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

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

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Rubi [A]  time = 0.133196, 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, Rules used = {12, 2028, 2008, 206} \[ \frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3} \]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2028

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*p*(n - j)), x] + Dist[a/c^j, Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c,
j, m, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2008

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a x^2+b x^n\right )^{3/2}}{c^4 x^4} \, dx &=\frac{\int \frac{\left (a x^2+b x^n\right )^{3/2}}{x^4} \, dx}{c^4}\\ &=-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3}+\frac{a \int \frac{\sqrt{a x^2+b x^n}}{x^2} \, dx}{c^4}\\ &=-\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}+\frac{a^2 \int \frac{1}{\sqrt{a x^2+b x^n}} \, dx}{c^4}\\ &=-\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}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{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}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.157735, 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)*Sqrt[1 + (a*x^(2 - n))/b]*ArcSin
h[(Sqrt[a]*x^(1 - n/2))/Sqrt[b]]))/(3*c^4*(-2 + n)*x^3*Sqrt[a*x^2 + b*x^n])

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Maple [F]  time = 0.324, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{c}^{4}{x}^{4}} \left ( a{x}^{2}+b{x}^{n} \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

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)

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

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

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

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

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)