Integrand size = 18, antiderivative size = 109 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {2 (d x)^{5/2} (a+b \arcsin (c x))^2}{5 d}-\frac {8 b c (d x)^{7/2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},c^2 x^2\right )}{35 d^2}+\frac {16 b^2 c^2 (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )}{315 d^3} \] Output:
2/5*(d*x)^(5/2)*(a+b*arcsin(c*x))^2/d-8/35*b*c*(d*x)^(7/2)*(a+b*arcsin(c*x ))*hypergeom([1/2, 7/4],[11/4],c^2*x^2)/d^2+16/315*b^2*c^2*(d*x)^(9/2)*hyp ergeom([1, 9/4, 9/4],[11/4, 13/4],c^2*x^2)/d^3
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {2}{315} x (d x)^{3/2} \left (9 (a+b \arcsin (c x)) \left (7 (a+b \arcsin (c x))-4 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},c^2 x^2\right )\right )+8 b^2 c^2 x^2 \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )\right ) \] Input:
Integrate[(d*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(2*x*(d*x)^(3/2)*(9*(a + b*ArcSin[c*x])*(7*(a + b*ArcSin[c*x]) - 4*b*c*x*H ypergeometric2F1[1/2, 7/4, 11/4, c^2*x^2]) + 8*b^2*c^2*x^2*HypergeometricP FQ[{1, 9/4, 9/4}, {11/4, 13/4}, c^2*x^2]))/315
Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5138, 5220}
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.
\(\displaystyle \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {2 (d x)^{5/2} (a+b \arcsin (c x))^2}{5 d}-\frac {4 b c \int \frac {(d x)^{5/2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 d}\) |
\(\Big \downarrow \) 5220 |
\(\displaystyle \frac {2 (d x)^{5/2} (a+b \arcsin (c x))^2}{5 d}-\frac {4 b c \left (\frac {2 (d x)^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},c^2 x^2\right ) (a+b \arcsin (c x))}{7 d}-\frac {4 b c (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )}{63 d^2}\right )}{5 d}\) |
Input:
Int[(d*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(2*(d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(5*d) - (4*b*c*((2*(d*x)^(7/2)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 7/4, 11/4, c^2*x^2])/(7*d) - (4*b*c* (d*x)^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, c^2*x^2])/(63*d ^2)))/(5*d)
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[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]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. )*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2* x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m /2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[m]
\[\int \left (d x \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
Input:
int((d*x)^(3/2)*(a+b*arcsin(c*x))^2,x)
Output:
int((d*x)^(3/2)*(a+b*arcsin(c*x))^2,x)
\[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*x)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")
Output:
integral((b^2*d*x*arcsin(c*x)^2 + 2*a*b*d*x*arcsin(c*x) + a^2*d*x)*sqrt(d* x), x)
\[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int \left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate((d*x)**(3/2)*(a+b*asin(c*x))**2,x)
Output:
Integral((d*x)**(3/2)*(a + b*asin(c*x))**2, x)
\[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*x)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")
Output:
2/5*b^2*d^(3/2)*x^(5/2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 1/1 0*a^2*c^2*d^(3/2)*(4*(c^2*x^(5/2) + 5*sqrt(x))/c^4 - 10*arctan(sqrt(c)*sqr t(x))/c^(9/2) + 5*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(9/2) ) + 10*a*b*c^2*d^(3/2)*integrate(1/5*x^(7/2)*arctan(c*x/(sqrt(c*x + 1)*sqr t(-c*x + 1)))/(c^2*x^2 - 1), x) + 4*b^2*c*d^(3/2)*integrate(1/5*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^(5/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2 *x^2 - 1), x) - 1/2*a^2*d^(3/2)*(4*sqrt(x)/c^2 - 2*arctan(sqrt(c)*sqrt(x)) /c^(5/2) + log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(5/2)) - 10* a*b*d^(3/2)*integrate(1/5*x^(3/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1) ))/(c^2*x^2 - 1), x)
Exception generated. \[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((d*x)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d\,x\right )}^{3/2} \,d x \] Input:
int((a + b*asin(c*x))^2*(d*x)^(3/2),x)
Output:
int((a + b*asin(c*x))^2*(d*x)^(3/2), x)
\[ \int (d x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, d \left (2 \sqrt {x}\, a^{2} x^{2}+10 \left (\int \sqrt {x}\, \mathit {asin} \left (c x \right ) x d x \right ) a b +5 \left (\int \sqrt {x}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2}\right )}{5} \] Input:
int((d*x)^(3/2)*(a+b*asin(c*x))^2,x)
Output:
(sqrt(d)*d*(2*sqrt(x)*a**2*x**2 + 10*int(sqrt(x)*asin(c*x)*x,x)*a*b + 5*in t(sqrt(x)*asin(c*x)**2*x,x)*b**2))/5