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

Optimal. Leaf size=271 \[ \frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{18 a^{4/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{9 a^{4/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{4/3} b^{11/3}}+\frac{x^2 (b e-2 a f)}{2 b^3}+\frac{f x^5}{5 b^2} \]

[Out]

((b*e - 2*a*f)*x^2)/(2*b^3) + (f*x^5)/(5*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*x^2)/(3*a*b^3*(a + b*x^3)) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Arc
Tan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(4/3)*b^(11/3)) - (
(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b
^(11/3)) + ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1
/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(11/3))

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Rubi [A]  time = 0.617647, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{18 a^{4/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{9 a^{4/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{4/3} b^{11/3}}+\frac{x^2 (b e-2 a f)}{2 b^3}+\frac{f x^5}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*e - 2*a*f)*x^2)/(2*b^3) + (f*x^5)/(5*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*x^2)/(3*a*b^3*(a + b*x^3)) - ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Arc
Tan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(4/3)*b^(11/3)) - (
(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b
^(11/3)) + ((b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1
/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(11/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{f x^{5}}{5 b^{2}} - \frac{\left (2 a f - b e\right ) \int x\, dx}{b^{3}} - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{\left (8 a^{3} f - 5 a^{2} b e + 2 a b^{2} d + b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{11}{3}}} + \frac{\left (8 a^{3} f - 5 a^{2} b e + 2 a b^{2} d + b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{11}{3}}} - \frac{\sqrt{3} \left (8 a^{3} f - 5 a^{2} b e + 2 a b^{2} d + b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{11}{3}}} \]

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)**2,x)

[Out]

f*x**5/(5*b**2) - (2*a*f - b*e)*Integral(x, x)/b**3 - x**2*(a**3*f - a**2*b*e +
a*b**2*d - b**3*c)/(3*a*b**3*(a + b*x**3)) - (8*a**3*f - 5*a**2*b*e + 2*a*b**2*d
+ b**3*c)*log(a**(1/3) + b**(1/3)*x)/(9*a**(4/3)*b**(11/3)) + (8*a**3*f - 5*a**
2*b*e + 2*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**(4/3)*b**(11/3)) - sqrt(3)*(8*a**3*f - 5*a**2*b*e + 2*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**(4/3)*b**(11/3))

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Mathematica [A]  time = 0.275454, size = 255, normalized size = 0.94 \[ \frac{\frac{30 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{4/3}}+\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{4/3}}+45 b^{2/3} x^2 (b e-2 a f)+18 b^{5/3} f x^5}{90 b^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

(45*b^(2/3)*(b*e - 2*a*f)*x^2 + 18*b^(5/3)*f*x^5 + (30*b^(2/3)*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^2)/(a*(a + b*x^3)) - (10*Sqrt[3]*(b^3*c + 2*a*b^2*d - 5*a^2
*b*e + 8*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (10*(b^3*
c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3
*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/a^(4/3))/(90*b^(11/3))

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Maple [B]  time = 0.014, size = 495, 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)^2,x)

[Out]

1/5*f*x^5/b^2-1/b^3*x^2*a*f+1/2/b^2*x^2*e-1/3/b^3*x^2*a^2/(b*x^3+a)*f+1/3/b^2*x^
2*a/(b*x^3+a)*e-1/3*x^2*d/(b*x^3+a)/b+1/3*x^2/a/(b*x^3+a)*c-8/9/b^4*a^2/(a/b)^(1
/3)*ln(x+(a/b)^(1/3))*f+5/9/b^3*a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-2/9/b^2/(a/b)^
(1/3)*ln(x+(a/b)^(1/3))*d-1/9/b/a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+4/9/b^4*a^2/(a
/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-5/18/b^3*a/(a/b)^(1/3)*ln(x^2-x*(a
/b)^(1/3)+(a/b)^(2/3))*e+1/9/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d
+1/18/b/a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+8/9/b^4*a^2*3^(1/2)/(a
/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-5/9/b^3*a*3^(1/2)/(a/b)^(1/3
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+2/9/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+1/9/b/a*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23619, size = 513, normalized size = 1.89 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f +{\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 10 \, \sqrt{3}{\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f +{\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 30 \,{\left (a b^{3} c + 2 \, a^{2} b^{2} d - 5 \, a^{3} b e + 8 \, a^{4} f +{\left (b^{4} c + 2 \, a b^{3} d - 5 \, a^{2} b^{2} e + 8 \, a^{3} b f\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (6 \, a b^{2} f x^{8} + 3 \,{\left (5 \, a b^{2} e - 8 \, a^{2} b f\right )} x^{5} + 5 \,{\left (2 \, b^{3} c - 2 \, a b^{2} d + 5 \, a^{2} b e - 8 \, a^{3} f\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{270 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

-1/270*sqrt(3)*(5*sqrt(3)*(a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c
+ 2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*log((-a*b^2)^(1/3)*b*x^2 - a*b + (-a
*b^2)^(2/3)*x) - 10*sqrt(3)*(a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*
c + 2*a*b^3*d - 5*a^2*b^2*e + 8*a^3*b*f)*x^3)*log(a*b + (-a*b^2)^(2/3)*x) + 30*(
a*b^3*c + 2*a^2*b^2*d - 5*a^3*b*e + 8*a^4*f + (b^4*c + 2*a*b^3*d - 5*a^2*b^2*e +
8*a^3*b*f)*x^3)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x)/(a*b)) -
3*sqrt(3)*(6*a*b^2*f*x^8 + 3*(5*a*b^2*e - 8*a^2*b*f)*x^5 + 5*(2*b^3*c - 2*a*b^2
*d + 5*a^2*b*e - 8*a^3*f)*x^2)*(-a*b^2)^(1/3))/((a*b^4*x^3 + a^2*b^3)*(-a*b^2)^(
1/3))

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Sympy [A]  time = 28.3682, size = 461, normalized size = 1.7 \[ - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{11} + 512 a^{9} f^{3} - 960 a^{8} b e f^{2} + 384 a^{7} b^{2} d f^{2} + 600 a^{7} b^{2} e^{2} f + 192 a^{6} b^{3} c f^{2} - 480 a^{6} b^{3} d e f - 125 a^{6} b^{3} e^{3} - 240 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f + 150 a^{5} b^{4} d e^{2} + 96 a^{4} b^{5} c d f + 75 a^{4} b^{5} c e^{2} - 60 a^{4} b^{5} d^{2} e + 24 a^{3} b^{6} c^{2} f - 60 a^{3} b^{6} c d e + 8 a^{3} b^{6} d^{3} - 15 a^{2} b^{7} c^{2} e + 12 a^{2} b^{7} c d^{2} + 6 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{3} b^{7}}{64 a^{6} f^{2} - 80 a^{5} b e f + 32 a^{4} b^{2} d f + 25 a^{4} b^{2} e^{2} + 16 a^{3} b^{3} c f - 20 a^{3} b^{3} d e - 10 a^{2} b^{4} c e + 4 a^{2} b^{4} d^{2} + 4 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{5}}{5 b^{2}} - \frac{x^{2} \left (2 a f - b e\right )}{2 b^{3}} \]

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)**2,x)

[Out]

-x**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*a**2*b**3 + 3*a*b**4*x**3) + Ro
otSum(729*_t**3*a**4*b**11 + 512*a**9*f**3 - 960*a**8*b*e*f**2 + 384*a**7*b**2*d
*f**2 + 600*a**7*b**2*e**2*f + 192*a**6*b**3*c*f**2 - 480*a**6*b**3*d*e*f - 125*
a**6*b**3*e**3 - 240*a**5*b**4*c*e*f + 96*a**5*b**4*d**2*f + 150*a**5*b**4*d*e**
2 + 96*a**4*b**5*c*d*f + 75*a**4*b**5*c*e**2 - 60*a**4*b**5*d**2*e + 24*a**3*b**
6*c**2*f - 60*a**3*b**6*c*d*e + 8*a**3*b**6*d**3 - 15*a**2*b**7*c**2*e + 12*a**2
*b**7*c*d**2 + 6*a*b**8*c**2*d + b**9*c**3, Lambda(_t, _t*log(81*_t**2*a**3*b**7
/(64*a**6*f**2 - 80*a**5*b*e*f + 32*a**4*b**2*d*f + 25*a**4*b**2*e**2 + 16*a**3*
b**3*c*f - 20*a**3*b**3*d*e - 10*a**2*b**4*c*e + 4*a**2*b**4*d**2 + 4*a*b**5*c*d
+ b**6*c**2) + x))) + f*x**5/(5*b**2) - x**2*(2*a*f - b*e)/(2*b**3)

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GIAC/XCAS [A]  time = 0.218952, size = 494, normalized size = 1.82 \[ -\frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 8 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{5}} + \frac{2 \, b^{8} f x^{5} - 10 \, a b^{7} f x^{2} + 5 \, b^{8} x^{2} e}{10 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-1/9*(b^3*c*(-a/b)^(1/3) + 2*a*b^2*d*(-a/b)^(1/3) + 8*a^3*f*(-a/b)^(1/3) - 5*a^2
*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/3*(b^3*c
*x^2 - a*b^2*d*x^2 - a^3*f*x^2 + a^2*b*x^2*e)/((b*x^3 + a)*a*b^3) - 1/9*sqrt(3)*
((-a*b^2)^(2/3)*b^3*c + 2*(-a*b^2)^(2/3)*a*b^2*d + 8*(-a*b^2)^(2/3)*a^3*f - 5*(-
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^2
*b^5) + 1/18*((-a*b^2)^(2/3)*b^3*c + 2*(-a*b^2)^(2/3)*a*b^2*d + 8*(-a*b^2)^(2/3)
*a^3*f - 5*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*
b^5) + 1/10*(2*b^8*f*x^5 - 10*a*b^7*f*x^2 + 5*b^8*x^2*e)/b^10