Integrand size = 30, antiderivative size = 291 \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{5/2} d^{9/2} n} \]
1/64*(-a*d+b*c)^2*(3*a^2*d^2+10*a*b*c*d+35*b^2*c^2)*arctanh(d^(1/2)*(a+b*x ^n)^(1/2)/b^(1/2)/(c+d*x^n)^(1/2))/b^(5/2)/d^(9/2)/n+1/96*(3*a^2*d^2+10*a* b*c*d+35*b^2*c^2)*(a+b*x^n)^(3/2)*(c+d*x^n)^(1/2)/b^2/d^3/n-1/24*(3*a*d+7* b*c)*(a+b*x^n)^(5/2)*(c+d*x^n)^(1/2)/b^2/d^2/n+1/4*x^n*(a+b*x^n)^(5/2)*(c+ d*x^n)^(1/2)/b/d/n-1/64*(-a*d+b*c)*(3*a^2*d^2+10*a*b*c*d+35*b^2*c^2)*(a+b* x^n)^(1/2)*(c+d*x^n)^(1/2)/b^2/d^4/n
Time = 1.24 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.83 \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=\frac {-b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (9 a^3 d^3+3 a^2 b d^2 \left (5 c-2 d x^n\right )-a b^2 d \left (145 c^2-92 c d x^n+72 d^2 x^{2 n}\right )+b^3 \left (105 c^3-70 c^2 d x^n+56 c d^2 x^{2 n}-48 d^3 x^{3 n}\right )\right )+3 (b c-a d)^{5/2} \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{192 b^3 d^{9/2} n \sqrt {c+d x^n}} \]
(-(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(9*a^3*d^3 + 3*a^2*b*d^2*(5*c - 2 *d*x^n) - a*b^2*d*(145*c^2 - 92*c*d*x^n + 72*d^2*x^(2*n)) + b^3*(105*c^3 - 70*c^2*d*x^n + 56*c*d^2*x^(2*n) - 48*d^3*x^(3*n)))) + 3*(b*c - a*d)^(5/2) *(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*A rcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(192*b^3*d^(9/2)*n*Sqrt [c + d*x^n])
Time = 0.34 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {948, 101, 27, 90, 60, 60, 66, 221}
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.
| \(\displaystyle \int \frac {x^{3 n-1} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {\int \frac {x^{2 n} \left (b x^n+a\right )^{3/2}}{\sqrt {d x^n+c}}dx^n}{n}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\frac {\int -\frac {\left (b x^n+a\right )^{3/2} \left ((7 b c+3 a d) x^n+2 a c\right )}{2 \sqrt {d x^n+c}}dx^n}{4 b d}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}-\frac {\int \frac {\left (b x^n+a\right )^{3/2} \left ((7 b c+3 a d) x^n+2 a c\right )}{\sqrt {d x^n+c}}dx^n}{8 b d}}{n}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}-\frac {\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d}-\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \int \frac {\left (b x^n+a\right )^{3/2}}{\sqrt {d x^n+c}}dx^n}{6 b d}}{8 b d}}{n}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}-\frac {\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d}-\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {b x^n+a}}{\sqrt {d x^n+c}}dx^n}{4 d}\right )}{6 b d}}{8 b d}}{n}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}-\frac {\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d}-\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{2 d}\right )}{4 d}\right )}{6 b d}}{8 b d}}{n}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}-\frac {\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d}-\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{d}-\frac {(b c-a d) \int \frac {1}{b-d x^{2 n}}d\frac {\sqrt {b x^n+a}}{\sqrt {d x^n+c}}}{d}\right )}{4 d}\right )}{6 b d}}{8 b d}}{n}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d}-\frac {\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{3 b d}-\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 b d}}{8 b d}}{n}\) |
((x^n*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(4*b*d) - (((7*b*c + 3*a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(3*b*d) - ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2* d^2)*(((a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n ])/(Sqrt[b]*Sqrt[c + d*x^n])])/(Sqrt[b]*d^(3/2))))/(4*d)))/(6*b*d))/(8*b*d ))/n
3.11.73.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
\[\int \frac {x^{-1+3 n} \left (a +b \,x^{n}\right )^{\frac {3}{2}}}{\sqrt {c +d \,x^{n}}}d x\]
Time = 0.32 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.09 \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{768 \, b^{3} d^{5} n}, -\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{384 \, b^{3} d^{5} n}\right ] \]
[1/768*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^ 3 + 3*a^4*d^4)*sqrt(b*d)*log(8*b^2*d^2*x^(2*n) + b^2*c^2 + 6*a*b*c*d + a^2 *d^2 + 4*(2*sqrt(b*d)*b*d*x^n + (b*c + a*d)*sqrt(b*d))*sqrt(b*x^n + a)*sqr t(d*x^n + c) + 8*(b^2*c*d + a*b*d^2)*x^n) + 4*(48*b^4*d^4*x^(3*n) - 105*b^ 4*c^3*d + 145*a*b^3*c^2*d^2 - 15*a^2*b^2*c*d^3 - 9*a^3*b*d^4 - 8*(7*b^4*c* d^3 - 9*a*b^3*d^4)*x^(2*n) + 2*(35*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 3*a^2*b^ 2*d^4)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^3*d^5*n), -1/384*(3*(35*b^ 4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*s qrt(-b*d)*arctan(1/2*(2*sqrt(-b*d)*b*d*x^n + (b*c + a*d)*sqrt(-b*d))*sqrt( b*x^n + a)*sqrt(d*x^n + c)/(b^2*d^2*x^(2*n) + a*b*c*d + (b^2*c*d + a*b*d^2 )*x^n)) - 2*(48*b^4*d^4*x^(3*n) - 105*b^4*c^3*d + 145*a*b^3*c^2*d^2 - 15*a ^2*b^2*c*d^3 - 9*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 9*a*b^3*d^4)*x^(2*n) + 2*(35 *b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^n)*sqrt(b*x^n + a)*sqrt(d *x^n + c))/(b^3*d^5*n)]
Timed out. \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=\text {Timed out} \]
\[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}} \,d x } \]
\[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}} \,d x } \]
Timed out. \[ \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx=\int \frac {x^{3\,n-1}\,{\left (a+b\,x^n\right )}^{3/2}}{\sqrt {c+d\,x^n}} \,d x \]