3.206 \(\int \frac{a+b \cos ^{-1}(c x)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{4 b \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right ),-1\right )}{\sqrt{c} \sqrt{d}}+\frac{2 \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )}{d}+\frac{4 b E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{\sqrt{c} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0739902, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4628, 329, 307, 221, 1199, 424} \[ \frac{2 \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )}{d}-\frac{4 b F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{\sqrt{c} \sqrt{d}}+\frac{4 b E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{\sqrt{c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])/Sqrt[d*x],x]

[Out]

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

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 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

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]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

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

Mathematica [C]  time = 0.034045, size = 45, normalized size = 0.51 \[ \frac{2 x \left (2 b c x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},c^2 x^2\right )+3 \left (a+b \cos ^{-1}(c x)\right )\right )}{3 \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])/Sqrt[d*x],x]

[Out]

(2*x*(3*(a + b*ArcCos[c*x]) + 2*b*c*x*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2]))/(3*Sqrt[d*x])

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Maple [A]  time = 0.008, size = 98, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( a\sqrt{dx}+b \left ( \sqrt{dx}\arccos \left ( cx \right ) -2\,{\frac{\sqrt{-cx+1}\sqrt{cx+1}}{\sqrt{-{c}^{2}{x}^{2}+1}} \left ({\it EllipticF} \left ( \sqrt{dx}\sqrt{{\frac{c}{d}}},i \right ) -{\it EllipticE} \left ( \sqrt{dx}\sqrt{{\frac{c}{d}}},i \right ) \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)^(1/2),x)

[Out]

2/d*(a*(d*x)^(1/2)+b*((d*x)^(1/2)*arccos(c*x)-2/(c/d)^(1/2)*(-c*x+1)^(1/2)*(c*x+1)^(1/2)/(-c^2*x^2+1)^(1/2)*(E
llipticF((d*x)^(1/2)*(c/d)^(1/2),I)-EllipticE((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)^(1/2),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 x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/(d*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)*(b*arccos(c*x) + a)/(d*x), 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)**(1/2),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}{\sqrt{d x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccos(c*x) + a)/sqrt(d*x), x)