3.201 \(\int x^{14} \left (a+b x^3+c x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=293 \[ \frac{\left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{32768 c^{13/2}}-\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{16384 c^6}+\frac{\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c x^3 \left (33 b^2-28 a c\right )\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c} \]

[Out]

-((b^2 - 4*a*c)*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*Sqrt[a + b*x^3
+ c*x^6])/(16384*c^6) + ((33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*(a + b
*x^3 + c*x^6)^(3/2))/(6144*c^5) - (11*b*x^6*(a + b*x^3 + c*x^6)^(5/2))/(336*c^2)
+ (x^9*(a + b*x^3 + c*x^6)^(5/2))/(24*c) - ((3*b*(77*b^2 - 124*a*c) - 10*c*(33*
b^2 - 28*a*c)*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(13440*c^4) + ((b^2 - 4*a*c)^2*(33
*b^4 - 72*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3
+ c*x^6])])/(32768*c^(13/2))

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Rubi [A]  time = 0.799884, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{32768 c^{13/2}}-\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{16384 c^6}+\frac{\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c x^3 \left (33 b^2-28 a c\right )\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c} \]

Antiderivative was successfully verified.

[In]  Int[x^14*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

-((b^2 - 4*a*c)*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*Sqrt[a + b*x^3
+ c*x^6])/(16384*c^6) + ((33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*(a + b
*x^3 + c*x^6)^(3/2))/(6144*c^5) - (11*b*x^6*(a + b*x^3 + c*x^6)^(5/2))/(336*c^2)
+ (x^9*(a + b*x^3 + c*x^6)^(5/2))/(24*c) - ((3*b*(77*b^2 - 124*a*c) - 10*c*(33*
b^2 - 28*a*c)*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(13440*c^4) + ((b^2 - 4*a*c)^2*(33
*b^4 - 72*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3
+ c*x^6])])/(32768*c^(13/2))

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Rubi in Sympy [A]  time = 67.2398, size = 284, normalized size = 0.97 \[ - \frac{11 b x^{6} \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{336 c^{2}} + \frac{x^{9} \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{24 c} - \frac{\left (\frac{9 b \left (- 124 a c + 77 b^{2}\right )}{8} - \frac{15 c x^{3} \left (- 28 a c + 33 b^{2}\right )}{4}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{5040 c^{4}} + \frac{\left (b + 2 c x^{3}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}} \left (16 a^{2} c^{2} - 72 a b^{2} c + 33 b^{4}\right )}{6144 c^{5}} - \frac{\left (b + 2 c x^{3}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}} \left (16 a^{2} c^{2} - 72 a b^{2} c + 33 b^{4}\right )}{16384 c^{6}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (16 a^{2} c^{2} - 72 a b^{2} c + 33 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{32768 c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

-11*b*x**6*(a + b*x**3 + c*x**6)**(5/2)/(336*c**2) + x**9*(a + b*x**3 + c*x**6)*
*(5/2)/(24*c) - (9*b*(-124*a*c + 77*b**2)/8 - 15*c*x**3*(-28*a*c + 33*b**2)/4)*(
a + b*x**3 + c*x**6)**(5/2)/(5040*c**4) + (b + 2*c*x**3)*(a + b*x**3 + c*x**6)**
(3/2)*(16*a**2*c**2 - 72*a*b**2*c + 33*b**4)/(6144*c**5) - (b + 2*c*x**3)*(-4*a*
c + b**2)*sqrt(a + b*x**3 + c*x**6)*(16*a**2*c**2 - 72*a*b**2*c + 33*b**4)/(1638
4*c**6) + (-4*a*c + b**2)**2*(16*a**2*c**2 - 72*a*b**2*c + 33*b**4)*atanh((b + 2
*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(32768*c**(13/2))

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Mathematica [A]  time = 0.313281, size = 289, normalized size = 0.99 \[ \frac{105 \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )-2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (16 b^3 c^2 \left (5103 a^2-780 a c x^6+88 c^2 x^{12}\right )-32 b^2 c^3 x^3 \left (1181 a^2-284 a c x^6+40 c^2 x^{12}\right )-64 b c^3 \left (919 a^3-302 a^2 c x^6+104 a c^2 x^{12}+1360 c^3 x^{18}\right )-4480 c^4 x^3 \left (-3 a^3+2 a^2 c x^6+24 a c^2 x^{12}+16 c^3 x^{18}\right )+84 b^5 c \left (22 c x^6-365 a\right )+24 b^4 c^2 x^3 \left (749 a-66 c x^6\right )+3465 b^7-2310 b^6 c x^3\right )}{3440640 c^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(3465*b^7 - 2310*b^6*c*x^3 + 24*b^4*c^2*x^3*
(749*a - 66*c*x^6) + 84*b^5*c*(-365*a + 22*c*x^6) - 32*b^2*c^3*x^3*(1181*a^2 - 2
84*a*c*x^6 + 40*c^2*x^12) + 16*b^3*c^2*(5103*a^2 - 780*a*c*x^6 + 88*c^2*x^12) -
4480*c^4*x^3*(-3*a^3 + 2*a^2*c*x^6 + 24*a*c^2*x^12 + 16*c^3*x^18) - 64*b*c^3*(91
9*a^3 - 302*a^2*c*x^6 + 104*a*c^2*x^12 + 1360*c^3*x^18)) + 105*(b^2 - 4*a*c)^2*(
33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c
*x^6]])/(3440640*c^(13/2))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{x}^{14} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^14*(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^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.334089, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (71680 \, c^{7} x^{21} + 87040 \, b c^{6} x^{18} + 1280 \,{\left (b^{2} c^{5} + 84 \, a c^{6}\right )} x^{15} - 128 \,{\left (11 \, b^{3} c^{4} - 52 \, a b c^{5}\right )} x^{12} + 16 \,{\left (99 \, b^{4} c^{3} - 568 \, a b^{2} c^{4} + 560 \, a^{2} c^{5}\right )} x^{9} - 3465 \, b^{7} + 30660 \, a b^{5} c - 81648 \, a^{2} b^{3} c^{2} + 58816 \, a^{3} b c^{3} - 8 \,{\left (231 \, b^{5} c^{2} - 1560 \, a b^{3} c^{3} + 2416 \, a^{2} b c^{4}\right )} x^{6} + 2 \,{\left (1155 \, b^{6} c - 8988 \, a b^{4} c^{2} + 18896 \, a^{2} b^{2} c^{3} - 6720 \, a^{3} c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{6881280 \, c^{\frac{13}{2}}}, \frac{2 \,{\left (71680 \, c^{7} x^{21} + 87040 \, b c^{6} x^{18} + 1280 \,{\left (b^{2} c^{5} + 84 \, a c^{6}\right )} x^{15} - 128 \,{\left (11 \, b^{3} c^{4} - 52 \, a b c^{5}\right )} x^{12} + 16 \,{\left (99 \, b^{4} c^{3} - 568 \, a b^{2} c^{4} + 560 \, a^{2} c^{5}\right )} x^{9} - 3465 \, b^{7} + 30660 \, a b^{5} c - 81648 \, a^{2} b^{3} c^{2} + 58816 \, a^{3} b c^{3} - 8 \,{\left (231 \, b^{5} c^{2} - 1560 \, a b^{3} c^{3} + 2416 \, a^{2} b c^{4}\right )} x^{6} + 2 \,{\left (1155 \, b^{6} c - 8988 \, a b^{4} c^{2} + 18896 \, a^{2} b^{2} c^{3} - 6720 \, a^{3} c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{3440640 \, \sqrt{-c} c^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^14,x, algorithm="fricas")

[Out]

[1/6881280*(4*(71680*c^7*x^21 + 87040*b*c^6*x^18 + 1280*(b^2*c^5 + 84*a*c^6)*x^1
5 - 128*(11*b^3*c^4 - 52*a*b*c^5)*x^12 + 16*(99*b^4*c^3 - 568*a*b^2*c^4 + 560*a^
2*c^5)*x^9 - 3465*b^7 + 30660*a*b^5*c - 81648*a^2*b^3*c^2 + 58816*a^3*b*c^3 - 8*
(231*b^5*c^2 - 1560*a*b^3*c^3 + 2416*a^2*b*c^4)*x^6 + 2*(1155*b^6*c - 8988*a*b^4
*c^2 + 18896*a^2*b^2*c^3 - 6720*a^3*c^4)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) +
105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*l
og(-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^(13/2), 1/3440640*(2*(71680*c^7*x^21 + 87040*b*c^6*x^18 + 12
80*(b^2*c^5 + 84*a*c^6)*x^15 - 128*(11*b^3*c^4 - 52*a*b*c^5)*x^12 + 16*(99*b^4*c
^3 - 568*a*b^2*c^4 + 560*a^2*c^5)*x^9 - 3465*b^7 + 30660*a*b^5*c - 81648*a^2*b^3
*c^2 + 58816*a^3*b*c^3 - 8*(231*b^5*c^2 - 1560*a*b^3*c^3 + 2416*a^2*b*c^4)*x^6 +
2*(1155*b^6*c - 8988*a*b^4*c^2 + 18896*a^2*b^2*c^3 - 6720*a^3*c^4)*x^3)*sqrt(c*
x^6 + b*x^3 + a)*sqrt(-c) + 105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*
a^3*b^2*c^3 + 256*a^4*c^4)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3
+ a)*c)))/(sqrt(-c)*c^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{14} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**14*(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{14}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^14,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)*x^14, x)