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

Optimal. Leaf size=336 \[ \frac{x \left (6 a^2 f-3 a b e+b^2 d\right )}{b^5}-\frac{x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{18 b^5 \left (a+b x^3\right )}+\frac{a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{54 a^{2/3} b^{16/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{27 a^{2/3} b^{16/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{2/3} b^{16/3}}+\frac{x^4 (b e-3 a f)}{4 b^4}+\frac{f x^7}{7 b^3} \]

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x)/b^5 + ((b*e - 3*a*f)*x^4)/(4*b^4) + (f*x^7)/(7*b
^3) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^5*(a + b*x^3)^2) - ((7*b^3*
c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(18*b^5*(a + b*x^3)) - ((2*b^3*c - 14
*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3
))])/(9*Sqrt[3]*a^(2/3)*b^(16/3)) + ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3
*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(2/3)*b^(16/3)) - ((2*b^3*c - 14*a*b^2*d + 3
5*a^2*b*e - 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(2/3
)*b^(16/3))

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Rubi [A]  time = 1.03791, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{x \left (6 a^2 f-3 a b e+b^2 d\right )}{b^5}-\frac{x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{18 b^5 \left (a+b x^3\right )}+\frac{a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{54 a^{2/3} b^{16/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{27 a^{2/3} b^{16/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{2/3} b^{16/3}}+\frac{x^4 (b e-3 a f)}{4 b^4}+\frac{f x^7}{7 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x)/b^5 + ((b*e - 3*a*f)*x^4)/(4*b^4) + (f*x^7)/(7*b
^3) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^5*(a + b*x^3)^2) - ((7*b^3*
c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(18*b^5*(a + b*x^3)) - ((2*b^3*c - 14
*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3
))])/(9*Sqrt[3]*a^(2/3)*b^(16/3)) + ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3
*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(2/3)*b^(16/3)) - ((2*b^3*c - 14*a*b^2*d + 3
5*a^2*b*e - 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(2/3
)*b^(16/3))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.755561, size = 323, normalized size = 0.96 \[ \frac{756 \sqrt [3]{b} x \left (6 a^2 f-3 a b e+b^2 d\right )-\frac{42 \sqrt [3]{b} x \left (-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right )}{a+b x^3}+\frac{126 a \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (65 a^3 f-35 a^2 b e+14 a b^2 d-2 b^3 c\right )}{a^{2/3}}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^3 f-35 a^2 b e+14 a b^2 d-2 b^3 c\right )}{a^{2/3}}+189 b^{4/3} x^4 (b e-3 a f)+108 b^{7/3} f x^7}{756 b^{16/3}} \]

Antiderivative was successfully verified.

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

[Out]

(756*b^(1/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x + 189*b^(4/3)*(b*e - 3*a*f)*x^4 + 108
*b^(7/3)*f*x^7 + (126*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^
3)^2 - (42*b^(1/3)*(7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(a + b*x^3)
+ (28*Sqrt[3]*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*ArcTan[(1 - (2*b^
(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 6
5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-2*b^3*c + 14*a*b^2*d - 35*a^2
*b*e + 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(756*b
^(16/3))

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Maple [B]  time = 0.019, size = 596, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-65/27/b^6*a^3*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+35/
27/b^5*a^2*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/27/b
^4*a*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-19/18/b^3/(b*
x^3+a)^2*x^4*a^2*e+1/4/b^3*x^4*e+1/b^3*d*x-3/b^4*a*e*x-7/18/b/(b*x^3+a)^2*x^4*c-
1/27/b^3*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27/b^3*c/(a/b)^(2/3)*
ln(x+(a/b)^(1/3))+1/7*f*x^7/b^3-3/4/b^4*x^4*a*f+6/b^5*a^2*f*x+25/18/b^4/(b*x^3+a
)^2*x^4*a^3*f-8/9/b^4/(b*x^3+a)^2*a^3*e*x+5/9/b^3/(b*x^3+a)^2*a^2*d*x-2/9/b^2/(b
*x^3+a)^2*a*c*x-65/27/b^6*a^3*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+65/54/b^6*a^3*f/(a
/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+35/27/b^5*a^2*e/(a/b)^(2/3)*ln(x+(a/
b)^(1/3))-35/54/b^5*a^2*e/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-14/27/b^
4*a*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+7/27/b^4*a*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3
)+(a/b)^(2/3))+13/18/b^2/(b*x^3+a)^2*x^4*a*d+11/9/b^5/(b*x^3+a)^2*a^4*f*x+2/27/b
^3*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^6/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234425, size = 779, normalized size = 2.32 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left ({\left (2 \, b^{5} c - 14 \, a b^{4} d + 35 \, a^{2} b^{3} e - 65 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c - 14 \, a^{3} b^{2} d + 35 \, a^{4} b e - 65 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c - 14 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 65 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left ({\left (2 \, b^{5} c - 14 \, a b^{4} d + 35 \, a^{2} b^{3} e - 65 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c - 14 \, a^{3} b^{2} d + 35 \, a^{4} b e - 65 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c - 14 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 65 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left ({\left (2 \, b^{5} c - 14 \, a b^{4} d + 35 \, a^{2} b^{3} e - 65 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c - 14 \, a^{3} b^{2} d + 35 \, a^{4} b e - 65 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c - 14 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 65 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (36 \, b^{4} f x^{13} + 9 \,{\left (7 \, b^{4} e - 13 \, a b^{3} f\right )} x^{10} + 18 \,{\left (14 \, b^{4} d - 35 \, a b^{3} e + 65 \, a^{2} b^{2} f\right )} x^{7} - 49 \,{\left (2 \, b^{4} c - 14 \, a b^{3} d + 35 \, a^{2} b^{2} e - 65 \, a^{3} b f\right )} x^{4} - 28 \,{\left (2 \, a b^{3} c - 14 \, a^{2} b^{2} d + 35 \, a^{3} b e - 65 \, a^{4} f\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{2268 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2268*sqrt(3)*(14*sqrt(3)*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)
*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^
2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*log((-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3
)*a*x + a^2) - 28*sqrt(3)*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*
x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2
*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*log((-a^2*b)^(1/3)*x - a) + 84*((2*b^5*
c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d +
35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f
)*x^3)*arctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/3)*x + sqrt(3)*a)/a) + 3*sqrt(3)*(36*b^
4*f*x^13 + 9*(7*b^4*e - 13*a*b^3*f)*x^10 + 18*(14*b^4*d - 35*a*b^3*e + 65*a^2*b^
2*f)*x^7 - 49*(2*b^4*c - 14*a*b^3*d + 35*a^2*b^2*e - 65*a^3*b*f)*x^4 - 28*(2*a*b
^3*c - 14*a^2*b^2*d + 35*a^3*b*e - 65*a^4*f)*x)*(-a^2*b)^(1/3))/((b^7*x^6 + 2*a*
b^6*x^3 + a^2*b^5)*(-a^2*b)^(1/3))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218029, size = 539, normalized size = 1.6 \[ -\frac{{\left (2 \, b^{3} c - 14 \, a b^{2} d - 65 \, a^{3} f + 35 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{5}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{6}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a b^{6}} - \frac{7 \, b^{4} c x^{4} - 13 \, a b^{3} d x^{4} - 25 \, a^{3} b f x^{4} + 19 \, a^{2} b^{2} x^{4} e + 4 \, a b^{3} c x - 10 \, a^{2} b^{2} d x - 22 \, a^{4} f x + 16 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{5}} + \frac{4 \, b^{18} f x^{7} - 21 \, a b^{17} f x^{4} + 7 \, b^{18} x^{4} e + 28 \, b^{18} d x + 168 \, a^{2} b^{16} f x - 84 \, a b^{17} x e}{28 \, b^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(2*b^3*c - 14*a*b^2*d - 65*a^3*f + 35*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a
/b)^(1/3)))/(a*b^5) + 1/27*sqrt(3)*(2*(-a*b^2)^(1/3)*b^3*c - 14*(-a*b^2)^(1/3)*a
*b^2*d - 65*(-a*b^2)^(1/3)*a^3*f + 35*(-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^6) + 1/54*(2*(-a*b^2)^(1/3)*b^3*c - 14*
(-a*b^2)^(1/3)*a*b^2*d - 65*(-a*b^2)^(1/3)*a^3*f + 35*(-a*b^2)^(1/3)*a^2*b*e)*ln
(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^6) - 1/18*(7*b^4*c*x^4 - 13*a*b^3*d*x
^4 - 25*a^3*b*f*x^4 + 19*a^2*b^2*x^4*e + 4*a*b^3*c*x - 10*a^2*b^2*d*x - 22*a^4*f
*x + 16*a^3*b*x*e)/((b*x^3 + a)^2*b^5) + 1/28*(4*b^18*f*x^7 - 21*a*b^17*f*x^4 +
7*b^18*x^4*e + 28*b^18*d*x + 168*a^2*b^16*f*x - 84*a*b^17*x*e)/b^21