3.286 \(\int \frac{x^{12} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=416 \[ \frac{x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}-\frac{a^2 x \left (-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac{a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a x \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{b^7}+\frac{x^4 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{4 b^6}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{54 b^{22/3}}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{27 b^{22/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} b^{22/3}}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3} \]

[Out]

-((a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x)/b^7) + ((b^3*c - 3*a*b^2*d
+ 6*a^2*b*e - 10*a^3*f)*x^4)/(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5
) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3*(b^3*c - a*b^2*d +
a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) - (a^2*(19*b^3*c - 25*a*b^2*d + 31*a^2
*b*e - 37*a^3*f)*x)/(18*b^7*(a + b*x^3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104
*a^2*b*e - 152*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt
[3]*b^(22/3)) + (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*Log[a
^(1/3) + b^(1/3)*x])/(27*b^(22/3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b
*e - 152*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))

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Rubi [A]  time = 1.4612, antiderivative size = 416, 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 \[ \frac{x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}-\frac{a^2 x \left (-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac{a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a x \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{b^7}+\frac{x^4 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{4 b^6}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{54 b^{22/3}}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{27 b^{22/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} b^{22/3}}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

-((a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x)/b^7) + ((b^3*c - 3*a*b^2*d
+ 6*a^2*b*e - 10*a^3*f)*x^4)/(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5
) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3*(b^3*c - a*b^2*d +
a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) - (a^2*(19*b^3*c - 25*a*b^2*d + 31*a^2
*b*e - 37*a^3*f)*x)/(18*b^7*(a + b*x^3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104
*a^2*b*e - 152*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt
[3]*b^(22/3)) + (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b*e - 152*a^3*f)*Log[a
^(1/3) + b^(1/3)*x])/(27*b^(22/3)) - (a^(4/3)*(35*b^3*c - 65*a*b^2*d + 104*a^2*b
*e - 152*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))

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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**12*(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 = 0.973115, size = 411, normalized size = 0.99 \[ \frac{x^7 \left (6 a^2 f-3 a b e+b^2 d\right )}{7 b^5}+\frac{a^2 x \left (37 a^3 f-31 a^2 b e+25 a b^2 d-19 b^3 c\right )}{18 b^7 \left (a+b x^3\right )}+\frac{a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}+\frac{a x \left (15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right )}{b^7}+\frac{x^4 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{4 b^6}+\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right )}{54 b^{22/3}}-\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right )}{27 b^{22/3}}+\frac{a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right )}{9 \sqrt{3} b^{22/3}}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b^2*d +
6*a^2*b*e - 10*a^3*f)*x^4)/(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5)
+ ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3*(b^3*c - a*b^2*d + a^
2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) + (a^2*(-19*b^3*c + 25*a*b^2*d - 31*a^2*
b*e + 37*a^3*f)*x)/(18*b^7*(a + b*x^3)) + (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104
*a^2*b*e + 152*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^
(22/3)) - (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*Log[a^(1/3
) + b^(1/3)*x])/(27*b^(22/3)) + (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e +
152*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))

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Maple [A]  time = 0.021, size = 706, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

35/27*a^2/b^5*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-152/
27*a^5/b^8*f/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+104/27*
a^4/b^7*e/(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*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/9*a^5/b^6/(
b*x^3+a)^2*e*x+11/9*a^4/b^5/(b*x^3+a)^2*d*x+6/7/b^5*x^7*a^2*f-3/7/b^4*x^7*a*e-5/
2/b^6*x^4*a^3*f+3/2/b^5*x^4*a^2*e-3/4/b^4*x^4*a*d+15/b^7*a^4*f*x-10/b^6*a^3*e*x+
6/b^5*a^2*d*x-3/b^4*a*c*x+1/13*f*x^13/b^3-3/10/b^4*x^10*a*f+37/18*a^5/b^6/(b*x^3
+a)^2*x^4*f-31/18*a^4/b^5/(b*x^3+a)^2*x^4*e+25/18*a^3/b^4/(b*x^3+a)^2*x^4*d-19/1
8*a^2/b^3/(b*x^3+a)^2*x^4*c+104/27*a^4/b^7*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-8/9*a
^3/b^4/(b*x^3+a)^2*c*x-152/27*a^5/b^8*f/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-52/27*a^4/
b^7*e/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-65/27*a^3/b^6*d/(a/b)^(2/3)*
ln(x+(a/b)^(1/3))+65/54*a^3/b^6*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+
35/27*a^2/b^5*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+76/27*a^5/b^8*f/(a/b)^(2/3)*ln(x^2
-x*(a/b)^(1/3)+(a/b)^(2/3))+1/7/b^3*x^7*d+1/4/b^3*x^4*c+1/10/b^3*x^10*e-35/54*a^
2/b^5*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+17/9*a^6/b^7/(b*x^3+a)^2*f
*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^12/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245609, size = 919, normalized size = 2.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^12/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/147420*sqrt(3)*(910*sqrt(3)*(35*a^3*b^3*c - 65*a^4*b^2*d + 104*a^5*b*e - 152*a
^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b^2*f)*x^6 + 2*(35*a
^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^3)*(-a/b)^(1/3)*log(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*sqrt(3)*(35*a^3*b^3*c - 65*a^4*b^2*d +
104*a^5*b*e - 152*a^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b
^2*f)*x^6 + 2*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^3)*(
-a/b)^(1/3)*log(x - (-a/b)^(1/3)) + 5460*(35*a^3*b^3*c - 65*a^4*b^2*d + 104*a^5*
b*e - 152*a^6*f + (35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b^2*f)*x^
6 + 2*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*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)*(1
260*b^6*f*x^19 + 126*(13*b^6*e - 19*a*b^5*f)*x^16 + 36*(65*b^6*d - 104*a*b^5*e +
152*a^2*b^4*f)*x^13 + 117*(35*b^6*c - 65*a*b^5*d + 104*a^2*b^4*e - 152*a^3*b^3*
f)*x^10 - 1170*(35*a*b^5*c - 65*a^2*b^4*d + 104*a^3*b^3*e - 152*a^4*b^2*f)*x^7 -
3185*(35*a^2*b^4*c - 65*a^3*b^3*d + 104*a^4*b^2*e - 152*a^5*b*f)*x^4 - 1820*(35
*a^3*b^3*c - 65*a^4*b^2*d + 104*a^5*b*e - 152*a^6*f)*x))/(b^9*x^6 + 2*a*b^8*x^3
+ a^2*b^7)

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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**12*(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 = 0.217243, size = 675, normalized size = 1.62 \[ \frac{\sqrt{3}{\left (35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 152 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, b^{8}} - \frac{{\left (35 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d - 152 \, a^{5} f + 104 \, a^{4} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{7}} + \frac{{\left (35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 152 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, b^{8}} - \frac{19 \, a^{2} b^{4} c x^{4} - 25 \, a^{3} b^{3} d x^{4} - 37 \, a^{5} b f x^{4} + 31 \, a^{4} b^{2} x^{4} e + 16 \, a^{3} b^{3} c x - 22 \, a^{4} b^{2} d x - 34 \, a^{6} f x + 28 \, a^{5} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{7}} + \frac{140 \, b^{36} f x^{13} - 546 \, a b^{35} f x^{10} + 182 \, b^{36} x^{10} e + 260 \, b^{36} d x^{7} + 1560 \, a^{2} b^{34} f x^{7} - 780 \, a b^{35} x^{7} e + 455 \, b^{36} c x^{4} - 1365 \, a b^{35} d x^{4} - 4550 \, a^{3} b^{33} f x^{4} + 2730 \, a^{2} b^{34} x^{4} e - 5460 \, a b^{35} c x + 10920 \, a^{2} b^{34} d x + 27300 \, a^{4} b^{32} f x - 18200 \, a^{3} b^{33} x e}{1820 \, b^{39}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^12/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

1/27*sqrt(3)*(35*(-a*b^2)^(1/3)*a*b^3*c - 65*(-a*b^2)^(1/3)*a^2*b^2*d - 152*(-a*
b^2)^(1/3)*a^4*f + 104*(-a*b^2)^(1/3)*a^3*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^
(1/3))/(-a/b)^(1/3))/b^8 - 1/27*(35*a^2*b^3*c - 65*a^3*b^2*d - 152*a^5*f + 104*a
^4*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/54*(35*(-a*b^2)^(1/3)
*a*b^3*c - 65*(-a*b^2)^(1/3)*a^2*b^2*d - 152*(-a*b^2)^(1/3)*a^4*f + 104*(-a*b^2)
^(1/3)*a^3*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^8 - 1/18*(19*a^2*b^4*c
*x^4 - 25*a^3*b^3*d*x^4 - 37*a^5*b*f*x^4 + 31*a^4*b^2*x^4*e + 16*a^3*b^3*c*x - 2
2*a^4*b^2*d*x - 34*a^6*f*x + 28*a^5*b*x*e)/((b*x^3 + a)^2*b^7) + 1/1820*(140*b^3
6*f*x^13 - 546*a*b^35*f*x^10 + 182*b^36*x^10*e + 260*b^36*d*x^7 + 1560*a^2*b^34*
f*x^7 - 780*a*b^35*x^7*e + 455*b^36*c*x^4 - 1365*a*b^35*d*x^4 - 4550*a^3*b^33*f*
x^4 + 2730*a^2*b^34*x^4*e - 5460*a*b^35*c*x + 10920*a^2*b^34*d*x + 27300*a^4*b^3
2*f*x - 18200*a^3*b^33*x*e)/b^39