3.787 \(\int x^5 \left (a+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{32 c^{3/2}}+\frac{a^2 x^2 \sqrt{a+c x^4}}{32 c}+\frac{1}{12} x^6 \left (a+c x^4\right )^{3/2}+\frac{1}{16} a x^6 \sqrt{a+c x^4} \]

[Out]

(a^2*x^2*Sqrt[a + c*x^4])/(32*c) + (a*x^6*Sqrt[a + c*x^4])/16 + (x^6*(a + c*x^4)
^(3/2))/12 - (a^3*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(32*c^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.14832, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{32 c^{3/2}}+\frac{a^2 x^2 \sqrt{a+c x^4}}{32 c}+\frac{1}{12} x^6 \left (a+c x^4\right )^{3/2}+\frac{1}{16} a x^6 \sqrt{a+c x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + c*x^4)^(3/2),x]

[Out]

(a^2*x^2*Sqrt[a + c*x^4])/(32*c) + (a*x^6*Sqrt[a + c*x^4])/16 + (x^6*(a + c*x^4)
^(3/2))/12 - (a^3*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(32*c^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.3602, size = 82, normalized size = 0.86 \[ - \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{32 c^{\frac{3}{2}}} + \frac{a^{2} x^{2} \sqrt{a + c x^{4}}}{32 c} + \frac{a x^{6} \sqrt{a + c x^{4}}}{16} + \frac{x^{6} \left (a + c x^{4}\right )^{\frac{3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(c*x**4+a)**(3/2),x)

[Out]

-a**3*atanh(sqrt(c)*x**2/sqrt(a + c*x**4))/(32*c**(3/2)) + a**2*x**2*sqrt(a + c*
x**4)/(32*c) + a*x**6*sqrt(a + c*x**4)/16 + x**6*(a + c*x**4)**(3/2)/12

_______________________________________________________________________________________

Mathematica [A]  time = 0.0670835, size = 82, normalized size = 0.86 \[ \frac{1}{2} \sqrt{a+c x^4} \left (\frac{a^2 x^2}{16 c}+\frac{7 a x^6}{24}+\frac{c x^{10}}{6}\right )-\frac{a^3 \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{32 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + c*x^4)^(3/2),x]

[Out]

(Sqrt[a + c*x^4]*((a^2*x^2)/(16*c) + (7*a*x^6)/24 + (c*x^10)/6))/2 - (a^3*Log[c*
x^2 + Sqrt[c]*Sqrt[a + c*x^4]])/(32*c^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 78, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{2}}{32\,c}\sqrt{c{x}^{4}+a}}-{\frac{{a}^{3}}{32}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{c{x}^{10}}{12}\sqrt{c{x}^{4}+a}}+{\frac{7\,{x}^{6}a}{48}\sqrt{c{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(c*x^4+a)^(3/2),x)

[Out]

1/32*a^2*x^2*(c*x^4+a)^(1/2)/c-1/32*a^3/c^(3/2)*ln(x^2*c^(1/2)+(c*x^4+a)^(1/2))+
1/12*c*x^10*(c*x^4+a)^(1/2)+7/48*a*x^6*(c*x^4+a)^(1/2)

_______________________________________________________________________________________

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^4 + a)^(3/2)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.29038, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8 \, c^{2} x^{10} + 14 \, a c x^{6} + 3 \, a^{2} x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{c}}{192 \, c^{\frac{3}{2}}}, -\frac{3 \, a^{3} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) -{\left (8 \, c^{2} x^{10} + 14 \, a c x^{6} + 3 \, a^{2} x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{-c}}{96 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^5,x, algorithm="fricas")

[Out]

[1/192*(3*a^3*log(2*sqrt(c*x^4 + a)*c*x^2 - (2*c*x^4 + a)*sqrt(c)) + 2*(8*c^2*x^
10 + 14*a*c*x^6 + 3*a^2*x^2)*sqrt(c*x^4 + a)*sqrt(c))/c^(3/2), -1/96*(3*a^3*arct
an(sqrt(-c)*x^2/sqrt(c*x^4 + a)) - (8*c^2*x^10 + 14*a*c*x^6 + 3*a^2*x^2)*sqrt(c*
x^4 + a)*sqrt(-c))/(sqrt(-c)*c)]

_______________________________________________________________________________________

Sympy [A]  time = 18.403, size = 122, normalized size = 1.28 \[ \frac{a^{\frac{5}{2}} x^{2}}{32 c \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{17 a^{\frac{3}{2}} x^{6}}{96 \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{11 \sqrt{a} c x^{10}}{48 \sqrt{1 + \frac{c x^{4}}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{32 c^{\frac{3}{2}}} + \frac{c^{2} x^{14}}{12 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(c*x**4+a)**(3/2),x)

[Out]

a**(5/2)*x**2/(32*c*sqrt(1 + c*x**4/a)) + 17*a**(3/2)*x**6/(96*sqrt(1 + c*x**4/a
)) + 11*sqrt(a)*c*x**10/(48*sqrt(1 + c*x**4/a)) - a**3*asinh(sqrt(c)*x**2/sqrt(a
))/(32*c**(3/2)) + c**2*x**14/(12*sqrt(a)*sqrt(1 + c*x**4/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228599, size = 90, normalized size = 0.95 \[ \frac{1}{96} \,{\left (2 \,{\left (4 \, c x^{4} + 7 \, a\right )} x^{4} + \frac{3 \, a^{2}}{c}\right )} \sqrt{c x^{4} + a} x^{2} + \frac{a^{3}{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{32 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^5,x, algorithm="giac")

[Out]

1/96*(2*(4*c*x^4 + 7*a)*x^4 + 3*a^2/c)*sqrt(c*x^4 + a)*x^2 + 1/32*a^3*ln(abs(-sq
rt(c)*x^2 + sqrt(c*x^4 + a)))/c^(3/2)