3.7.91 \(\int (d+e x^2) (a+b \text {ArcSin}(c x))^{3/2} \, dx\) [691]

Optimal. Leaf size=482 \[ \frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{6 c}+d x (a+b \text {ArcSin}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {ArcSin}(c x))^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \]

[Out]

d*x*(a+b*arcsin(c*x))^(3/2)+1/3*e*x^3*(a+b*arcsin(c*x))^(3/2)+1/144*b^(3/2)*e*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(
1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/c^3+1/144*b^(3/2)*e*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcs
in(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/c^3-3/4*b^(3/2)*d*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*
arcsin(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c-3/16*b^(3/2)*e*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c
*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^3-3/4*b^(3/2)*d*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/
2))*sin(a/b)*2^(1/2)*Pi^(1/2)/c-3/16*b^(3/2)*e*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(
a/b)*2^(1/2)*Pi^(1/2)/c^3+3/2*b*d*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c+1/3*b*e*(-c^2*x^2+1)^(1/2)*(a+b
*arcsin(c*x))^(1/2)/c^3+1/6*b*e*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c

________________________________________________________________________________________

Rubi [A]
time = 0.95, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {4757, 4715, 4767, 4719, 3387, 3386, 3432, 3385, 3433, 4725, 4795, 4731, 4491} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{6 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcSin}(c x)}}{3 c^3}+d x (a+b \text {ArcSin}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {ArcSin}(c x))^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(3*b*d*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + (b*e*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c
^3) + (b*e*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(6*c) + d*x*(a + b*ArcSin[c*x])^(3/2) + (e*x^3*(a +
b*ArcSin[c*x])^(3/2))/3 - (3*b^(3/2)*d*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[
b]])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^3)
+ (b^(3/2)*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(24*c^3) - (3*b^(
3/2)*d*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c) - (3*b^(3/2)*e*Sqrt[P
i/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) + (b^(3/2)*e*Sqrt[Pi/6]*FresnelS
[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4725

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSin[c*x])^n/(m
+ 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4757

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx &=\int \left (d \left (a+b \sin ^{-1}(c x)\right )^{3/2}+e x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx+e \int x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {1}{2} (3 b c d) \int \frac {x \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} (b c e) \int \frac {x^3 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {1}{4} \left (3 b^2 d\right ) \int \frac {1}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{12} \left (b^2 e\right ) \int \frac {x^2}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {(b e) \int \frac {x \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx}{3 c}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {(3 b d) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {\left (b^2 e\right ) \int \frac {1}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{6 c^2}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {(b e) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {a+b x}}-\frac {\cos (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {\left (3 b d \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}-\frac {\left (3 b d \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac {\left (3 b d \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 c}-\frac {\left (b e \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac {\left (3 b d \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 c}-\frac {\left (b e \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {\left (b e \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{3 c^3}-\frac {\left (b^2 e \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 e \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac {\left (b e \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{3 c^3}-\frac {\left (b^2 e \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 e \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{3 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {b^{3/2} e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 c^3}-\frac {\left (b e \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}+\frac {\left (b e \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}-\frac {\left (b e \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}+\frac {\left (b e \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{24 c^3}\\ &=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 8.70, size = 873, normalized size = 1.81 \begin {gather*} \frac {a b d e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c x))}{b}\right )\right )}{2 c \sqrt {a+b \text {ArcSin}(c x)}}+\frac {a b e e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c x))}{b}\right )-\sqrt {3} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \text {ArcSin}(c x)}}+\frac {b d \left (2 \sqrt {a+b \text {ArcSin}(c x)} \left (3 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)\right )-\sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c}+\frac {b e \left (18 \sqrt {a+b \text {ArcSin}(c x)} \left (3 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)\right )-9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}\right ) \left (b \cos \left (\frac {3 a}{b}\right )+2 a \sin \left (\frac {3 a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {6 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}\right ) \left (-2 a \cos \left (\frac {3 a}{b}\right )+b \sin \left (\frac {3 a}{b}\right )\right )-6 \sqrt {a+b \text {ArcSin}(c x)} (\cos (3 \text {ArcSin}(c x))+2 \text {ArcSin}(c x) \sin (3 \text {ArcSin}(c x)))\right )}{144 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(a*b*d*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*
(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b]))/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (a
*b*e*(9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 9*E^(((4
*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*(Sqrt[((-I)*(a + b*A
rcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*G
amma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (b*d*(2*Sqrt[a
+ b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqr
t[2/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]
*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c) + (b*e*(18*Sqrt[a + b*ArcSin[c*x]]*
(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a
+ b*ArcSin[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*S
qrt[a + b*ArcSin[c*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/P
i]*Sqrt[a + b*ArcSin[c*x]]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a)/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)
]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(-2*a*Cos[(3*a)/b] + b*Sin[(3*a)/b]) - 6*Sqrt[a + b*ArcSin[c*x]]*(Cos[3*
ArcSin[c*x]] + 2*ArcSin[c*x]*Sin[3*ArcSin[c*x]])))/(144*c^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(849\) vs. \(2(374)=748\).
time = 0.47, size = 850, normalized size = 1.76

method	result	size
default	\(-\frac {108 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} c^{2} d -108 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} c^{2} d +27 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} e -27 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2} e -\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b^{2} e +\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b^{2} e +144 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} c^{2} d +288 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b \,c^{2} d -216 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} c^{2} d +36 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} e -12 \arcsin \left (c x \right )^{2} \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2} e +144 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} c^{2} d -216 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b \,c^{2} d +72 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b e -54 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2} e -24 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b e +6 \arcsin \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2} e +36 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} e -54 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b e -12 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2} e +6 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b e}{144 c^{3} \sqrt {a +b \arcsin \left (c x \right )}}\)	\(850\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/144/c^3*(108*2^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(
a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*b^2*c^2*d-108*2^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*Fresnel
S(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*b^2*c^2*d+27*2^(1/2)*(-1/b)^(1/2)*(a+b*arc
sin(c*x))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*b^2*e-27*2
^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^
(1/2)/b)*Pi^(1/2)*b^2*e-2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a
+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*(-3/b)^(1/2)*b^2*e+2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelS(3*2^(
1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*(-3/b)^(1/2)*b^2*e+144*arcsin(c*x)^2*sin(-(a+b*
arcsin(c*x))/b+a/b)*b^2*c^2*d+288*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b*c^2*d-216*arcsin(c*x)*cos(-(a+
b*arcsin(c*x))/b+a/b)*b^2*c^2*d+36*arcsin(c*x)^2*sin(-(a+b*arcsin(c*x))/b+a/b)*b^2*e-12*arcsin(c*x)^2*sin(-3*(
a+b*arcsin(c*x))/b+3*a/b)*b^2*e+144*sin(-(a+b*arcsin(c*x))/b+a/b)*a^2*c^2*d-216*cos(-(a+b*arcsin(c*x))/b+a/b)*
a*b*c^2*d+72*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b*e-54*arcsin(c*x)*cos(-(a+b*arcsin(c*x))/b+a/b)*b^2*
e-24*arcsin(c*x)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a*b*e+6*arcsin(c*x)*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2*e
+36*sin(-(a+b*arcsin(c*x))/b+a/b)*a^2*e-54*cos(-(a+b*arcsin(c*x))/b+a/b)*a*b*e-12*sin(-3*(a+b*arcsin(c*x))/b+3
*a/b)*a^2*e+6*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*a*b*e)/(a+b*arcsin(c*x))^(1/2)

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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((e*x^2+d)*(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate((x^2*e + d)*(b*arcsin(c*x) + a)^(3/2), x)

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Fricas [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((e*x^2+d)*(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*asin(c*x))**(3/2),x)

[Out]

Integral((a + b*asin(c*x))**(3/2)*(d + e*x**2), x)

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Giac [C] Result contains complex when optimal does not.
time = 1.76, size = 2814, normalized size = 5.84 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")

[Out]

1/96*(48*sqrt(2)*sqrt(pi)*a^2*b*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sq
rt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 48*I*sqrt(2)*sqrt(pi)*
a*b^2*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt
(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 48*sqrt(2)*sqrt(pi)*a^2*b*c^2*d*erf(1/2*I*sqrt(2
)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^
2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*I*sqrt(2)*sqrt(pi)*a*b^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)
/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(a
bs(b))) - 48*I*sqrt(2)*sqrt(pi)*a*b*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2
)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 36*sqrt(2)*sqrt(pi)*b^
2*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs
(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 48*I*sqrt(2)*sqrt(pi)*a*b*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b
*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs
(b)) + sqrt(abs(b))) + 36*sqrt(2)*sqrt(pi)*b^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) -
1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) - 48*I*sqrt(
b*arcsin(c*x) + a)*b*c^2*d*arcsin(c*x)*e^(I*arcsin(c*x)) + 48*I*sqrt(b*arcsin(c*x) + a)*b*c^2*d*arcsin(c*x)*e^
(-I*arcsin(c*x)) - 96*sqrt(pi)*a^2*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)
*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) - 96*sqrt
(pi)*a^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sq
rt(abs(b))/b)*e^(-I*a/b)/(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) + 12*sqrt(2)*sqrt(pi)*a^2*b*e*erf(
-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I
*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 12*I*sqrt(2)*sqrt(pi)*a*b^2*e*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c
*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*
sqrt(abs(b))) + 12*sqrt(2)*sqrt(pi)*a^2*b*e*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(
2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 12*I*sqrt(2)*sq
rt(pi)*a*b^2*e*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sq
rt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*I*sqrt(b*arcsin(c*x) + a)*a*c^2*d*e^(I*ar
csin(c*x)) + 72*sqrt(b*arcsin(c*x) + a)*b*c^2*d*e^(I*arcsin(c*x)) + 48*I*sqrt(b*arcsin(c*x) + a)*a*c^2*d*e^(-I
*arcsin(c*x)) + 72*sqrt(b*arcsin(c*x) + a)*b*c^2*d*e^(-I*arcsin(c*x)) - 24*sqrt(pi)*a^2*sqrt(b)*e*erf(-1/2*sqr
t(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt
(6)*b + I*sqrt(6)*b^2/abs(b)) - 8*I*sqrt(pi)*a*b^(3/2)*e*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/
2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b + I*sqrt(6)*b^2/abs(b)) - 12*I*sqrt
(2)*sqrt(pi)*a*b*e*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) +
a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 9*sqrt(2)*sqrt(pi)*b^2*e*erf(-1/2*I*sqrt(2)*s
qrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(
abs(b)) + sqrt(abs(b))) + 12*I*sqrt(2)*sqrt(pi)*a*b*e*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) -
1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) + 9*sqrt(2)
*sqrt(pi)*b^2*e*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*s
qrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) - 24*sqrt(pi)*a^2*sqrt(b)*e*erf(-1/2*sqrt(6)*sqrt
(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b -
I*sqrt(6)*b^2/abs(b)) + 8*I*sqrt(pi)*a*b^(3/2)*e*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt
(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b - I*sqrt(6)*b^2/abs(b)) + 4*I*sqrt(b*arcsi
n(c*x) + a)*b*e*arcsin(c*x)*e^(3*I*arcsin(c*x)) - 12*I*sqrt(b*arcsin(c*x) + a)*b*e*arcsin(c*x)*e^(I*arcsin(c*x
)) + 12*I*sqrt(b*arcsin(c*x) + a)*b*e*arcsin(c*x)*e^(-I*arcsin(c*x)) - 4*I*sqrt(b*arcsin(c*x) + a)*b*e*arcsin(
c*x)*e^(-3*I*arcsin(c*x)) + 24*sqrt(pi)*a^2*e*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)
*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) + 8*I*sqrt(p
i)*a*b*e*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + ...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))^(3/2)*(d + e*x^2),x)

[Out]

int((a + b*asin(c*x))^(3/2)*(d + e*x^2), x)

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