3.202 \(\int x^{11} \left (a+b x^3+c x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=223 \[ -\frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{2048 c^{11/2}}+\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{\left (-16 a c+21 b^2-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \]

[Out]

(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1024*c^
5) - (b*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*c^4) + (x^
6*(a + b*x^3 + c*x^6)^(5/2))/(21*c) + ((21*b^2 - 16*a*c - 30*b*c*x^3)*(a + b*x^3
+ c*x^6)^(5/2))/(840*c^3) - (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(b + 2*c
*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(2048*c^(11/2))

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Rubi [A]  time = 0.43652, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{2048 c^{11/2}}+\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac{\left (-16 a c+21 b^2-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}+\frac{x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1024*c^
5) - (b*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*c^4) + (x^
6*(a + b*x^3 + c*x^6)^(5/2))/(21*c) + ((21*b^2 - 16*a*c - 30*b*c*x^3)*(a + b*x^3
+ c*x^6)^(5/2))/(840*c^3) - (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(b + 2*c
*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(2048*c^(11/2))

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Rubi in Sympy [A]  time = 41.3591, size = 214, normalized size = 0.96 \[ - \frac{b \left (b + 2 c x^{3}\right ) \left (- 4 a c + 3 b^{2}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{384 c^{4}} + \frac{b \left (b + 2 c x^{3}\right ) \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 3 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{1024 c^{5}} - \frac{b \left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{2048 c^{\frac{11}{2}}} + \frac{x^{6} \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{21 c} + \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}} \left (- 12 a c + \frac{63 b^{2}}{4} - \frac{45 b c x^{3}}{2}\right )}{630 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

-b*(b + 2*c*x**3)*(-4*a*c + 3*b**2)*(a + b*x**3 + c*x**6)**(3/2)/(384*c**4) + b*
(b + 2*c*x**3)*(-4*a*c + b**2)*(-4*a*c + 3*b**2)*sqrt(a + b*x**3 + c*x**6)/(1024
*c**5) - b*(-4*a*c + b**2)**2*(-4*a*c + 3*b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*
sqrt(a + b*x**3 + c*x**6)))/(2048*c**(11/2)) + x**6*(a + b*x**3 + c*x**6)**(5/2)
/(21*c) + (a + b*x**3 + c*x**6)**(5/2)*(-12*a*c + 63*b**2/4 - 45*b*c*x**3/2)/(63
0*c**3)

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Mathematica [A]  time = 0.23929, size = 222, normalized size = 1. \[ \frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (16 b^2 c^2 \left (343 a^2-62 a c x^6+8 c^2 x^{12}\right )+32 b c^3 x^3 \left (-73 a^2+22 a c x^6+200 c^2 x^{12}\right )+168 b^4 c \left (c x^6-15 a\right )+16 b^3 c^2 x^3 \left (91 a-9 c x^6\right )+1024 c^3 \left (a+c x^6\right )^2 \left (5 c x^6-2 a\right )+315 b^6-210 b^5 c x^3\right )-105 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{215040 c^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(315*b^6 - 210*b^5*c*x^3 + 16*b^3*c^2*x^3*(91
*a - 9*c*x^6) + 168*b^4*c*(-15*a + c*x^6) + 1024*c^3*(a + c*x^6)^2*(-2*a + 5*c*x
^6) + 16*b^2*c^2*(343*a^2 - 62*a*c*x^6 + 8*c^2*x^12) + 32*b*c^3*x^3*(-73*a^2 + 2
2*a*c*x^6 + 200*c^2*x^12)) - 105*b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*Log[b + 2*c*x
^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(215040*c^(11/2))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{x}^{11} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^11*(c*x^6+b*x^3+a)^(3/2),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((c*x^6 + b*x^3 + a)^(3/2)*x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.311414, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (5120 \, c^{6} x^{18} + 6400 \, b c^{5} x^{15} + 128 \,{\left (b^{2} c^{4} + 64 \, a c^{5}\right )} x^{12} - 16 \,{\left (9 \, b^{3} c^{3} - 44 \, a b c^{4}\right )} x^{9} + 8 \,{\left (21 \, b^{4} c^{2} - 124 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} x^{6} + 315 \, b^{6} - 2520 \, a b^{4} c + 5488 \, a^{2} b^{2} c^{2} - 2048 \, a^{3} c^{3} - 2 \,{\left (105 \, b^{5} c - 728 \, a b^{3} c^{2} + 1168 \, a^{2} b c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} - 105 \,{\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{430080 \, c^{\frac{11}{2}}}, \frac{2 \,{\left (5120 \, c^{6} x^{18} + 6400 \, b c^{5} x^{15} + 128 \,{\left (b^{2} c^{4} + 64 \, a c^{5}\right )} x^{12} - 16 \,{\left (9 \, b^{3} c^{3} - 44 \, a b c^{4}\right )} x^{9} + 8 \,{\left (21 \, b^{4} c^{2} - 124 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} x^{6} + 315 \, b^{6} - 2520 \, a b^{4} c + 5488 \, a^{2} b^{2} c^{2} - 2048 \, a^{3} c^{3} - 2 \,{\left (105 \, b^{5} c - 728 \, a b^{3} c^{2} + 1168 \, a^{2} b c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 105 \,{\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{215040 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^11,x, algorithm="fricas")

[Out]

[1/430080*(4*(5120*c^6*x^18 + 6400*b*c^5*x^15 + 128*(b^2*c^4 + 64*a*c^5)*x^12 -
16*(9*b^3*c^3 - 44*a*b*c^4)*x^9 + 8*(21*b^4*c^2 - 124*a*b^2*c^3 + 128*a^2*c^4)*x
^6 + 315*b^6 - 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^3 - 2*(105*b^5*c - 7
28*a*b^3*c^2 + 1168*a^2*b*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) - 105*(3*b^7
- 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*log(-4*sqrt(c*x^6 + b*x^3 + a)*(2
*c^2*x^3 + b*c) - (8*c^2*x^6 + 8*b*c*x^3 + b^2 + 4*a*c)*sqrt(c)))/c^(11/2), 1/21
5040*(2*(5120*c^6*x^18 + 6400*b*c^5*x^15 + 128*(b^2*c^4 + 64*a*c^5)*x^12 - 16*(9
*b^3*c^3 - 44*a*b*c^4)*x^9 + 8*(21*b^4*c^2 - 124*a*b^2*c^3 + 128*a^2*c^4)*x^6 +
315*b^6 - 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^3 - 2*(105*b^5*c - 728*a*
b^3*c^2 + 1168*a^2*b*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) - 105*(3*b^7 - 2
8*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sq
rt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{11} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**11*(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{11}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^11,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)*x^11, x)