Integrand size = 16, antiderivative size = 89 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}}+\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}} \] Output:
2*(d*x)^(1/2)*(a+b*arcsin(c*x))/d-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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\frac {2 x \left (3 (a+b \arcsin (c x))-2 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{3 \sqrt {d x}} \] Input:
Integrate[(a + b*ArcSin[c*x])/Sqrt[d*x],x]
Output:
(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])
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5138, 266, 836, 27, 762, 1389, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {2 b c \int \frac {\sqrt {d x}}{\sqrt {1-c^2 x^2}}dx}{d}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b c \int \frac {d x}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{d^2}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b c \left (\frac {d \int \frac {c x d+d}{d \sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}-\frac {d \int \frac {1}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}\right )}{d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b c \left (\frac {\int \frac {c x d+d}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}-\frac {d \int \frac {1}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}\right )}{d^2}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b c \left (\frac {\int \frac {c x d+d}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}-\frac {d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{c^{3/2}}\right )}{d^2}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b c \left (\frac {d \int \frac {\sqrt {c x+1}}{\sqrt {1-c x}}d\sqrt {d x}}{c}-\frac {d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{c^{3/2}}\right )}{d^2}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))}{d}-\frac {4 b c \left (\frac {d^{3/2} E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{c^{3/2}}-\frac {d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{c^{3/2}}\right )}{d^2}\) |
Input:
Int[(a + b*ArcSin[c*x])/Sqrt[d*x],x]
Output:
(2*Sqrt[d*x]*(a + b*ArcSin[c*x]))/d - (4*b*c*((d^(3/2)*EllipticE[ArcSin[(S qrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/c^(3/2) - (d^(3/2)*EllipticF[ArcSin[(Sqrt [c]*Sqrt[d*x])/Sqrt[d]], -1])/c^(3/2)))/d^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[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]
Time = 0.30 (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 | \(\frac {2 a \sqrt {d x}}{d}+\frac {2 b \left (\sqrt {d x}\, \arcsin \left (c x \right )+\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\) | \(101\) |
Input:
int((a+b*arcsin(c*x))/(d*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
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)*(EllipticF((d*x)^(1/2)*(c/d)^(1/2),I)-El lipticE((d*x)^(1/2)*(c/d)^(1/2),I))))
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) - {\left (b c \arcsin \left (c x\right ) + a c\right )} \sqrt {d x}\right )}}{c d} \] Input:
integrate((a+b*arcsin(c*x))/(d*x)^(1/2),x, algorithm="fricas")
Output:
-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)*sqrt(d*x))/(c*d)
Exception generated. \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*asin(c*x))/(d*x)**(1/2),x)
Output:
Exception raised: TypeError >> Invalid comparison of non-real zoo
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {d x}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/(d*x)^(1/2),x, algorithm="maxima")
Output:
2*(b*sqrt(d)*sqrt(x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (b*c*d*i ntegrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(c^2*d*x^2 - d), x) + a*sqrt (x))*sqrt(d))/d
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {d x}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/(d*x)^(1/2),x, algorithm="giac")
Output:
integrate((b*arcsin(c*x) + a)/sqrt(d*x), x)
Timed out. \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\sqrt {d\,x}} \,d x \] Input:
int((a + b*asin(c*x))/(d*x)^(1/2),x)
Output:
int((a + b*asin(c*x))/(d*x)^(1/2), x)
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {x}\, a +\left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {x}}d x \right ) b}{\sqrt {d}} \] Input:
int((a+b*asin(c*x))/(d*x)^(1/2),x)
Output:
(2*sqrt(x)*a + int(asin(c*x)/sqrt(x),x)*b)/sqrt(d)