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

Optimal. Leaf size=87 \[ \frac{2 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c (1-n) \sqrt{c x}}-\frac{2 \sqrt{a x+b x^n}}{c (1-n) \sqrt{c x}} \]

[Out]

(-2*Sqrt[a*x + b*x^n])/(c*(1 - n)*Sqrt[c*x]) + (2*Sqrt[a]*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b*x^n]]
)/(c*(1 - n)*Sqrt[c*x])

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Rubi [A]  time = 0.137653, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2028, 2031, 2029, 206} \[ \frac{2 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c (1-n) \sqrt{c x}}-\frac{2 \sqrt{a x+b x^n}}{c (1-n) \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x + b*x^n]/(c*x)^(3/2),x]

[Out]

(-2*Sqrt[a*x + b*x^n])/(c*(1 - n)*Sqrt[c*x]) + (2*Sqrt[a]*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b*x^n]]
)/(c*(1 - n)*Sqrt[c*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 2031

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

Rule 2029

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

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

Mathematica [A]  time = 0.250924, size = 100, normalized size = 1.15 \[ \frac{x \left (-2 \sqrt{a} \sqrt{b} x^{\frac{n+1}{2}} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )+2 a x+2 b x^n\right )}{(n-1) (c x)^{3/2} \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.383, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{ax+b{x}^{n}} \left ( cx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b*x^n)^(1/2)/(c*x)^(3/2),x)

[Out]

int((a*x+b*x^n)^(1/2)/(c*x)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b*x^n)^(1/2)/(c*x)^(3/2),x, algorithm="maxima")

[Out]

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

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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+b*x^n)^(1/2)/(c*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b*x**n)**(1/2)/(c*x)**(3/2),x)

[Out]

Integral(sqrt(a*x + b*x**n)/(c*x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b*x^n)^(1/2)/(c*x)^(3/2),x, algorithm="giac")

[Out]

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