Integrand size = 14, antiderivative size = 86 \[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\frac {10 b x \sqrt {1-c^2 x^4}}{147 c^3}+\frac {2 b x^5 \sqrt {1-c^2 x^4}}{49 c}+\frac {1}{7} x^7 \left (a+b \arcsin \left (c x^2\right )\right )-\frac {10 b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {c} x\right ),-1\right )}{147 c^{7/2}} \]
1/7*x^7*(a+b*arcsin(c*x^2))-10/147*b*EllipticF(x*c^(1/2),I)/c^(7/2)+10/147 *b*x*(-c^2*x^4+1)^(1/2)/c^3+2/49*b*x^5*(-c^2*x^4+1)^(1/2)/c
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\frac {1}{147} \left (21 a x^7+\frac {2 b x \sqrt {1-c^2 x^4} \left (5+3 c^2 x^4\right )}{c^3}+21 b x^7 \arcsin \left (c x^2\right )-\frac {10 i b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c} x\right ),-1\right )}{(-c)^{7/2}}\right ) \]
(21*a*x^7 + (2*b*x*Sqrt[1 - c^2*x^4]*(5 + 3*c^2*x^4))/c^3 + 21*b*x^7*ArcSi n[c*x^2] - ((10*I)*b*EllipticF[I*ArcSinh[Sqrt[-c]*x], -1])/(-c)^(7/2))/147
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5341, 27, 843, 843, 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.
\(\displaystyle \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 5341 |
\(\displaystyle \frac {1}{7} x^7 \left (a+b \arcsin \left (c x^2\right )\right )-\frac {1}{7} b \int \frac {2 c x^8}{\sqrt {1-c^2 x^4}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} x^7 \left (a+b \arcsin \left (c x^2\right )\right )-\frac {2}{7} b c \int \frac {x^8}{\sqrt {1-c^2 x^4}}dx\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{7} x^7 \left (a+b \arcsin \left (c x^2\right )\right )-\frac {2}{7} b c \left (\frac {5 \int \frac {x^4}{\sqrt {1-c^2 x^4}}dx}{7 c^2}-\frac {x^5 \sqrt {1-c^2 x^4}}{7 c^2}\right )\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{7} x^7 \left (a+b \arcsin \left (c x^2\right )\right )-\frac {2}{7} b c \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^4}}dx}{3 c^2}-\frac {x \sqrt {1-c^2 x^4}}{3 c^2}\right )}{7 c^2}-\frac {x^5 \sqrt {1-c^2 x^4}}{7 c^2}\right )\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {1}{7} x^7 \left (a+b \arcsin \left (c x^2\right )\right )-\frac {2}{7} b c \left (\frac {5 \left (\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {c} x\right ),-1\right )}{3 c^{5/2}}-\frac {x \sqrt {1-c^2 x^4}}{3 c^2}\right )}{7 c^2}-\frac {x^5 \sqrt {1-c^2 x^4}}{7 c^2}\right )\) |
(x^7*(a + b*ArcSin[c*x^2]))/7 - (2*b*c*(-1/7*(x^5*Sqrt[1 - c^2*x^4])/c^2 + (5*(-1/3*(x*Sqrt[1 - c^2*x^4])/c^2 + EllipticF[ArcSin[Sqrt[c]*x], -1]/(3* c^(5/2))))/(7*c^2)))/7
3.4.52.3.1 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[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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(c + d*x)^(m + 1)*((a + b*ArcSin[u])/(d*(m + 1))), x] - Simp[b/(d*(m + 1) ) Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/Sqrt[1 - u^2]), x], x] , x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
Time = 0.94 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {x^{7} a}{7}+b \left (\frac {x^{7} \arcsin \left (c \,x^{2}\right )}{7}-\frac {2 c \left (-\frac {x^{5} \sqrt {-c^{2} x^{4}+1}}{7 c^{2}}-\frac {5 x \sqrt {-c^{2} x^{4}+1}}{21 c^{4}}+\frac {5 \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {c}, i\right )}{21 c^{\frac {9}{2}} \sqrt {-c^{2} x^{4}+1}}\right )}{7}\right )\) | \(108\) |
parts | \(\frac {x^{7} a}{7}+b \left (\frac {x^{7} \arcsin \left (c \,x^{2}\right )}{7}-\frac {2 c \left (-\frac {x^{5} \sqrt {-c^{2} x^{4}+1}}{7 c^{2}}-\frac {5 x \sqrt {-c^{2} x^{4}+1}}{21 c^{4}}+\frac {5 \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {c}, i\right )}{21 c^{\frac {9}{2}} \sqrt {-c^{2} x^{4}+1}}\right )}{7}\right )\) | \(108\) |
1/7*x^7*a+b*(1/7*x^7*arcsin(c*x^2)-2/7*c*(-1/7/c^2*x^5*(-c^2*x^4+1)^(1/2)- 5/21/c^4*x*(-c^2*x^4+1)^(1/2)+5/21/c^(9/2)*(-c*x^2+1)^(1/2)*(c*x^2+1)^(1/2 )/(-c^2*x^4+1)^(1/2)*EllipticF(x*c^(1/2),I)))
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\frac {21 \, b c^{3} x^{7} \arcsin \left (c x^{2}\right ) + 21 \, a c^{3} x^{7} + 2 \, {\left (3 \, b c^{2} x^{5} + 5 \, b x\right )} \sqrt {-c^{2} x^{4} + 1}}{147 \, c^{3}} \]
1/147*(21*b*c^3*x^7*arcsin(c*x^2) + 21*a*c^3*x^7 + 2*(3*b*c^2*x^5 + 5*b*x) *sqrt(-c^2*x^4 + 1))/c^3
Time = 1.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\frac {a x^{7}}{7} - \frac {b c x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{14 \Gamma \left (\frac {13}{4}\right )} + \frac {b x^{7} \operatorname {asin}{\left (c x^{2} \right )}}{7} \]
a*x**7/7 - b*c*x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), c**2*x**4*exp_po lar(2*I*pi))/(14*gamma(13/4)) + b*x**7*asin(c*x**2)/7
\[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\int { {\left (b \arcsin \left (c x^{2}\right ) + a\right )} x^{6} \,d x } \]
1/7*a*x^7 + 1/7*(x^7*arctan2(c*x^2, sqrt(c*x^2 + 1)*sqrt(-c*x^2 + 1)) + 14 *c*integrate(1/7*x^8*e^(1/2*log(c*x^2 + 1) + 1/2*log(-c*x^2 + 1))/(c^4*x^8 - c^2*x^4 + (c^2*x^4 - 1)*e^(log(c*x^2 + 1) + log(-c*x^2 + 1))), x))*b
\[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\int { {\left (b \arcsin \left (c x^{2}\right ) + a\right )} x^{6} \,d x } \]
Timed out. \[ \int x^6 \left (a+b \arcsin \left (c x^2\right )\right ) \, dx=\int x^6\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \]