Integrand size = 27, antiderivative size = 351 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \] Output:
5/256*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-59/768*b*c*d^2*x ^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+17/288*b*c^3*d^2*x^6*(-c^2*d*x^ 2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/64*b*c^5*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^ 2*x^2+1)^(1/2)-5/128*d^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c^2+5/64 *d^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))+5/48*d*x^3*(-c^2*d*x^2+d)^ (3/2)*(a+b*arcsin(c*x))+1/8*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))+5/2 56*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/b/c^3/(-c^2*x^2+1)^(1/2)
Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.56 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (45 a^2+b^2 c^2 x^2 \left (45-177 c^2 x^2+136 c^4 x^4-36 c^6 x^6\right )+6 a b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )+6 b \left (15 a+b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right ) \arcsin (c x)+45 b^2 \arcsin (c x)^2\right )}{2304 b c^3 \sqrt {1-c^2 x^2}} \] Input:
Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*(45*a^2 + b^2*c^2*x^2*(45 - 177*c^2*x^2 + 136*c^4 *x^4 - 36*c^6*x^6) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136* c^4*x^4 + 48*c^6*x^6) + 6*b*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2 *x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x] + 45*b^2*ArcSin[c*x]^2))/(23 04*b*c^3*Sqrt[1 - c^2*x^2])
Time = 1.54 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5202, 243, 49, 2009, 5202, 244, 2009, 5198, 15, 5210, 15, 5152}
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.
\(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx\) |
\(\Big \downarrow \) 5202 |
\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2dx}{8 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^2 \left (1-c^2 x^2\right )^2dx^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^6-2 c^2 x^4+x^2\right )dx^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5202 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^3-c^2 x^5\right )dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5198 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )-\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\) |
Input:
Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
Output:
-1/16*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x^4/2 - (2*c^2*x^6)/3 + (c^4*x^8)/4))/ Sqrt[1 - c^2*x^2] + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/8 + (5 *d*(-1/6*(b*c*d*Sqrt[d - c^2*d*x^2]*(x^4/4 - (c^2*x^6)/6))/Sqrt[1 - c^2*x^ 2] + (x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/6 + (d*(-1/16*(b*c*x^ 4*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (x^3*Sqrt[d - c^2*d*x^2]*(a + b *ArcSin[c*x]))/4 + (Sqrt[d - c^2*d*x^2]*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x ^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*Sq rt[1 - c^2*x^2])))/2))/8
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
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] + (Simp[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] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(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]
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 907, normalized size of antiderivative = 2.58
method | result | size |
default | \(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{48 c^{2}}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{192 c^{2}}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}+128 c^{9} x^{9}+256 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}-320 c^{7} x^{7}-160 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+272 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-88 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+8 c x \right ) \left (8 \arcsin \left (c x \right )+i\right ) d^{2}}{16384 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (73 i+312 \arcsin \left (c x \right )\right ) \cos \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (55 i+456 \arcsin \left (c x \right )\right ) \sin \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (13 i+12 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (i+12 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (i+4 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (5 i+4 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(907\) |
parts | \(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{48 c^{2}}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{192 c^{2}}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 i \sqrt {-c^{2} x^{2}+1}\, x^{8} c^{8}+128 c^{9} x^{9}+256 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}-320 c^{7} x^{7}-160 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+272 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-88 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+8 c x \right ) \left (8 \arcsin \left (c x \right )+i\right ) d^{2}}{16384 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2}}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (73 i+312 \arcsin \left (c x \right )\right ) \cos \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (55 i+456 \arcsin \left (c x \right )\right ) \sin \left (7 \arcsin \left (c x \right )\right ) d^{2}}{147456 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (13 i+12 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (i+12 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2}}{9216 c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (i+4 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (5 i+4 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2}}{1024 c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(907\) |
Input:
int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/19 2*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/ 128*a/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b *(-5/256*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin( c*x)^2*d^2+1/16384*(-d*(c^2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^8*c ^8+128*c^9*x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-160*I*(-c^2*x^ 2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3- I*(-c^2*x^2+1)^(1/2)+8*c*x)*(8*arcsin(c*x)+I)*d^2/c^3/(c^2*x^2-1)+1/256*(- d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2 +1)^(1/2)-2*c*x)*(-I+2*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)+1/147456*(-d*(c^2* x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(73*I+312*arcsin(c*x))* cos(7*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-1/147456*(-d*(c^2*x^2-1))^(1/2)*(I* (-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(55*I+456*arcsin(c*x))*sin(7*arcsin(c*x) )*d^2/c^3/(c^2*x^2-1)+1/9216*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x ^2+1)^(1/2)-I)*(13*I+12*arcsin(c*x))*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1 )-5/9216*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(I+12 *arcsin(c*x))*sin(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-3/1024*(-d*(c^2*x^2-1 ))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(I+4*arcsin(c*x))*cos(3*arcs in(c*x))*d^2/c^3/(c^2*x^2-1)+1/1024*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1) ^(1/2)*c*x+c^2*x^2-1)*(5*I+4*arcsin(c*x))*sin(3*arcsin(c*x))*d^2/c^3/(c...
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas" )
Output:
integral((a*c^4*d^2*x^6 - 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 - 2 *b*c^2*d^2*x^4 + b*d^2*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \] Input:
integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)
Output:
Timed out
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima" )
Output:
b*sqrt(d)*integrate((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(c*x + 1)* sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 1/384*(8*( -c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c ^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/ 2)*arcsin(c*x)/c^3)*a
Time = 0.83 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.03 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {1}{8} \, \sqrt {-c^{2} d x^{2} + d} a c^{4} d^{2} x^{7} - \frac {17}{48} \, \sqrt {-c^{2} d x^{2} + d} a c^{2} d^{2} x^{5} + \frac {59}{192} \, \sqrt {-c^{2} d x^{2} + d} a d^{2} x^{3} - \frac {5 \, \sqrt {-c^{2} d x^{2} + d} a d^{2} x}{128 \, c^{2}} - \frac {5 \, a d^{3} \log \left ({\left | -c \sqrt {-d} x + \sqrt {c^{2} x^{2} - 1} \sqrt {-d} \right |}\right )}{128 \, c^{3} \sqrt {-d}} + \frac {9216 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{\frac {5}{2}} x \arcsin \left (c x\right ) + 1536 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{\frac {5}{2}} x \arcsin \left (c x\right ) + 1920 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{\frac {5}{2}} x \arcsin \left (c x\right ) - \frac {1152 \, {\left (c^{2} x^{2} - 1\right )}^{4} b d^{\frac {5}{2}}}{c} + 2880 \, \sqrt {-c^{2} x^{2} + 1} b d^{\frac {5}{2}} x \arcsin \left (c x\right ) - \frac {256 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d^{\frac {5}{2}}}{c} + \frac {480 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{\frac {5}{2}}}{c} + \frac {1440 \, b d^{\frac {5}{2}} \arcsin \left (c x\right )^{2}}{c} - \frac {1440 \, {\left (c^{2} x^{2} - 1\right )} b d^{\frac {5}{2}}}{c} - \frac {665 \, b d^{\frac {5}{2}}}{c}}{73728 \, c^{2}} \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
1/8*sqrt(-c^2*d*x^2 + d)*a*c^4*d^2*x^7 - 17/48*sqrt(-c^2*d*x^2 + d)*a*c^2* d^2*x^5 + 59/192*sqrt(-c^2*d*x^2 + d)*a*d^2*x^3 - 5/128*sqrt(-c^2*d*x^2 + d)*a*d^2*x/c^2 - 5/128*a*d^3*log(abs(-c*sqrt(-d)*x + sqrt(c^2*x^2 - 1)*sqr t(-d)))/(c^3*sqrt(-d)) + 1/73728*(9216*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)* b*d^(5/2)*x*arcsin(c*x) + 1536*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^(5/2 )*x*arcsin(c*x) + 1920*(-c^2*x^2 + 1)^(3/2)*b*d^(5/2)*x*arcsin(c*x) - 1152 *(c^2*x^2 - 1)^4*b*d^(5/2)/c + 2880*sqrt(-c^2*x^2 + 1)*b*d^(5/2)*x*arcsin( c*x) - 256*(c^2*x^2 - 1)^3*b*d^(5/2)/c + 480*(c^2*x^2 - 1)^2*b*d^(5/2)/c + 1440*b*d^(5/2)*arcsin(c*x)^2/c - 1440*(c^2*x^2 - 1)*b*d^(5/2)/c - 665*b*d ^(5/2)/c)/c^2
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int(x^2*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2),x)
Output:
int(x^2*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a +48 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} x^{7}-136 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} x^{5}+118 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} x^{3}-15 \sqrt {-c^{2} x^{2}+1}\, a c x +384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{6}d x \right ) b \,c^{7}-768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,c^{5}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) b \,c^{3}\right )}{384 c^{3}} \] Input:
int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x)),x)
Output:
(sqrt(d)*d**2*(15*asin(c*x)*a + 48*sqrt( - c**2*x**2 + 1)*a*c**7*x**7 - 13 6*sqrt( - c**2*x**2 + 1)*a*c**5*x**5 + 118*sqrt( - c**2*x**2 + 1)*a*c**3*x **3 - 15*sqrt( - c**2*x**2 + 1)*a*c*x + 384*int(sqrt( - c**2*x**2 + 1)*asi n(c*x)*x**6,x)*b*c**7 - 768*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**4,x)*b *c**5 + 384*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**2,x)*b*c**3))/(384*c** 3)