\(\int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx\) [206]

Optimal result

Integrand size = 16, antiderivative size = 89 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}}+\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4723, 335, 313, 227, 1213, 435} \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}+\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}}-\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}} \]

[In]

Int[(a + b*ArcSin[c*x])/Sqrt[d*x],x]

[Out]

(2*Sqrt[d*x]*(a + b*ArcSin[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 227

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 313

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 335

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 435

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

Rule 1213

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 4723

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\frac {2 x \left (3 (a+b \arcsin (c x))-2 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{3 \sqrt {d x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \arcsin \left (c x \right )+\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(98\)
default	\(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \arcsin \left (c x \right )+\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(98\)
parts	\(\frac {2 a \sqrt {d x}}{d}+\frac {2 b \left (\sqrt {d x}\, \arcsin \left (c x \right )+\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(101\)

[In]

int((a+b*arcsin(c*x))/(d*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d*((d*x)^(1/2)*a+b*((d*x)^(1/2)*arcsin(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))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) - {\left (b c \arcsin \left (c x\right ) + a c\right )} \sqrt {d x}\right )}}{c d} \]

[In]

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

[Out]

-2*(2*sqrt(-c^2*d)*b*weierstrassZeta(4/c^2, 0, weierstrassPInverse(4/c^2, 0, x)) - (b*c*arcsin(c*x) + a*c)*sqr
t(d*x))/(c*d)

Sympy [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*asin(c*x))/(d*x)**(1/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {d x}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {d x}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\sqrt {d\,x}} \,d x \]

[In]

int((a + b*asin(c*x))/(d*x)^(1/2),x)

[Out]

int((a + b*asin(c*x))/(d*x)^(1/2), x)