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

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

[Out]

4*b*EllipticF(c^(1/2)*(d*x)^(1/2)/d^(1/2),I)*c^(1/2)/d^(3/2)-2*(a+b*arcsin(c*x))/d/(d*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 = {4627, 329, 221} \[ \frac {4 b \sqrt {c} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[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]

Rubi steps

\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \sin ^{-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 \sin ^{-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 \sin ^{-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*}

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Mathematica [C]  time = 0.01, size = 40, normalized size = 0.73 \[ -\frac {2 x \left (a-2 b c x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )+b \sin ^{-1}(c x)\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*(a + b*ArcSin[c*x] - 2*b*c*x*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2]))/(d*x)^(3/2)

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\left (b \arcsin \left (c x\right ) + a\right )}}{d^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x\right ) + a}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 85, normalized size = 1.55 \[ \frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arcsin \left (c x \right )}{\sqrt {d x}}+\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-a/(d*x)^(1/2)+b*(-1/(d*x)^(1/2)*arcsin(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]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (b \sqrt {d} \sqrt {x} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (b c \sqrt {x} \int \frac {\sqrt {-c x + 1}}{\sqrt {c x + 1} c x^{\frac {3}{2}} - \sqrt {c x + 1} \sqrt {x}}\,{d x} + a\right )} \sqrt {d} \sqrt {x}\right )}}{d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b*sqrt(d)*sqrt(x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (b*c*d^2*sqrt(x)*integrate(sqrt(c*x + 1)*sq
rt(-c*x + 1)*sqrt(x)/(c^2*d^2*x^3 - d^2*x), x) + a)*sqrt(d)*sqrt(x))/(d^2*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))/(d*x)^(3/2),x)

[Out]

int((a + b*asin(c*x))/(d*x)^(3/2), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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