3.7.52 \(\int \frac {a+b \text {ArcSin}(c x)}{(d+e x^2)^{5/2}} \, dx\) [652]

Optimal. Leaf size=146 \[ \frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \text {ArcSin}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {2 b \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {198, 197, 4755, 12, 585, 79, 65, 223, 209} \begin {gather*} \frac {2 x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {ArcSin}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 b \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}+\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

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

Rule 585

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 4755

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {x \left (3 d+2 e x^2\right )}{3 d^2 \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x \left (3 d+2 e x^2\right )}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {3 d+2 e x}{\sqrt {1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d^2}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c d^2}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{3 c d^2}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {2 b \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.16, size = 190, normalized size = 1.30 \begin {gather*} \frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\sqrt {d+e x^2} \left (\frac {a x}{3 d \left (d+e x^2\right )^2}+\frac {2 a x}{3 d^2 \left (d+e x^2\right )}\right )-\frac {b c x^2 \sqrt {\frac {d+e x^2}{d}} F_1\left (1;\frac {1}{2},\frac {1}{2};2;c^2 x^2,-\frac {e x^2}{d}\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b x \left (3 d+2 e x^2\right ) \text {ArcSin}(c x)}{3 d^2 \left (d+e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2]) + Sqrt[d + e*x^2]*((a*x)/(3*d*(d + e*x^2)^2) + (2*a*
x)/(3*d^2*(d + e*x^2))) - (b*c*x^2*Sqrt[(d + e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/(3*d^2
*Sqrt[d + e*x^2]) + (b*x*(3*d + 2*e*x^2)*ArcSin[c*x])/(3*d^2*(d + e*x^2)^(3/2))

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Maple [F]
time = 0.84, size = 0, normalized size = 0.00 \[\int \frac {a +b \arcsin \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

1/3*a*(2*x/(sqrt(x^2*e + d)*d^2) + x/((x^2*e + d)^(3/2)*d)) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1))/((x^4*e^2 + 2*d*x^2*e + d^2)*sqrt(x^2*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (125) = 250\).
time = 1.72, size = 333, normalized size = 2.28 \begin {gather*} \frac {{\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} {\left (c^{2} d + {\left (2 \, c^{2} x^{2} - 1\right )} e\right )} \sqrt {x^{2} e + d} e^{\frac {1}{2}}}{2 \, {\left ({\left (c^{3} x^{4} - c x^{2}\right )} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) e^{\frac {1}{2}} + {\left (3 \, a c^{2} d^{2} x e + 2 \, a x^{3} e^{3} + {\left (3 \, b c^{2} d^{2} x e + 2 \, b x^{3} e^{3} + {\left (2 \, b c^{2} d x^{3} + 3 \, b d x\right )} e^{2}\right )} \arcsin \left (c x\right ) + {\left (2 \, a c^{2} d x^{3} + 3 \, a d x\right )} e^{2} + {\left (b c d x^{2} e^{2} + b c d^{2} e\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {x^{2} e + d}}{3 \, {\left (c^{2} d^{5} e + d^{2} x^{4} e^{4} + {\left (c^{2} d^{3} x^{4} + 2 \, d^{3} x^{2}\right )} e^{3} + {\left (2 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((b*c^2*d^3 + b*x^4*e^3 + (b*c^2*d*x^4 + 2*b*d*x^2)*e^2 + (2*b*c^2*d^2*x^2 + b*d^2)*e)*arctan(1/2*sqrt(-c^
2*x^2 + 1)*(c^2*d + (2*c^2*x^2 - 1)*e)*sqrt(x^2*e + d)*e^(1/2)/((c^3*x^4 - c*x^2)*e^2 + (c^3*d*x^2 - c*d)*e))*
e^(1/2) + (3*a*c^2*d^2*x*e + 2*a*x^3*e^3 + (3*b*c^2*d^2*x*e + 2*b*x^3*e^3 + (2*b*c^2*d*x^3 + 3*b*d*x)*e^2)*arc
sin(c*x) + (2*a*c^2*d*x^3 + 3*a*d*x)*e^2 + (b*c*d*x^2*e^2 + b*c*d^2*e)*sqrt(-c^2*x^2 + 1))*sqrt(x^2*e + d))/(c
^2*d^5*e + d^2*x^4*e^4 + (c^2*d^3*x^4 + 2*d^3*x^2)*e^3 + (2*c^2*d^4*x^2 + d^4)*e^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((b*arcsin(c*x) + a)/(e*x^2 + d)^(5/2), 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\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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