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