3.3.85 \(\int \frac {a+b \text {ArcSin}(c+d x)}{\sqrt {c e+d e x}} \, dx\) [285]

Optimal. Leaf size=81 \[ \frac {2 \sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4889, 4723, 326, 324, 435} \begin {gather*} \frac {2 \sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 324

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Dist[-2/(Sqrt[a]*(-b/a)^(3/4)), Subst[Int[Sqrt[1 - 2*x^2]
/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-b/a]*x]/Sqrt[2]], x] /; FreeQ[{a, b}, x] && GtQ[-b/a, 0] && GtQ[a, 0]

Rule 326

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

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 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]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.02, size = 59, normalized size = 0.73 \begin {gather*} -\frac {2 \sqrt {e (c+d x)} \left (-3 (a+b \text {ArcSin}(c+d x))+2 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )\right )}{3 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 149, normalized size = 1.84

method	result	size
derivativedivides	\(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \arcsin \left (\frac {d e x +c e}{e}\right )+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{\sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{d e}\)	\(149\)
default	\(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \arcsin \left (\frac {d e x +c e}{e}\right )+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{\sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{d e}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.85, size = 66, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left ({\left (b d \arcsin \left (d x + c\right ) + a d\right )} \sqrt {d x + c} e^{\frac {1}{2}} - 2 \, \sqrt {-d^{3} e} b {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )\right )} e^{\left (-1\right )}}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((b*d*arcsin(d*x + c) + a*d)*sqrt(d*x + c)*e^(1/2) - 2*sqrt(-d^3*e)*b*weierstrassZeta(4/d^2, 0, weierstrassP
Inverse(4/d^2, 0, (d*x + c)/d)))*e^(-1)/d^2

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{\sqrt {c\,e+d\,e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(1/2),x)

[Out]

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(1/2), x)

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