Integrand size = 26, antiderivative size = 217 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac {2 a (7 b c-4 a d) \sqrt {c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}-\frac {\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^3 \left (a+b x^2\right )^{3/2}}+\frac {2 d \left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^4 \sqrt {a+b x^2}} \]
-1/7*a^2*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)/(b*x^2+a)^(7/2)+2/35*a*(-4*a*d+7*b *c)*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^2/(b*x^2+a)^(5/2)-1/105*(3*a^2*d^2-14*a *b*c*d+35*b^2*c^2)*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^3/(b*x^2+a)^(3/2)+2/105* d*(3*a^2*d^2-14*a*b*c*d+35*b^2*c^2)*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^4/(b*x^ 2+a)^(1/2)
Time = 3.49 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-35 b^3 c^2 x^4 \left (c-2 d x^2\right )+7 a^3 d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )-7 a b^2 c x^2 \left (4 c^2-37 c d x^2+4 d^2 x^4\right )+a^2 b \left (-8 c^3+200 c^2 d x^2-101 c d^2 x^4+6 d^3 x^6\right )\right )}{105 (b c-a d)^4 \left (a+b x^2\right )^{7/2}} \]
(Sqrt[c + d*x^2]*(-35*b^3*c^2*x^4*(c - 2*d*x^2) + 7*a^3*d*(8*c^2 - 4*c*d*x ^2 + 3*d^2*x^4) - 7*a*b^2*c*x^2*(4*c^2 - 37*c*d*x^2 + 4*d^2*x^4) + a^2*b*( -8*c^3 + 200*c^2*d*x^2 - 101*c*d^2*x^4 + 6*d^3*x^6)))/(105*(b*c - a*d)^4*( a + b*x^2)^(7/2))
Time = 0.33 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {354, 100, 27, 87, 55, 48}
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.
| \(\displaystyle \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (b x^2+a\right )^{9/2} \sqrt {d x^2+c}}dx^2\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \int -\frac {a (7 b c-a d)-7 b (b c-a d) x^2}{2 \left (b x^2+a\right )^{7/2} \sqrt {d x^2+c}}dx^2}{7 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {a (7 b c-a d)-7 b (b c-a d) x^2}{\left (b x^2+a\right )^{7/2} \sqrt {d x^2+c}}dx^2}{7 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right ) \int \frac {1}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c}}dx^2}{5 (b c-a d)}-\frac {4 a \sqrt {c+d x^2} (7 b c-4 a d)}{5 \left (a+b x^2\right )^{5/2} (b c-a d)}}{7 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right ) \left (-\frac {2 d \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx^2}{3 (b c-a d)}-\frac {2 \sqrt {c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4 a \sqrt {c+d x^2} (7 b c-4 a d)}{5 \left (a+b x^2\right )^{5/2} (b c-a d)}}{7 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right ) \left (\frac {4 d \sqrt {c+d x^2}}{3 \sqrt {a+b x^2} (b c-a d)^2}-\frac {2 \sqrt {c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4 a \sqrt {c+d x^2} (7 b c-4 a d)}{5 \left (a+b x^2\right )^{5/2} (b c-a d)}}{7 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}\right )\) |
((-2*a^2*Sqrt[c + d*x^2])/(7*b^2*(b*c - a*d)*(a + b*x^2)^(7/2)) - ((-4*a*( 7*b*c - 4*a*d)*Sqrt[c + d*x^2])/(5*(b*c - a*d)*(a + b*x^2)^(5/2)) - ((35*b ^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*((-2*Sqrt[c + d*x^2])/(3*(b*c - a*d)*(a + b*x^2)^(3/2)) + (4*d*Sqrt[c + d*x^2])/(3*(b*c - a*d)^2*Sqrt[a + b*x^2]))) /(5*(b*c - a*d)))/(7*b^2*(b*c - a*d)))/2
3.10.89.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 3.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(\frac {\sqrt {d \,x^{2}+c}\, \left (6 a^{2} b \,d^{3} x^{6}-28 a \,b^{2} c \,d^{2} x^{6}+70 b^{3} c^{2} d \,x^{6}+21 a^{3} d^{3} x^{4}-101 a^{2} b c \,d^{2} x^{4}+259 a \,b^{2} c^{2} d \,x^{4}-35 c^{3} b^{3} x^{4}-28 a^{3} c \,d^{2} x^{2}+200 a^{2} b \,c^{2} d \,x^{2}-28 a \,b^{2} c^{3} x^{2}+56 a^{3} c^{2} d -8 a^{2} b \,c^{3}\right )}{105 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}\) | \(205\) |
| gosper | \(\frac {\sqrt {d \,x^{2}+c}\, \left (6 a^{2} b \,d^{3} x^{6}-28 a \,b^{2} c \,d^{2} x^{6}+70 b^{3} c^{2} d \,x^{6}+21 a^{3} d^{3} x^{4}-101 a^{2} b c \,d^{2} x^{4}+259 a \,b^{2} c^{2} d \,x^{4}-35 c^{3} b^{3} x^{4}-28 a^{3} c \,d^{2} x^{2}+200 a^{2} b \,c^{2} d \,x^{2}-28 a \,b^{2} c^{3} x^{2}+56 a^{3} c^{2} d -8 a^{2} b \,c^{3}\right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(213\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {d \,x^{2}+c}\, \left (6 a^{2} b \,d^{3} x^{6}-28 a \,b^{2} c \,d^{2} x^{6}+70 b^{3} c^{2} d \,x^{6}+21 a^{3} d^{3} x^{4}-101 a^{2} b c \,d^{2} x^{4}+259 a \,b^{2} c^{2} d \,x^{4}-35 c^{3} b^{3} x^{4}-28 a^{3} c \,d^{2} x^{2}+200 a^{2} b \,c^{2} d \,x^{2}-28 a \,b^{2} c^{3} x^{2}+56 a^{3} c^{2} d -8 a^{2} b \,c^{3}\right )}{105 \sqrt {b \,x^{2}+a}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(285\) |
1/105*(d*x^2+c)^(1/2)*(6*a^2*b*d^3*x^6-28*a*b^2*c*d^2*x^6+70*b^3*c^2*d*x^6 +21*a^3*d^3*x^4-101*a^2*b*c*d^2*x^4+259*a*b^2*c^2*d*x^4-35*b^3*c^3*x^4-28* a^3*c*d^2*x^2+200*a^2*b*c^2*d*x^2-28*a*b^2*c^3*x^2+56*a^3*c^2*d-8*a^2*b*c^ 3)/(b*x^2+a)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (193) = 386\).
Time = 0.69 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.08 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {{\left (2 \, {\left (35 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{6} - 8 \, a^{2} b c^{3} + 56 \, a^{3} c^{2} d - {\left (35 \, b^{3} c^{3} - 259 \, a b^{2} c^{2} d + 101 \, a^{2} b c d^{2} - 21 \, a^{3} d^{3}\right )} x^{4} - 4 \, {\left (7 \, a b^{2} c^{3} - 50 \, a^{2} b c^{2} d + 7 \, a^{3} c d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{8} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{6} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{4} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x^{2}\right )}} \]
1/105*(2*(35*b^3*c^2*d - 14*a*b^2*c*d^2 + 3*a^2*b*d^3)*x^6 - 8*a^2*b*c^3 + 56*a^3*c^2*d - (35*b^3*c^3 - 259*a*b^2*c^2*d + 101*a^2*b*c*d^2 - 21*a^3*d ^3)*x^4 - 4*(7*a*b^2*c^3 - 50*a^2*b*c^2*d + 7*a^3*c*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4 *a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4* a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^8 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3* b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^6 + 6*(a^2*b^6*c^4 - 4*a^3* b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^4 + 4*(a^ 3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b* d^4)*x^2)
\[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{5}}{\left (a + b x^{2}\right )^{\frac {9}{2}} \sqrt {c + d x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (193) = 386\).
Time = 0.42 (sec) , antiderivative size = 1036, normalized size of antiderivative = 4.77 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {4 \, {\left (35 \, \sqrt {b d} b^{10} c^{5} d - 119 \, \sqrt {b d} a b^{9} c^{4} d^{2} + 150 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{3} - 86 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{4} + 23 \, \sqrt {b d} a^{4} b^{6} c d^{5} - 3 \, \sqrt {b d} a^{5} b^{5} d^{6} - 245 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{8} c^{4} d + 588 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{3} d^{2} - 462 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c^{2} d^{3} + 140 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} c d^{4} - 21 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{4} b^{4} d^{5} + 630 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{6} c^{3} d - 714 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{5} c^{2} d^{2} + 42 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} c d^{3} + 42 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{3} b^{3} d^{4} - 770 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{4} c^{2} d + 140 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a b^{3} c d^{2} - 210 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a^{2} b^{2} d^{3} + 455 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{8} b^{2} c d + 105 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{8} a b d^{2} - 105 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{10} d\right )}}{105 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{7} {\left | b \right |}} \]
4/105*(35*sqrt(b*d)*b^10*c^5*d - 119*sqrt(b*d)*a*b^9*c^4*d^2 + 150*sqrt(b* d)*a^2*b^8*c^3*d^3 - 86*sqrt(b*d)*a^3*b^7*c^2*d^4 + 23*sqrt(b*d)*a^4*b^6*c *d^5 - 3*sqrt(b*d)*a^5*b^5*d^6 - 245*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*b^8*c^4*d + 588*sqrt(b*d)*(sqrt (b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*a*b^7*c^3 *d^2 - 462*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a) *b*d - a*b*d))^2*a^2*b^6*c^2*d^3 + 140*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d ) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*a^3*b^5*c*d^4 - 21*sqrt(b*d)* (sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*a^4* b^4*d^5 + 630*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*b^6*c^3*d - 714*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a*b^5*c^2*d^2 + 42*sqrt(b*d)*(sq rt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a^2*b^4 *c*d^3 + 42*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a )*b*d - a*b*d))^4*a^3*b^3*d^4 - 770*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^6*b^4*c^2*d + 140*sqrt(b*d)*(sqrt( b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^6*a*b^3*c*d^ 2 - 210*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b* d - a*b*d))^6*a^2*b^2*d^3 + 455*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqr t(b^2*c + (b*x^2 + a)*b*d - a*b*d))^8*b^2*c*d + 105*sqrt(b*d)*(sqrt(b*x...
Time = 6.46 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.55 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b\,x^2+a}\,\left (\frac {x^6\,\left (21\,a^3\,d^4-95\,a^2\,b\,c\,d^3+231\,a\,b^2\,c^2\,d^2+35\,b^3\,c^3\,d\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}-\frac {x^4\,\left (7\,a^3\,c\,d^3-99\,a^2\,b\,c^2\,d^2-231\,a\,b^2\,c^3\,d+35\,b^3\,c^4\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {8\,a^2\,c^3\,\left (7\,a\,d-b\,c\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,d^2\,x^8\,\left (3\,a^2\,d^2-14\,a\,b\,c\,d+35\,b^2\,c^2\right )}{105\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,a\,c^2\,x^2\,\left (7\,a^2\,d^2+48\,a\,b\,c\,d-7\,b^2\,c^2\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^8\,\sqrt {d\,x^2+c}+\frac {a^4\,\sqrt {d\,x^2+c}}{b^4}+\frac {4\,a\,x^6\,\sqrt {d\,x^2+c}}{b}+\frac {6\,a^2\,x^4\,\sqrt {d\,x^2+c}}{b^2}+\frac {4\,a^3\,x^2\,\sqrt {d\,x^2+c}}{b^3}} \]
((a + b*x^2)^(1/2)*((x^6*(21*a^3*d^4 + 35*b^3*c^3*d + 231*a*b^2*c^2*d^2 - 95*a^2*b*c*d^3))/(105*b^4*(a*d - b*c)^4) - (x^4*(35*b^3*c^4 + 7*a^3*c*d^3 - 99*a^2*b*c^2*d^2 - 231*a*b^2*c^3*d))/(105*b^4*(a*d - b*c)^4) + (8*a^2*c^ 3*(7*a*d - b*c))/(105*b^4*(a*d - b*c)^4) + (2*d^2*x^8*(3*a^2*d^2 + 35*b^2* c^2 - 14*a*b*c*d))/(105*b^3*(a*d - b*c)^4) + (4*a*c^2*x^2*(7*a^2*d^2 - 7*b ^2*c^2 + 48*a*b*c*d))/(105*b^4*(a*d - b*c)^4)))/(x^8*(c + d*x^2)^(1/2) + ( a^4*(c + d*x^2)^(1/2))/b^4 + (4*a*x^6*(c + d*x^2)^(1/2))/b + (6*a^2*x^4*(c + d*x^2)^(1/2))/b^2 + (4*a^3*x^2*(c + d*x^2)^(1/2))/b^3)