3.204 \(\int x^5 \left (a+b x^3+c x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=150 \[ -\frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{256 c^{7/2}}+\frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]

[Out]

(b*(b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(128*c^3) - (b*(b + 2*c*
x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*c^2) + (a + b*x^3 + c*x^6)^(5/2)/(15*c) - (b
*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(25
6*c^(7/2))

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Rubi [A]  time = 0.231477, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{256 c^{7/2}}+\frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(b*(b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(128*c^3) - (b*(b + 2*c*
x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*c^2) + (a + b*x^3 + c*x^6)^(5/2)/(15*c) - (b
*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(25
6*c^(7/2))

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Rubi in Sympy [A]  time = 23.6512, size = 138, normalized size = 0.92 \[ - \frac{b \left (b + 2 c x^{3}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{48 c^{2}} + \frac{b \left (b + 2 c x^{3}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{128 c^{3}} - \frac{b \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{256 c^{\frac{7}{2}}} + \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{15 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

-b*(b + 2*c*x**3)*(a + b*x**3 + c*x**6)**(3/2)/(48*c**2) + b*(b + 2*c*x**3)*(-4*
a*c + b**2)*sqrt(a + b*x**3 + c*x**6)/(128*c**3) - b*(-4*a*c + b**2)**2*atanh((b
+ 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(256*c**(7/2)) + (a + b*x**3
+ c*x**6)**(5/2)/(15*c)

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Mathematica [A]  time = 0.155749, size = 142, normalized size = 0.95 \[ \frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (4 b^2 c \left (2 c x^6-25 a\right )+8 b c^2 x^3 \left (7 a+22 c x^6\right )+128 c^2 \left (a+c x^6\right )^2+15 b^4-10 b^3 c x^3\right )-15 b \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{3840 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(15*b^4 - 10*b^3*c*x^3 + 128*c^2*(a + c*x^6)^
2 + 4*b^2*c*(-25*a + 2*c*x^6) + 8*b*c^2*x^3*(7*a + 22*c*x^6)) - 15*b*(b^2 - 4*a*
c)^2*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(3840*c^(7/2))

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^5*(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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305644, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (128 \, c^{4} x^{12} + 176 \, b c^{3} x^{9} + 8 \,{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} x^{6} + 15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2} - 2 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{7680 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (128 \, c^{4} x^{12} + 176 \, b c^{3} x^{9} + 8 \,{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} x^{6} + 15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2} - 2 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{3840 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^5,x, algorithm="fricas")

[Out]

[1/7680*(4*(128*c^4*x^12 + 176*b*c^3*x^9 + 8*(b^2*c^2 + 32*a*c^3)*x^6 + 15*b^4 -
100*a*b^2*c + 128*a^2*c^2 - 2*(5*b^3*c - 28*a*b*c^2)*x^3)*sqrt(c*x^6 + b*x^3 +
a)*sqrt(c) + 15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*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^(7/2), 1/38
40*(2*(128*c^4*x^12 + 176*b*c^3*x^9 + 8*(b^2*c^2 + 32*a*c^3)*x^6 + 15*b^4 - 100*
a*b^2*c + 128*a^2*c^2 - 2*(5*b^3*c - 28*a*b*c^2)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sq
rt(-c) - 15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(
sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**5*(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [A]  time = 0.294707, size = 232, normalized size = 1.55 \[ \frac{1}{1920} \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x^{3} + 11 \, b\right )} x^{3} + \frac{b^{2} c^{3} + 32 \, a c^{4}}{c^{4}}\right )} x^{3} - \frac{5 \, b^{3} c^{2} - 28 \, a b c^{3}}{c^{4}}\right )} x^{3} + \frac{15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}}{c^{4}}\right )} + \frac{{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^5,x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^6 + b*x^3 + a)*(2*(4*(2*(8*c*x^3 + 11*b)*x^3 + (b^2*c^3 + 32*a*c
^4)/c^4)*x^3 - (5*b^3*c^2 - 28*a*b*c^3)/c^4)*x^3 + (15*b^4*c - 100*a*b^2*c^2 + 1
28*a^2*c^3)/c^4) + 1/256*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*ln(abs(-2*(sqrt(c)*x^3
- sqrt(c*x^6 + b*x^3 + a))*sqrt(c) - b))/c^(7/2)