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

Optimal. Leaf size=375 \[ \frac{x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} b^{19/3}}+\frac{a x \left (-31 a^3 f+25 a^2 b e-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac{a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{x \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{b^6}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3} \]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f
)*x^4)/(4*b^5) + ((b*e - 3*a*f)*x^7)/(7*b^4) + (f*x^10)/(10*b^3) - (a^2*(b^3*c -
a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^6*(a + b*x^3)^2) + (a*(13*b^3*c - 19*a*b^2*d
+ 25*a^2*b*e - 31*a^3*f)*x)/(18*b^6*(a + b*x^3)) + (a^(1/3)*(14*b^3*c - 35*a*b^
2*d + 65*a^2*b*e - 104*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*b^(19/3)) - (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65*a^2*b*e - 104*a^3*f
)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(19/3)) + (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65
*a^2*b*e - 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(19/
3))

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Rubi [A]  time = 1.25337, antiderivative size = 375, 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 \[ \frac{x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} b^{19/3}}+\frac{a x \left (-31 a^3 f+25 a^2 b e-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac{a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{x \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{b^6}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f
)*x^4)/(4*b^5) + ((b*e - 3*a*f)*x^7)/(7*b^4) + (f*x^10)/(10*b^3) - (a^2*(b^3*c -
a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^6*(a + b*x^3)^2) + (a*(13*b^3*c - 19*a*b^2*d
+ 25*a^2*b*e - 31*a^3*f)*x)/(18*b^6*(a + b*x^3)) + (a^(1/3)*(14*b^3*c - 35*a*b^
2*d + 65*a^2*b*e - 104*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*b^(19/3)) - (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65*a^2*b*e - 104*a^3*f
)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(19/3)) + (a^(1/3)*(14*b^3*c - 35*a*b^2*d + 65
*a^2*b*e - 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(19/
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**9*(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 = 0.960212, size = 362, normalized size = 0.97 \[ \frac{945 b^{4/3} x^4 \left (6 a^2 f-3 a b e+b^2 d\right )+\frac{210 a \sqrt [3]{b} x \left (-31 a^3 f+25 a^2 b e-19 a b^2 d+13 b^3 c\right )}{a+b x^3}+\frac{630 a^2 \sqrt [3]{b} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+3780 \sqrt [3]{b} x \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (104 a^3 f-65 a^2 b e+35 a b^2 d-14 b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (104 a^3 f-65 a^2 b e+35 a b^2 d-14 b^3 c\right )-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (104 a^3 f-65 a^2 b e+35 a b^2 d-14 b^3 c\right )+540 b^{7/3} x^7 (b e-3 a f)+378 b^{10/3} f x^{10}}{3780 b^{19/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3780*b^(1/3)*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x + 945*b^(4/3)*(b^2*d
- 3*a*b*e + 6*a^2*f)*x^4 + 540*b^(7/3)*(b*e - 3*a*f)*x^7 + 378*b^(10/3)*f*x^10 +
(630*a^2*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3)^2 + (210
*a*b^(1/3)*(13*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(a + b*x^3) - 140*
Sqrt[3]*a^(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*ArcTan[(1 - (2
*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e
+ 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-14*b^3*c + 35*a*b^2*d - 65
*a^2*b*e + 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3780*b^(1
9/3))

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Maple [A]  time = 0.02, size = 651, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

104/27*a^4/b^7*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-65/
27*a^3/b^6*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+35/27*a
^2/b^5*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/27*a/b^4
*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/10*f*x^10/b^3-3
/7/b^4*x^7*a*f+3/2/b^5*x^4*a^2*f-3/4/b^4*x^4*a*e-10/b^6*a^3*f*x+6/b^5*a^2*e*x-3/
b^4*a*d*x-31/18*a^4/b^5/(b*x^3+a)^2*x^4*f+25/18*a^3/b^4/(b*x^3+a)^2*x^4*e-19/18*
a^2/b^3/(b*x^3+a)^2*x^4*d+13/18*a/b^2/(b*x^3+a)^2*x^4*c+65/54*a^3/b^6*e/(a/b)^(2
/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+35/27*a^2/b^5*d/(a/b)^(2/3)*ln(x+(a/b)^(1/
3))-35/54*a^2/b^5*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-14/27*a/b^4*c/
(a/b)^(2/3)*ln(x+(a/b)^(1/3))+7/27*a/b^4*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b
)^(2/3))+1/7/b^3*x^7*e+1/4/b^3*x^4*d+1/b^3*c*x+104/27*a^4/b^7*f/(a/b)^(2/3)*ln(x
+(a/b)^(1/3))-52/27*a^4/b^7*f/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-65/2
7*a^3/b^6*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-14/9*a^5/b^6/(b*x^3+a)^2*f*x+11/9*a^4/
b^5/(b*x^3+a)^2*e*x-8/9*a^3/b^4/(b*x^3+a)^2*d*x+5/9*a^2/b^3/(b*x^3+a)^2*c*x

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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^9/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241863, size = 832, normalized size = 2.22 \[ \frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 140 \, \sqrt{3}{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (126 \, b^{5} f x^{16} + 36 \,{\left (5 \, b^{5} e - 8 \, a b^{4} f\right )} x^{13} + 9 \,{\left (35 \, b^{5} d - 65 \, a b^{4} e + 104 \, a^{2} b^{3} f\right )} x^{10} + 90 \,{\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{7} + 245 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{4} + 140 \,{\left (14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f\right )} x\right )}\right )}}{11340 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11340*sqrt(3)*(70*sqrt(3)*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a^3*b^2
*f)*x^6 + 14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c -
35*a^2*b^3*d + 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(
1/3) + (a/b)^(2/3)) - 140*sqrt(3)*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a
^3*b^2*f)*x^6 + 14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b
^4*c - 35*a^2*b^3*d + 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*log(x + (a/b)
^(1/3)) + 420*((14*b^5*c - 35*a*b^4*d + 65*a^2*b^3*e - 104*a^3*b^2*f)*x^6 + 14*a
^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 104*a^5*f + 2*(14*a*b^4*c - 35*a^2*b^3*d
+ 65*a^3*b^2*e - 104*a^4*b*f)*x^3)*(a/b)^(1/3)*arctan(-1/3*(2*sqrt(3)*x - sqrt(3
)*(a/b)^(1/3))/(a/b)^(1/3)) + 3*sqrt(3)*(126*b^5*f*x^16 + 36*(5*b^5*e - 8*a*b^4*
f)*x^13 + 9*(35*b^5*d - 65*a*b^4*e + 104*a^2*b^3*f)*x^10 + 90*(14*b^5*c - 35*a*b
^4*d + 65*a^2*b^3*e - 104*a^3*b^2*f)*x^7 + 245*(14*a*b^4*c - 35*a^2*b^3*d + 65*a
^3*b^2*e - 104*a^4*b*f)*x^4 + 140*(14*a^2*b^3*c - 35*a^3*b^2*d + 65*a^4*b*e - 10
4*a^5*f)*x))/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6)

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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**9*(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 = 0.217133, size = 598, normalized size = 1.59 \[ -\frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, b^{7}} + \frac{{\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d - 104 \, a^{4} f + 65 \, a^{3} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{6}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, b^{7}} + \frac{13 \, a b^{4} c x^{4} - 19 \, a^{2} b^{3} d x^{4} - 31 \, a^{4} b f x^{4} + 25 \, a^{3} b^{2} x^{4} e + 10 \, a^{2} b^{3} c x - 16 \, a^{3} b^{2} d x - 28 \, a^{5} f x + 22 \, a^{4} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{14 \, b^{27} f x^{10} - 60 \, a b^{26} f x^{7} + 20 \, b^{27} x^{7} e + 35 \, b^{27} d x^{4} + 210 \, a^{2} b^{25} f x^{4} - 105 \, a b^{26} x^{4} e + 140 \, b^{27} c x - 420 \, a b^{26} d x - 1400 \, a^{3} b^{24} f x + 840 \, a^{2} b^{25} x e}{140 \, b^{30}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*sqrt(3)*(14*(-a*b^2)^(1/3)*b^3*c - 35*(-a*b^2)^(1/3)*a*b^2*d - 104*(-a*b^2
)^(1/3)*a^3*f + 65*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3
))/(-a/b)^(1/3))/b^7 + 1/27*(14*a*b^3*c - 35*a^2*b^2*d - 104*a^4*f + 65*a^3*b*e)
*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^6) - 1/54*(14*(-a*b^2)^(1/3)*b^3*c
- 35*(-a*b^2)^(1/3)*a*b^2*d - 104*(-a*b^2)^(1/3)*a^3*f + 65*(-a*b^2)^(1/3)*a^2*b
*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^7 + 1/18*(13*a*b^4*c*x^4 - 19*a^2*
b^3*d*x^4 - 31*a^4*b*f*x^4 + 25*a^3*b^2*x^4*e + 10*a^2*b^3*c*x - 16*a^3*b^2*d*x
- 28*a^5*f*x + 22*a^4*b*x*e)/((b*x^3 + a)^2*b^6) + 1/140*(14*b^27*f*x^10 - 60*a*
b^26*f*x^7 + 20*b^27*x^7*e + 35*b^27*d*x^4 + 210*a^2*b^25*f*x^4 - 105*a*b^26*x^4
*e + 140*b^27*c*x - 420*a*b^26*d*x - 1400*a^3*b^24*f*x + 840*a^2*b^25*x*e)/b^30