3.375 \(\int \frac{\left (a x^2+b x^n\right )^{3/2}}{c^4 x^4} \, dx\)

Optimal. Leaf size=104 \[ \frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.224533, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^n)^(3/2)/(c^4*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.5031, size = 87, normalized size = 0.84 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{n}}} \right )}}{c^{4} \left (- n + 2\right )} - \frac{2 a \sqrt{a x^{2} + b x^{n}}}{c^{4} x \left (- n + 2\right )} - \frac{2 \left (a x^{2} + b x^{n}\right )^{\frac{3}{2}}}{3 c^{4} x^{3} \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(3/2)*atanh(sqrt(a)*x/sqrt(a*x**2 + b*x**n))/(c**4*(-n + 2)) - 2*a*sqrt(a*x
**2 + b*x**n)/(c**4*x*(-n + 2)) - 2*(a*x**2 + b*x**n)**(3/2)/(3*c**4*x**3*(-n +
2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.188749, size = 117, normalized size = 1.12 \[ \frac{2 \left (-3 a^{3/2} \sqrt{b} x^{\frac{n}{2}+3} \sqrt{\frac{a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{1-\frac{n}{2}}}{\sqrt{b}}\right )+4 a^2 x^4+5 a b x^{n+2}+b^2 x^{2 n}\right )}{3 c^4 (n-2) x^3 \sqrt{a x^2+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(4*a^2*x^4 + b^2*x^(2*n) + 5*a*b*x^(2 + n) - 3*a^(3/2)*Sqrt[b]*x^(3 + n/2)*Sq
rt[1 + (a*x^(2 - n))/b]*ArcSinh[(Sqrt[a]*x^(1 - n/2))/Sqrt[b]]))/(3*c^4*(-2 + n)
*x^3*Sqrt[a*x^2 + b*x^n])

_______________________________________________________________________________________

Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{1}{{c}^{4}{x}^{4}} \left ( a{x}^{2}+b{x}^{n} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^2+b*x^n)^(3/2)/c^4/x^4,x)

[Out]

int((a*x^2+b*x^n)^(3/2)/c^4/x^4,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{{\left (a x^{2} + b x^{n}\right )}^{\frac{3}{2}}}{x^{4}}\,{d x}}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^2 + b*x^n)^(3/2)/x^4, x)/c^4

_______________________________________________________________________________________

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((a*x^2 + b*x^n)^(3/2)/(c^4*x^4),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a \sqrt{a x^{2} + b x^{n}}}{x^{2}}\, dx + \int \frac{b x^{n} \sqrt{a x^{2} + b x^{n}}}{x^{4}}\, dx}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(Integral(a*sqrt(a*x**2 + b*x**n)/x**2, x) + Integral(b*x**n*sqrt(a*x**2 + b*x**
n)/x**4, x))/c**4

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a x^{2} + b x^{n}\right )}^{\frac{3}{2}}}{c^{4} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^2 + b*x^n)^(3/2)/(c^4*x^4), x)