Integrand size = 23, antiderivative size = 136 \[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\frac {20 b e^2 \sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}{147 d}+\frac {4 b (e (c+d x))^{5/2} \sqrt {1-(c+d x)^2}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \arcsin (c+d x))}{7 d e}-\frac {20 b e^{5/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{147 d} \]
2/7*(e*(d*x+c))^(7/2)*(a+b*arcsin(d*x+c))/d/e-20/147*b*e^(5/2)*EllipticF(( e*(d*x+c))^(1/2)/e^(1/2),I)/d+4/49*b*(e*(d*x+c))^(5/2)*(1-(d*x+c)^2)^(1/2) /d+20/147*b*e^2*(e*(d*x+c))^(1/2)*(1-(d*x+c)^2)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.85 \[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\frac {2 (e (c+d x))^{5/2} \left (21 a (c+d x)^3+10 b \sqrt {1-(c+d x)^2}+6 b (c+d x)^2 \sqrt {1-(c+d x)^2}+21 b (c+d x)^3 \arcsin (c+d x)-10 b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{147 d (c+d x)^2} \]
(2*(e*(c + d*x))^(5/2)*(21*a*(c + d*x)^3 + 10*b*Sqrt[1 - (c + d*x)^2] + 6* b*(c + d*x)^2*Sqrt[1 - (c + d*x)^2] + 21*b*(c + d*x)^3*ArcSin[c + d*x] - 1 0*b*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(147*d*(c + d*x)^2)
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5304, 5138, 262, 262, 266, 762}
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.
\(\displaystyle \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx\) |
\(\Big \downarrow \) 5304 |
\(\displaystyle \frac {\int (e (c+d x))^{5/2} (a+b \arcsin (c+d x))d(c+d x)}{d}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \arcsin (c+d x))}{7 e}-\frac {2 b \int \frac {(e (c+d x))^{7/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)}{7 e}}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \arcsin (c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \int \frac {(e (c+d x))^{3/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {2}{7} e \sqrt {1-(c+d x)^2} (e (c+d x))^{5/2}\right )}{7 e}}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \arcsin (c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {2}{3} e \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}\right )-\frac {2}{7} e \sqrt {1-(c+d x)^2} (e (c+d x))^{5/2}\right )}{7 e}}{d}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \arcsin (c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {2}{3} e \int \frac {1}{\sqrt {1-(c+d x)^2}}d\sqrt {e (c+d x)}-\frac {2}{3} e \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}\right )-\frac {2}{7} e \sqrt {1-(c+d x)^2} (e (c+d x))^{5/2}\right )}{7 e}}{d}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \arcsin (c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {2}{3} e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )-\frac {2}{3} e \sqrt {1-(c+d x)^2} \sqrt {e (c+d x)}\right )-\frac {2}{7} e \sqrt {1-(c+d x)^2} (e (c+d x))^{5/2}\right )}{7 e}}{d}\) |
((2*(e*(c + d*x))^(7/2)*(a + b*ArcSin[c + d*x]))/(7*e) - (2*b*((-2*e*(e*(c + d*x))^(5/2)*Sqrt[1 - (c + d*x)^2])/7 + (5*e^2*((-2*e*Sqrt[e*(c + d*x)]* Sqrt[1 - (c + d*x)^2])/3 + (2*e^(3/2)*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/S qrt[e]], -1])/3))/7))/(7*e))/d
3.3.82.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
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_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 2.88 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.51
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{21 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{d e}\) | \(206\) |
default | \(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{21 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{d e}\) | \(206\) |
parts | \(\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{21 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{e d}\) | \(211\) |
2/d/e*(1/7*a*(d*e*x+c*e)^(7/2)+b*(1/7*(d*e*x+c*e)^(7/2)*arcsin(1/e*(d*e*x+ c*e))-2/7/e*(-1/7*e^2*(d*e*x+c*e)^(5/2)*(-1/e^2*(d*e*x+c*e)^2+1)^(1/2)-5/2 1*e^4*(d*e*x+c*e)^(1/2)*(-1/e^2*(d*e*x+c*e)^2+1)^(1/2)+5/21*e^4/(1/e)^(1/2 )*(1-1/e*(d*e*x+c*e))^(1/2)*(1+1/e*(d*e*x+c*e))^(1/2)/(-1/e^2*(d*e*x+c*e)^ 2+1)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(1/e)^(1/2),I))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.62 \[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\frac {2 \, {\left (10 \, \sqrt {-d^{3} e} b e^{2} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (21 \, a d^{5} e^{2} x^{3} + 63 \, a c d^{4} e^{2} x^{2} + 63 \, a c^{2} d^{3} e^{2} x + 21 \, a c^{3} d^{2} e^{2} + 21 \, {\left (b d^{5} e^{2} x^{3} + 3 \, b c d^{4} e^{2} x^{2} + 3 \, b c^{2} d^{3} e^{2} x + b c^{3} d^{2} e^{2}\right )} \arcsin \left (d x + c\right ) + 2 \, {\left (3 \, b d^{4} e^{2} x^{2} + 6 \, b c d^{3} e^{2} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2} e^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d e x + c e}\right )}}{147 \, d^{3}} \]
2/147*(10*sqrt(-d^3*e)*b*e^2*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) + (21*a*d^5*e^2*x^3 + 63*a*c*d^4*e^2*x^2 + 63*a*c^2*d^3*e^2*x + 21*a*c^3*d^2 *e^2 + 21*(b*d^5*e^2*x^3 + 3*b*c*d^4*e^2*x^2 + 3*b*c^2*d^3*e^2*x + b*c^3*d ^2*e^2)*arcsin(d*x + c) + 2*(3*b*d^4*e^2*x^2 + 6*b*c*d^3*e^2*x + (3*b*c^2 + 5*b)*d^2*e^2)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))*sqrt(d*e*x + c*e))/d^3
Timed out. \[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\text {Timed out} \]
Exception generated. \[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\int { {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (c e+d e x)^{5/2} (a+b \arcsin (c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^{5/2}\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]