Integrand size = 25, antiderivative size = 131 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=-\frac {2 a^2 (5+4 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}} \]
-2*a^2*(5+4*n)*cos(f*x+e)*hypergeom([1/2, -n],[3/2],1-sin(f*x+e))*(d*sin(f *x+e))^n/f/(3+2*n)/(sin(f*x+e)^n)/(a+a*sin(f*x+e))^(1/2)-2*a^2*cos(f*x+e)* (d*sin(f*x+e))^(1+n)/d/f/(3+2*n)/(a+a*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 6.53 (sec) , antiderivative size = 5131, normalized size of antiderivative = 39.17 \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=\text {Result too large to show} \]
Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3242, 27, 2011, 3042, 3255, 77, 75}
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.
\(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (d \sin (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (d \sin (e+f x))^ndx\) |
\(\Big \downarrow \) 3242 |
\(\displaystyle \frac {2 \int \frac {(d \sin (e+f x))^n \left (d (4 n+5) a^2+d (4 n+5) \sin (e+f x) a^2\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{d (2 n+3)}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(d \sin (e+f x))^n \left (d (4 n+5) a^2+d (4 n+5) \sin (e+f x) a^2\right )}{\sqrt {\sin (e+f x) a+a}}dx}{d (2 n+3)}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 2011 |
\(\displaystyle \frac {a (4 n+5) \int (d \sin (e+f x))^n \sqrt {\sin (e+f x) a+a}dx}{2 n+3}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a (4 n+5) \int (d \sin (e+f x))^n \sqrt {\sin (e+f x) a+a}dx}{2 n+3}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 3255 |
\(\displaystyle \frac {a^3 (4 n+5) \cos (e+f x) \int \frac {(d \sin (e+f x))^n}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f (2 n+3) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 77 |
\(\displaystyle \frac {a^3 (4 n+5) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \int \frac {\sin ^n(e+f x)}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f (2 n+3) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {2 a^2 (4 n+5) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\) |
(-2*a^2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(f*(3 + 2*n)*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt [a + a*Sin[e + f*x]])
3.2.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-b)*(c/ d))^IntPart[m]*((b*x)^FracPart[m]/((-d)*(x/c))^FracPart[m]) Int[((-d)*(x/ c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0]
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(c + d*x)^n/Sqrt[a - b*x], x] , x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[2*n]
\[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (d \sin {\left (e + f x \right )}\right )^{n}\, dx \]
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (d \sin \left (f x + e\right )\right )^{n} \,d x } \]
Timed out. \[ \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]