3.207 \(\int \frac{a+b \cos ^{-1}(c x)}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{4 b \sqrt{c} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right ),-1\right )}{d^{3/2}}-\frac{2 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt{d x}} \]

[Out]

(-2*(a + b*ArcCos[c*x]))/(d*Sqrt[d*x]) - (4*b*Sqrt[c]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/d^(3
/2)

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Rubi [A]  time = 0.0340409, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4628, 329, 221} \[ -\frac{2 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt{d x}}-\frac{4 b \sqrt{c} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{d^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])/(d*x)^(3/2),x]

[Out]

(-2*(a + b*ArcCos[c*x]))/(d*Sqrt[d*x]) - (4*b*Sqrt[c]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/d^(3
/2)

Rule 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

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

Mathematica [C]  time = 0.150651, size = 93, normalized size = 1.69 \[ \frac{2 x \left (\frac{2 i b \sqrt{-\frac{1}{c}} c^2 x^{3/2} \sqrt{1-\frac{1}{c^2 x^2}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right ),-1\right )}{\sqrt{1-c^2 x^2}}-a-b \cos ^{-1}(c x)\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-a - b*ArcCos[c*x] + ((2*I)*b*Sqrt[-c^(-1)]*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-
c^(-1)]/Sqrt[x]], -1])/Sqrt[1 - c^2*x^2]))/(d*x)^(3/2)

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Maple [A]  time = 0.006, size = 85, normalized size = 1.6 \begin{align*} 2\,{\frac{1}{d} \left ( -{\frac{a}{\sqrt{dx}}}+b \left ( -{\frac{\arccos \left ( cx \right ) }{\sqrt{dx}}}-2\,{\frac{c\sqrt{-cx+1}\sqrt{cx+1}}{d\sqrt{-{c}^{2}{x}^{2}+1}}{\it EllipticF} \left ( \sqrt{dx}\sqrt{{\frac{c}{d}}},i \right ){\frac{1}{\sqrt{{\frac{c}{d}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))/(d*x)^(3/2),x)

[Out]

2/d*(-1/(d*x)^(1/2)*a+b*(-1/(d*x)^(1/2)*arccos(c*x)-2*c/d/(c/d)^(1/2)*(-c*x+1)^(1/2)*(c*x+1)^(1/2)/(-c^2*x^2+1
)^(1/2)*EllipticF((d*x)^(1/2)*(c/d)^(1/2),I)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}{\left (b \arccos \left (c x\right ) + a\right )}}{d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/(d*x)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)*(b*arccos(c*x) + a)/(d^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))/(d*x)**(3/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccos(c*x) + a)/(d*x)^(3/2), x)