3.380 \(\int (c x)^{7/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2} \, dx\)

Optimal. Leaf size=122 \[ -\frac{2 a^{3/2} c^4 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}}+\frac{2 a c^2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{n+3}+\frac{2 (c x)^{9/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)} \]

[Out]

(2*a*c^2*(c*x)^(3/2)*Sqrt[a/x^3 + b*x^n])/(3 + n) + (2*(c*x)^(9/2)*(a/x^3 + b*x^
n)^(3/2))/(3*c*(3 + n)) - (2*a^(3/2)*c^4*Sqrt[x]*ArcTanh[Sqrt[a]/(x^(3/2)*Sqrt[a
/x^3 + b*x^n])])/((3 + n)*Sqrt[c*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.42938, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 a^{3/2} c^4 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}}+\frac{2 a c^2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{n+3}+\frac{2 (c x)^{9/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2),x]

[Out]

(2*a*c^2*(c*x)^(3/2)*Sqrt[a/x^3 + b*x^n])/(3 + n) + (2*(c*x)^(9/2)*(a/x^3 + b*x^
n)^(3/2))/(3*c*(3 + n)) - (2*a^(3/2)*c^4*Sqrt[x]*ArcTanh[Sqrt[a]/(x^(3/2)*Sqrt[a
/x^3 + b*x^n])])/((3 + n)*Sqrt[c*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.1237, size = 109, normalized size = 0.89 \[ - \frac{2 a^{\frac{3}{2}} c^{3} \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}} \right )}}{\sqrt{x} \left (n + 3\right )} + \frac{2 a c^{2} \left (c x\right )^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}}{n + 3} + \frac{2 \left (c x\right )^{\frac{9}{2}} \left (\frac{a}{x^{3}} + b x^{n}\right )^{\frac{3}{2}}}{3 c \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(7/2)*(a/x**3+b*x**n)**(3/2),x)

[Out]

-2*a**(3/2)*c**3*sqrt(c*x)*atanh(sqrt(a)/(x**(3/2)*sqrt(a/x**3 + b*x**n)))/(sqrt
(x)*(n + 3)) + 2*a*c**2*(c*x)**(3/2)*sqrt(a/x**3 + b*x**n)/(n + 3) + 2*(c*x)**(9
/2)*(a/x**3 + b*x**n)**(3/2)/(3*c*(n + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.432136, size = 0, normalized size = 0. \[ \int (c x)^{7/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2),x]

[Out]

Integrate[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2), x]

_______________________________________________________________________________________

Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{7}{2}}} \left ({\frac{a}{{x}^{3}}}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x)

[Out]

int((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2),x, algorithm="maxima")

[Out]

integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2), x)

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(7/2)*(a/x**3+b*x**n)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2),x, algorithm="giac")

[Out]

integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2), x)