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

Optimal. Leaf size=264 \[ \frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{18 a^{5/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^{5/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^4}{4 b^2} \]

[Out]

((b*e - 2*a*f)*x)/b^3 + (f*x^4)/(4*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x
)/(3*a*b^3*(a + b*x^3)) - ((2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*ArcTan[(a^(
1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(10/3)) + ((2*b^3*c
+ a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(10/3))
- ((2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(18*a^(5/3)*b^(10/3))

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Rubi [A]  time = 0.597331, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{18 a^{5/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^{5/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*e - 2*a*f)*x)/b^3 + (f*x^4)/(4*b^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x
)/(3*a*b^3*(a + b*x^3)) - ((2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*ArcTan[(a^(
1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(10/3)) + ((2*b^3*c
+ a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(10/3))
- ((2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(18*a^(5/3)*b^(10/3))

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Rubi in Sympy [A]  time = 105.473, size = 258, normalized size = 0.98 \[ \frac{f x^{4}}{4 b^{2}} - \frac{x \left (2 a f - b e\right )}{b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{3} \left (a + b x^{3}\right )} + \frac{\left (7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{10}{3}}} - \frac{\left (7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{10}{3}}} - \frac{\sqrt{3} \left (7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x**4/(4*b**2) - x*(2*a*f - b*e)/b**3 - x*(a**3*f - a**2*b*e + a*b**2*d - b**3*
c)/(3*a*b**3*(a + b*x**3)) + (7*a**3*f - 4*a**2*b*e + a*b**2*d + 2*b**3*c)*log(a
**(1/3) + b**(1/3)*x)/(9*a**(5/3)*b**(10/3)) - (7*a**3*f - 4*a**2*b*e + a*b**2*d
+ 2*b**3*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(5/3)*b*
*(10/3)) - sqrt(3)*(7*a**3*f - 4*a**2*b*e + a*b**2*d + 2*b**3*c)*atan(sqrt(3)*(a
**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(5/3)*b**(10/3))

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Mathematica [A]  time = 0.281821, size = 251, normalized size = 0.95 \[ \frac{\frac{12 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{5/3}}-\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{5/3}}+36 \sqrt [3]{b} x (b e-2 a f)+9 b^{4/3} f x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

(36*b^(1/3)*(b*e - 2*a*f)*x + 9*b^(4/3)*f*x^4 + (12*b^(1/3)*(b^3*c - a*b^2*d + a
^2*b*e - a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(2*b^3*c + a*b^2*d - 4*a^2*b*e +
7*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (4*(2*b^3*c + a
*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(2*b^3*c +
a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a
^(5/3))/(36*b^(10/3))

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Maple [B]  time = 0.013, size = 482, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235543, size = 494, normalized size = 1.87 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f +{\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f +{\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left (2 \, a b^{3} c + a^{2} b^{2} d - 4 \, a^{3} b e + 7 \, a^{4} f +{\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 7 \, a^{3} b f\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (3 \, a b^{2} f x^{7} + 3 \,{\left (4 \, a b^{2} e - 7 \, a^{2} b f\right )} x^{4} + 4 \,{\left (b^{3} c - a b^{2} d + 4 \, a^{2} b e - 7 \, a^{3} f\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{108 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/108*sqrt(3)*(2*sqrt(3)*(2*a*b^3*c + a^2*b^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*
c + a*b^3*d - 4*a^2*b^2*e + 7*a^3*b*f)*x^3)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3
)*a*x + a^2) - 4*sqrt(3)*(2*a*b^3*c + a^2*b^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*c
+ a*b^3*d - 4*a^2*b^2*e + 7*a^3*b*f)*x^3)*log((a^2*b)^(1/3)*x + a) - 12*(2*a*b^
3*c + a^2*b^2*d - 4*a^3*b*e + 7*a^4*f + (2*b^4*c + a*b^3*d - 4*a^2*b^2*e + 7*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)*(3*
a*b^2*f*x^7 + 3*(4*a*b^2*e - 7*a^2*b*f)*x^4 + 4*(b^3*c - a*b^2*d + 4*a^2*b*e - 7
*a^3*f)*x)*(a^2*b)^(1/3))/((a*b^4*x^3 + a^2*b^3)*(a^2*b)^(1/3))

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Sympy [A]  time = 11.7539, size = 376, normalized size = 1.42 \[ - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{10} - 343 a^{9} f^{3} + 588 a^{8} b e f^{2} - 147 a^{7} b^{2} d f^{2} - 336 a^{7} b^{2} e^{2} f - 294 a^{6} b^{3} c f^{2} + 168 a^{6} b^{3} d e f + 64 a^{6} b^{3} e^{3} + 336 a^{5} b^{4} c e f - 21 a^{5} b^{4} d^{2} f - 48 a^{5} b^{4} d e^{2} - 84 a^{4} b^{5} c d f - 96 a^{4} b^{5} c e^{2} + 12 a^{4} b^{5} d^{2} e - 84 a^{3} b^{6} c^{2} f + 48 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 48 a^{2} b^{7} c^{2} e - 6 a^{2} b^{7} c d^{2} - 12 a b^{8} c^{2} d - 8 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t a^{2} b^{3}}{7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c} + x \right )} \right )\right )} + \frac{f x^{4}}{4 b^{2}} - \frac{x \left (2 a f - b e\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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**2*b**3 + 3*a*b**4*x**3) + RootS
um(729*_t**3*a**5*b**10 - 343*a**9*f**3 + 588*a**8*b*e*f**2 - 147*a**7*b**2*d*f*
*2 - 336*a**7*b**2*e**2*f - 294*a**6*b**3*c*f**2 + 168*a**6*b**3*d*e*f + 64*a**6
*b**3*e**3 + 336*a**5*b**4*c*e*f - 21*a**5*b**4*d**2*f - 48*a**5*b**4*d*e**2 - 8
4*a**4*b**5*c*d*f - 96*a**4*b**5*c*e**2 + 12*a**4*b**5*d**2*e - 84*a**3*b**6*c**
2*f + 48*a**3*b**6*c*d*e - a**3*b**6*d**3 + 48*a**2*b**7*c**2*e - 6*a**2*b**7*c*
d**2 - 12*a*b**8*c**2*d - 8*b**9*c**3, Lambda(_t, _t*log(9*_t*a**2*b**3/(7*a**3*
f - 4*a**2*b*e + a*b**2*d + 2*b**3*c) + x))) + f*x**4/(4*b**2) - x*(2*a*f - b*e)
/b**3

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GIAC/XCAS [A]  time = 0.217001, size = 433, normalized size = 1.64 \[ -\frac{{\left (2 \, b^{3} c + a b^{2} d + 7 \, a^{3} f - 4 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{4}} + \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{4}} + \frac{b^{6} f x^{4} - 8 \, a b^{5} f x + 4 \, b^{6} x e}{4 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(2*b^3*c + a*b^2*d + 7*a^3*f - 4*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1
/3)))/(a^2*b^3) + 1/9*sqrt(3)*(2*(-a*b^2)^(1/3)*b^3*c + (-a*b^2)^(1/3)*a*b^2*d +
7*(-a*b^2)^(1/3)*a^3*f - 4*(-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^2*b^4) + 1/3*(b^3*c*x - a*b^2*d*x - a^3*f*x + a^2*b
*x*e)/((b*x^3 + a)*a*b^3) + 1/18*(2*(-a*b^2)^(1/3)*b^3*c + (-a*b^2)^(1/3)*a*b^2*
d + 7*(-a*b^2)^(1/3)*a^3*f - 4*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a^2*b^4) + 1/4*(b^6*f*x^4 - 8*a*b^5*f*x + 4*b^6*x*e)/b^8