3.264 \(\int \frac{x^3 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=288 \[ \frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2} \]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x)/b^4 + ((b*e - 2*a*f)*x^4)/(4*b^3) + (f*x^7)/(7*b
^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*b^4*(a + b*x^3)) - ((b^3*c - 4*
a*b^2*d + 7*a^2*b*e - 10*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(3*Sqrt[3]*a^(2/3)*b^(13/3)) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*Lo
g[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^(13/3)) - ((b^3*c - 4*a*b^2*d + 7*a^2*b*e -
10*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(13/3))

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Rubi [A]  time = 0.679117, antiderivative size = 288, 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 \[ \frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x)/b^4 + ((b*e - 2*a*f)*x^4)/(4*b^3) + (f*x^7)/(7*b
^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*b^4*(a + b*x^3)) - ((b^3*c - 4*
a*b^2*d + 7*a^2*b*e - 10*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(3*Sqrt[3]*a^(2/3)*b^(13/3)) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*Lo
g[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^(13/3)) - ((b^3*c - 4*a*b^2*d + 7*a^2*b*e -
10*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(13/3))

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Rubi in Sympy [A]  time = 141.744, size = 284, normalized size = 0.99 \[ \frac{f x^{7}}{7 b^{2}} - \frac{x^{4} \left (2 a f - b e\right )}{4 b^{3}} + \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{b^{4}} + \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{4} \left (a + b x^{3}\right )} - \frac{\left (10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{13}{3}}} + \frac{\left (10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{13}{3}}} + \frac{\sqrt{3} \left (10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{13}{3}}} \]

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)**2,x)

[Out]

f*x**7/(7*b**2) - x**4*(2*a*f - b*e)/(4*b**3) + x*(3*a**2*f - 2*a*b*e + b**2*d)/
b**4 + x*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*b**4*(a + b*x**3)) - (10*a**
3*f - 7*a**2*b*e + 4*a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(9*a**(2/3)*b
**(13/3)) + (10*a**3*f - 7*a**2*b*e + 4*a*b**2*d - b**3*c)*log(a**(2/3) - a**(1/
3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(2/3)*b**(13/3)) + sqrt(3)*(10*a**3*f - 7*
a**2*b*e + 4*a*b**2*d - b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1
/3))/(9*a**(2/3)*b**(13/3))

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Mathematica [A]  time = 0.299475, size = 277, normalized size = 0.96 \[ \frac{252 \sqrt [3]{b} x \left (3 a^2 f-2 a b e+b^2 d\right )-\frac{84 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right )}{a^{2/3}}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right )}{a^{2/3}}+63 b^{4/3} x^4 (b e-2 a f)+36 b^{7/3} f x^7}{252 b^{13/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(252*b^(1/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x + 63*b^(4/3)*(b*e - 2*a*f)*x^4 + 36*b
^(7/3)*f*x^7 - (84*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3) +
(28*Sqrt[3]*(-(b^3*c) + 4*a*b^2*d - 7*a^2*b*e + 10*a^3*f)*ArcTan[(1 - (2*b^(1/3)
*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(b^3*c) + 4*a*b^2*d - 7*a^2*b*e + 10*a
^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(252*b^(13/3))

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Maple [B]  time = 0.015, size = 514, 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)^2,x)

[Out]

1/7*f*x^7/b^2-1/2/b^3*x^4*a*f+1/4/b^2*x^4*e+3/b^4*a^2*f*x-2/b^3*a*e*x+x*d/b^2+1/
3/b^4*x/(b*x^3+a)*a^3*f-1/3/b^3*x/(b*x^3+a)*a^2*e+1/3/b^2*x/(b*x^3+a)*a*d-1/3/b*
x/(b*x^3+a)*c-10/9/b^5*a^3*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+5/9/b^5*a^3*f/(a/b)^(
2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-10/9/b^5*a^3*f/(a/b)^(2/3)*3^(1/2)*arctan
(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+7/9/b^4*a^2*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/
18/b^4*a^2*e/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/9/b^4*a^2*e/(a/b)^(
2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-4/9/b^3*a*d/(a/b)^(2/3)*ln(
x+(a/b)^(1/3))+2/9/b^3*a*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9/b^3
*a*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/9/b^2*c/(a/b)
^(2/3)*ln(x+(a/b)^(1/3))-1/18/b^2*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)+1/9/b^2*c/(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)*x^3/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235551, size = 531, normalized size = 1.84 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f +{\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f +{\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f +{\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (12 \, b^{3} f x^{10} + 3 \,{\left (7 \, b^{3} e - 10 \, a b^{2} f\right )} x^{7} + 21 \,{\left (4 \, b^{3} d - 7 \, a b^{2} e + 10 \, a^{2} b f\right )} x^{4} - 28 \,{\left (b^{3} c - 4 \, a b^{2} d + 7 \, a^{2} b e - 10 \, a^{3} f\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{756 \,{\left (b^{5} x^{3} + a b^{4}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

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)^2,x, algorithm="fricas")

[Out]

1/756*sqrt(3)*(14*sqrt(3)*(a*b^3*c - 4*a^2*b^2*d + 7*a^3*b*e - 10*a^4*f + (b^4*c
- 4*a*b^3*d + 7*a^2*b^2*e - 10*a^3*b*f)*x^3)*log((-a^2*b)^(2/3)*x^2 + (-a^2*b)^
(1/3)*a*x + a^2) - 28*sqrt(3)*(a*b^3*c - 4*a^2*b^2*d + 7*a^3*b*e - 10*a^4*f + (b
^4*c - 4*a*b^3*d + 7*a^2*b^2*e - 10*a^3*b*f)*x^3)*log((-a^2*b)^(1/3)*x - a) + 84
*(a*b^3*c - 4*a^2*b^2*d + 7*a^3*b*e - 10*a^4*f + (b^4*c - 4*a*b^3*d + 7*a^2*b^2*
e - 10*a^3*b*f)*x^3)*arctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/3)*x + sqrt(3)*a)/a) + 3*
sqrt(3)*(12*b^3*f*x^10 + 3*(7*b^3*e - 10*a*b^2*f)*x^7 + 21*(4*b^3*d - 7*a*b^2*e
+ 10*a^2*b*f)*x^4 - 28*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x)*(-a^2*b)^(1
/3))/((b^5*x^3 + a*b^4)*(-a^2*b)^(1/3))

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Sympy [A]  time = 17.6266, size = 398, normalized size = 1.38 \[ \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{13} + 1000 a^{9} f^{3} - 2100 a^{8} b e f^{2} + 1200 a^{7} b^{2} d f^{2} + 1470 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 1680 a^{6} b^{3} d e f - 343 a^{6} b^{3} e^{3} + 420 a^{5} b^{4} c e f + 480 a^{5} b^{4} d^{2} f + 588 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f - 147 a^{4} b^{5} c e^{2} - 336 a^{4} b^{5} d^{2} e + 30 a^{3} b^{6} c^{2} f + 168 a^{3} b^{6} c d e + 64 a^{3} b^{6} d^{3} - 21 a^{2} b^{7} c^{2} e - 48 a^{2} b^{7} c d^{2} + 12 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a b^{4}}{10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{7}}{7 b^{2}} - \frac{x^{4} \left (2 a f - b e\right )}{4 b^{3}} + \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{b^{4}} \]

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)**2,x)

[Out]

x*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*a*b**4 + 3*b**5*x**3) + RootSum(729
*_t**3*a**2*b**13 + 1000*a**9*f**3 - 2100*a**8*b*e*f**2 + 1200*a**7*b**2*d*f**2
+ 1470*a**7*b**2*e**2*f - 300*a**6*b**3*c*f**2 - 1680*a**6*b**3*d*e*f - 343*a**6
*b**3*e**3 + 420*a**5*b**4*c*e*f + 480*a**5*b**4*d**2*f + 588*a**5*b**4*d*e**2 -
240*a**4*b**5*c*d*f - 147*a**4*b**5*c*e**2 - 336*a**4*b**5*d**2*e + 30*a**3*b**
6*c**2*f + 168*a**3*b**6*c*d*e + 64*a**3*b**6*d**3 - 21*a**2*b**7*c**2*e - 48*a*
*2*b**7*c*d**2 + 12*a*b**8*c**2*d - b**9*c**3, Lambda(_t, _t*log(-9*_t*a*b**4/(1
0*a**3*f - 7*a**2*b*e + 4*a*b**2*d - b**3*c) + x))) + f*x**7/(7*b**2) - x**4*(2*
a*f - b*e)/(4*b**3) + x*(3*a**2*f - 2*a*b*e + b**2*d)/b**4

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GIAC/XCAS [A]  time = 0.219428, size = 471, normalized size = 1.64 \[ -\frac{{\left (b^{3} c - 4 \, a b^{2} d - 10 \, a^{3} f + 7 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{4}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{5}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{5}} + \frac{4 \, b^{12} f x^{7} - 14 \, a b^{11} f x^{4} + 7 \, b^{12} x^{4} e + 28 \, b^{12} d x + 84 \, a^{2} b^{10} f x - 56 \, a b^{11} x e}{28 \, b^{14}} \]

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)^2,x, algorithm="giac")

[Out]

-1/9*(b^3*c - 4*a*b^2*d - 10*a^3*f + 7*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(
1/3)))/(a*b^4) + 1/9*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - 4*(-a*b^2)^(1/3)*a*b^2*d -
10*(-a*b^2)^(1/3)*a^3*f + 7*(-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^5) - 1/3*(b^3*c*x - a*b^2*d*x - a^3*f*x + a^2*b*x
*e)/((b*x^3 + a)*b^4) + 1/18*((-a*b^2)^(1/3)*b^3*c - 4*(-a*b^2)^(1/3)*a*b^2*d -
10*(-a*b^2)^(1/3)*a^3*f + 7*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-
a/b)^(2/3))/(a*b^5) + 1/28*(4*b^12*f*x^7 - 14*a*b^11*f*x^4 + 7*b^12*x^4*e + 28*b
^12*d*x + 84*a^2*b^10*f*x - 56*a*b^11*x*e)/b^14