\(\int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx\) [287]

Optimal result

Integrand size = 23, antiderivative size = 122 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{3 d e^3 \sqrt {c+d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 4723, 331, 326, 324, 435} \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{3 d e^3 \sqrt {c+d x}}-\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}} \]

[In]

Int[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(5/2),x]

[Out]

(-4*b*Sqrt[1 - (c + d*x)^2])/(3*d*e^2*Sqrt[e*(c + d*x)]) - (2*(a + b*ArcSin[c + d*x]))/(3*d*e*(e*(c + d*x))^(3
/2)) + (4*b*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(3*d*e^3*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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \arcsin (x)}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}-\frac {(2 b) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e^3} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}-\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 )}{3 d e^3 \sqrt {c+d x}} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\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 )}{3 d e^3 \sqrt {c+d x}} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{3 d e^3 \sqrt {c+d x}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.46 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \left (a+b \arcsin (c+d x)+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},(c+d x)^2\right )\right )}{3 d e (e (c+d x))^{3/2}} \]

[In]

Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(5/2),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.72 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{3 e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(190\)
default	\(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{3 e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(190\)
parts	\(-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}} d e}+\frac {2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{3 e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{e d}\)	\(195\)

[In]

int((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/d/e*(-1/3*a/(d*e*x+c*e)^(3/2)+b*(-1/3/(d*e*x+c*e)^(3/2)*arcsin(1/e*(d*e*x+c*e))+2/3/e*(-(-1/e^2*(d*e*x+c*e)^
2+1)^(1/2)/(d*e*x+c*e)^(1/2)+1/e/(1/e)^(1/2)*(1-1/e*(d*e*x+c*e))^(1/2)*(1+1/e*(d*e*x+c*e))^(1/2)/(-1/e^2*(d*e*
x+c*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)))))

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + \sqrt {d e x + c e} {\left (b d \arcsin \left (d x + c\right ) + a d + 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )}\right )}}{3 \, {\left (d^{4} e^{3} x^{2} + 2 \, c d^{3} e^{3} x + c^{2} d^{2} e^{3}\right )}} \]

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((a+b*asin(d*x+c))/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*asin(c + d*x))/(e*(c + d*x))**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)/(d*e*x + c*e)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]

[In]

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(5/2),x)

[Out]

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(5/2), x)