4*b**3*d*e + 2*a**3*b**4*
c*e + a**3*b**4*d**2 - 2*a**2*b**5*c*d + a*b**6*c**2) + x))) + f*x**11/(11*b) -
x**8*(a*f - b*e)/(8*b**2) + x**5*(a**2*f - a*b*e + b**2*d)/(5*b**3) - x**2*(a**3
*f - a**2*b*e + a*b**2*d - b**3*c)/(2*b**4)

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GIAC/XCAS [A]  time = 0.219166, size = 521, normalized size = 1.87 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{6}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{6}} + \frac{{\left (a b^{10} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{9} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{4} b^{7} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{8} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{11}} + \frac{40 \, b^{10} f x^{11} - 55 \, a b^{9} f x^{8} + 55 \, b^{10} x^{8} e + 88 \, b^{10} d x^{5} + 88 \, a^{2} b^{8} f x^{5} - 88 \, a b^{9} x^{5} e + 220 \, b^{10} c x^{2} - 220 \, a b^{9} d x^{2} - 220 \, a^{3} b^{7} f x^{2} + 220 \, a^{2} b^{8} x^{2} e}{440 \, b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(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)
)/b^6 - 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))/b^6 + 1/3*(a
*b^10*c*(-a/b)^(1/3) - a^2*b^9*d*(-a/b)^(1/3) - a^4*b^7*f*(-a/b)^(1/3) + a^3*b^8
*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^11) + 1/440*(40*b^1
0*f*x^11 - 55*a*b^9*f*x^8 + 55*b^10*x^8*e + 88*b^10*d*x^5 + 88*a^2*b^8*f*x^5 - 8
8*a*b^9*x^5*e + 220*b^10*c*x^2 - 220*a*b^9*d*x^2 - 220*a^3*b^7*f*x^2 + 220*a^2*b
^8*x^2*e)/b^11