3.289 \(\int \frac{x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=345 \[ \frac{x^2 \left (6 a^2 f-3 a b e+b^2 d\right )}{2 b^5}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{27 \sqrt [3]{a} b^{17/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{9 \sqrt{3} \sqrt [3]{a} b^{17/3}}-\frac{x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{9 b^5 \left (a+b x^3\right )}+\frac{a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{54 \sqrt [3]{a} b^{17/3}}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^8}{8 b^3} \]

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x^2)/(2*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^8
)/(8*b^3) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^5*(a + b*x^3)^2) -
((4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(9*b^5*(a + b*x^3)) - ((5*b^
3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3
]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(17/3)) - ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e
- 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(1/3)*b^(17/3)) + ((5*b^3*c - 20*a*b
^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(5
4*a^(1/3)*b^(17/3))

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Rubi [A]  time = 1.52835, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^2 \left (6 a^2 f-3 a b e+b^2 d\right )}{2 b^5}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{27 \sqrt [3]{a} b^{17/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{9 \sqrt{3} \sqrt [3]{a} b^{17/3}}-\frac{x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{9 b^5 \left (a+b x^3\right )}+\frac{a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{54 \sqrt [3]{a} b^{17/3}}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(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^2)/(2*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^8
)/(8*b^3) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*b^5*(a + b*x^3)^2) -
((4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(9*b^5*(a + b*x^3)) - ((5*b^
3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3
]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(17/3)) - ((5*b^3*c - 20*a*b^2*d + 44*a^2*b*e
- 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(1/3)*b^(17/3)) + ((5*b^3*c - 20*a*b
^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(5
4*a^(1/3)*b^(17/3))

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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**7*(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 = 0.415864, size = 329, normalized size = 0.95 \[ \frac{540 b^{2/3} x^2 \left (6 a^2 f-3 a b e+b^2 d\right )+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (77 a^3 f-44 a^2 b e+20 a b^2 d-5 b^3 c\right )}{\sqrt [3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (77 a^3 f-44 a^2 b e+20 a b^2 d-5 b^3 c\right )}{\sqrt [3]{a}}-\frac{120 b^{2/3} x^2 \left (-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right )}{a+b x^3}+\frac{180 a b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right )}{\sqrt [3]{a}}+216 b^{5/3} x^5 (b e-3 a f)+135 b^{8/3} f x^8}{1080 b^{17/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(540*b^(2/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^2 + 216*b^(5/3)*(b*e - 3*a*f)*x^5 + 1
35*b^(8/3)*f*x^8 + (180*a*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a +
b*x^3)^2 - (120*b^(2/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(a +
b*x^3) + (40*Sqrt[3]*(-5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*ArcTan[(1 -
(2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-5*b^3*c + 20*a*b^2*d - 44*a^2*
b*e + 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(5*b^3*c - 20*a*b^2*d +
44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/
(1080*b^(17/3))

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Maple [B]  time = 0.019, size = 611, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

1/8*f*x^8/b^3+5/54/b^3*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/2/b^3*x
^2*d+1/5/b^3*x^5*e-3/5/b^4*x^5*a*f+3/b^5*x^2*a^2*f-3/2/b^4*x^2*a*e-4/9/b/(b*x^3+
a)^2*x^5*c-5/27/b^3*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+44/27/b^5*a^2*e*3^(1/2)/(a/b
)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-20/27/b^4*a*d*3^(1/2)/(a/b)^(1/3
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-77/27/b^6*a^3*f*3^(1/2)/(a/b)^(1/3)*ar
ctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-17/18/b^4/(b*x^3+a)^2*x^2*a^3*e+11/18/b^3/
(b*x^3+a)^2*x^2*a^2*d-5/18/b^2/(b*x^3+a)^2*x^2*a*c+77/27/b^6*a^3*f/(a/b)^(1/3)*l
n(x+(a/b)^(1/3))-77/54/b^6*a^3*f/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4
4/27/b^5*a^2*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+22/27/b^5*a^2*e/(a/b)^(1/3)*ln(x^2-
x*(a/b)^(1/3)+(a/b)^(2/3))+20/27/b^4*a*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-10/27/b^4
*a*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/27/b^3*c*3^(1/2)/(a/b)^(1/3
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9/b^4/(b*x^3+a)^2*x^5*f*a^3-10/9/b^
3/(b*x^3+a)^2*x^5*a^2*e+7/9/b^2/(b*x^3+a)^2*x^5*a*d+23/18/b^5/(b*x^3+a)^2*x^2*a^
4*f

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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^7/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238246, size = 780, normalized size = 2.26 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left ({\left (5 \, b^{5} c - 20 \, a b^{4} d + 44 \, a^{2} b^{3} e - 77 \, a^{3} b^{2} f\right )} x^{6} + 5 \, a^{2} b^{3} c - 20 \, a^{3} b^{2} d + 44 \, a^{4} b e - 77 \, a^{5} f + 2 \,{\left (5 \, a b^{4} c - 20 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 77 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 40 \, \sqrt{3}{\left ({\left (5 \, b^{5} c - 20 \, a b^{4} d + 44 \, a^{2} b^{3} e - 77 \, a^{3} b^{2} f\right )} x^{6} + 5 \, a^{2} b^{3} c - 20 \, a^{3} b^{2} d + 44 \, a^{4} b e - 77 \, a^{5} f + 2 \,{\left (5 \, a b^{4} c - 20 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 77 \, a^{4} b f\right )} x^{3}\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 120 \,{\left ({\left (5 \, b^{5} c - 20 \, a b^{4} d + 44 \, a^{2} b^{3} e - 77 \, a^{3} b^{2} f\right )} x^{6} + 5 \, a^{2} b^{3} c - 20 \, a^{3} b^{2} d + 44 \, a^{4} b e - 77 \, a^{5} f + 2 \,{\left (5 \, a b^{4} c - 20 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 77 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (45 \, b^{4} f x^{14} + 18 \,{\left (4 \, b^{4} e - 7 \, a b^{3} f\right )} x^{11} + 9 \,{\left (20 \, b^{4} d - 44 \, a b^{3} e + 77 \, a^{2} b^{2} f\right )} x^{8} - 32 \,{\left (5 \, b^{4} c - 20 \, a b^{3} d + 44 \, a^{2} b^{2} e - 77 \, a^{3} b f\right )} x^{5} - 20 \,{\left (5 \, a b^{3} c - 20 \, a^{2} b^{2} d + 44 \, a^{3} b e - 77 \, a^{4} f\right )} x^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{3240 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/3240*sqrt(3)*(20*sqrt(3)*((5*b^5*c - 20*a*b^4*d + 44*a^2*b^3*e - 77*a^3*b^2*f)
*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 20*a^
2*b^3*d + 44*a^3*b^2*e - 77*a^4*b*f)*x^3)*log((a*b^2)^(1/3)*b*x^2 + a*b - (a*b^2
)^(2/3)*x) - 40*sqrt(3)*((5*b^5*c - 20*a*b^4*d + 44*a^2*b^3*e - 77*a^3*b^2*f)*x^
6 + 5*a^2*b^3*c - 20*a^3*b^2*d + 44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 20*a^2*b
^3*d + 44*a^3*b^2*e - 77*a^4*b*f)*x^3)*log(a*b + (a*b^2)^(2/3)*x) + 120*((5*b^5*
c - 20*a*b^4*d + 44*a^2*b^3*e - 77*a^3*b^2*f)*x^6 + 5*a^2*b^3*c - 20*a^3*b^2*d +
44*a^4*b*e - 77*a^5*f + 2*(5*a*b^4*c - 20*a^2*b^3*d + 44*a^3*b^2*e - 77*a^4*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)*
(45*b^4*f*x^14 + 18*(4*b^4*e - 7*a*b^3*f)*x^11 + 9*(20*b^4*d - 44*a*b^3*e + 77*a
^2*b^2*f)*x^8 - 32*(5*b^4*c - 20*a*b^3*d + 44*a^2*b^2*e - 77*a^3*b*f)*x^5 - 20*(
5*a*b^3*c - 20*a^2*b^2*d + 44*a^3*b*e - 77*a^4*f)*x^2)*(a*b^2)^(1/3))/((b^7*x^6
+ 2*a*b^6*x^3 + a^2*b^5)*(a*b^2)^(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**7*(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 = 0.221355, size = 601, normalized size = 1.74 \[ -\frac{{\left (5 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 77 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 44 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{5}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{7}} - \frac{8 \, b^{4} c x^{5} - 14 \, a b^{3} d x^{5} - 26 \, a^{3} b f x^{5} + 20 \, a^{2} b^{2} x^{5} e + 5 \, a b^{3} c x^{2} - 11 \, a^{2} b^{2} d x^{2} - 23 \, a^{4} f x^{2} + 17 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{5}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 77 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a b^{7}} + \frac{5 \, b^{21} f x^{8} - 24 \, a b^{20} f x^{5} + 8 \, b^{21} x^{5} e + 20 \, b^{21} d x^{2} + 120 \, a^{2} b^{19} f x^{2} - 60 \, a b^{20} x^{2} e}{40 \, b^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

-1/27*(5*b^3*c*(-a/b)^(1/3) - 20*a*b^2*d*(-a/b)^(1/3) - 77*a^3*f*(-a/b)^(1/3) +
44*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^5) - 1/27*s
qrt(3)*(5*(-a*b^2)^(2/3)*b^3*c - 20*(-a*b^2)^(2/3)*a*b^2*d - 77*(-a*b^2)^(2/3)*a
^3*f + 44*(-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*b^7) - 1/18*(8*b^4*c*x^5 - 14*a*b^3*d*x^5 - 26*a^3*b*f*x^5 + 20*a^2*b
^2*x^5*e + 5*a*b^3*c*x^2 - 11*a^2*b^2*d*x^2 - 23*a^4*f*x^2 + 17*a^3*b*x^2*e)/((b
*x^3 + a)^2*b^5) + 1/54*(5*(-a*b^2)^(2/3)*b^3*c - 20*(-a*b^2)^(2/3)*a*b^2*d - 77
*(-a*b^2)^(2/3)*a^3*f + 44*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a
/b)^(2/3))/(a*b^7) + 1/40*(5*b^21*f*x^8 - 24*a*b^20*f*x^5 + 8*b^21*x^5*e + 20*b^
21*d*x^2 + 120*a^2*b^19*f*x^2 - 60*a*b^20*x^2*e)/b^24