3.203 \(\int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=204 \[ \frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{9/2}}-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c} \]

[Out]

-((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*c^4
) + ((7*b^2 - 4*a*c)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*c^3) - (7*b*(
a + b*x^3 + c*x^6)^(5/2))/(180*c^2) + (x^3*(a + b*x^3 + c*x^6)^(5/2))/(18*c) + (
(b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3
+ c*x^6])])/(3072*c^(9/2))

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Rubi [A]  time = 0.363897, antiderivative size = 204, 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 \[ \frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{9/2}}-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac{x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

-((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*c^4
) + ((7*b^2 - 4*a*c)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*c^3) - (7*b*(
a + b*x^3 + c*x^6)^(5/2))/(180*c^2) + (x^3*(a + b*x^3 + c*x^6)^(5/2))/(18*c) + (
(b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3
+ c*x^6])])/(3072*c^(9/2))

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Rubi in Sympy [A]  time = 39.2812, size = 190, normalized size = 0.93 \[ - \frac{7 b \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{180 c^{2}} + \frac{x^{3} \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{18 c} + \frac{\left (b + 2 c x^{3}\right ) \left (- 4 a c + 7 b^{2}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{576 c^{3}} - \frac{\left (b + 2 c x^{3}\right ) \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 7 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{1536 c^{4}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c + 7 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3072 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

-7*b*(a + b*x**3 + c*x**6)**(5/2)/(180*c**2) + x**3*(a + b*x**3 + c*x**6)**(5/2)
/(18*c) + (b + 2*c*x**3)*(-4*a*c + 7*b**2)*(a + b*x**3 + c*x**6)**(3/2)/(576*c**
3) - (b + 2*c*x**3)*(-4*a*c + b**2)*(-4*a*c + 7*b**2)*sqrt(a + b*x**3 + c*x**6)/
(1536*c**4) + (-4*a*c + b**2)**2*(-4*a*c + 7*b**2)*atanh((b + 2*c*x**3)/(2*sqrt(
c)*sqrt(a + b*x**3 + c*x**6)))/(3072*c**(9/2))

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Mathematica [A]  time = 0.194568, size = 194, normalized size = 0.95 \[ \frac{15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )-2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (-16 b c^2 \left (-81 a^2+18 a c x^6+104 c^2 x^{12}\right )-160 c^3 x^3 \left (3 a^2+14 a c x^6+8 c^2 x^{12}\right )+8 b^3 c \left (7 c x^6-95 a\right )-48 b^2 c^2 x^3 \left (c x^6-9 a\right )+105 b^5-70 b^4 c x^3\right )}{46080 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(105*b^5 - 70*b^4*c*x^3 - 48*b^2*c^2*x^3*(-9
*a + c*x^6) + 8*b^3*c*(-95*a + 7*c*x^6) - 160*c^3*x^3*(3*a^2 + 14*a*c*x^6 + 8*c^
2*x^12) - 16*b*c^2*(-81*a^2 + 18*a*c*x^6 + 104*c^2*x^12)) + 15*(b^2 - 4*a*c)^2*(
7*b^2 - 4*a*c)*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(46080*c^(9
/2))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{x}^{8} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^8*(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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310548, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (1280 \, c^{5} x^{15} + 1664 \, b c^{4} x^{12} + 16 \,{\left (3 \, b^{2} c^{3} + 140 \, a c^{4}\right )} x^{9} - 8 \,{\left (7 \, b^{3} c^{2} - 36 \, a b c^{3}\right )} x^{6} - 105 \, b^{5} + 760 \, a b^{3} c - 1296 \, a^{2} b c^{2} + 2 \,{\left (35 \, b^{4} c - 216 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} - 15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{92160 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (1280 \, c^{5} x^{15} + 1664 \, b c^{4} x^{12} + 16 \,{\left (3 \, b^{2} c^{3} + 140 \, a c^{4}\right )} x^{9} - 8 \,{\left (7 \, b^{3} c^{2} - 36 \, a b c^{3}\right )} x^{6} - 105 \, b^{5} + 760 \, a b^{3} c - 1296 \, a^{2} b c^{2} + 2 \,{\left (35 \, b^{4} c - 216 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{46080 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^8,x, algorithm="fricas")

[Out]

[1/92160*(4*(1280*c^5*x^15 + 1664*b*c^4*x^12 + 16*(3*b^2*c^3 + 140*a*c^4)*x^9 -
8*(7*b^3*c^2 - 36*a*b*c^3)*x^6 - 105*b^5 + 760*a*b^3*c - 1296*a^2*b*c^2 + 2*(35*
b^4*c - 216*a*b^2*c^2 + 240*a^2*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) - 15*(
7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*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^(9/2), 1/
46080*(2*(1280*c^5*x^15 + 1664*b*c^4*x^12 + 16*(3*b^2*c^3 + 140*a*c^4)*x^9 - 8*(
7*b^3*c^2 - 36*a*b*c^3)*x^6 - 105*b^5 + 760*a*b^3*c - 1296*a^2*b*c^2 + 2*(35*b^4
*c - 216*a*b^2*c^2 + 240*a^2*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) + 15*(7*
b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*arctan(1/2*(2*c*x^3 + b)*sqrt(-
c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{8} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**8*(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^8,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)*x^8, x)