Integrand size = 26, antiderivative size = 137 \[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{3 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {2 a (3 b c-2 a d) \sqrt {c+d x^2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{b^{5/2} \sqrt {d}} \]
arctanh(d^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/(d*x^2+c)^(1/2))/b^(5/2)/d^(1/2)-1 /3*a^2*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)/(b*x^2+a)^(3/2)+2/3*a*(-2*a*d+3*b*c) *(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^2/(b*x^2+a)^(1/2)
Time = 2.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {a \sqrt {c+d x^2} \left (-6 b c+3 a d+\frac {a b \left (c+d x^2\right )}{a+b x^2}\right )}{3 b^2 (b c-a d)^2 \sqrt {a+b x^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{b^{5/2} \sqrt {d}} \]
-1/3*(a*Sqrt[c + d*x^2]*(-6*b*c + 3*a*d + (a*b*(c + d*x^2))/(a + b*x^2)))/ (b^2*(b*c - a*d)^2*Sqrt[a + b*x^2]) + ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/(S qrt[d]*Sqrt[a + b*x^2])]/(b^(5/2)*Sqrt[d])
Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19, 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, 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.
| \(\displaystyle \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c}}dx^2\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \int -\frac {a (3 b c-a d)-3 b (b c-a d) x^2}{2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx^2}{3 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {a (3 b c-a d)-3 b (b c-a d) x^2}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx^2}{3 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-3 (b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2-\frac {4 a \sqrt {c+d x^2} (3 b c-2 a d)}{\sqrt {a+b x^2} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-6 (b c-a d) \int \frac {1}{b-d x^4}d\frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}-\frac {4 a \sqrt {c+d x^2} (3 b c-2 a d)}{\sqrt {a+b x^2} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 \sqrt {c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 a^2 \sqrt {c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac {-\frac {6 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}}-\frac {4 a \sqrt {c+d x^2} (3 b c-2 a d)}{\sqrt {a+b x^2} (b c-a d)}}{3 b^2 (b c-a d)}\right )\) |
((-2*a^2*Sqrt[c + d*x^2])/(3*b^2*(b*c - a*d)*(a + b*x^2)^(3/2)) - ((-4*a*( 3*b*c - 2*a*d)*Sqrt[c + d*x^2])/((b*c - a*d)*Sqrt[a + b*x^2]) - (6*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(Sqrt[b ]*Sqrt[d]))/(3*b^2*(b*c - a*d)))/2
3.10.83.3.1 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_)]*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*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[((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_)^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]
Leaf count of result is larger than twice the leaf count of optimal. \(284\) vs. \(2(113)=226\).
Time = 3.31 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.08
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 b^{2} \sqrt {b d}}-\frac {a^{2} \sqrt {b d \left (x^{2}+\frac {a}{b}\right )^{2}+\left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}}{3 b^{4} \left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {2 a^{2} d \sqrt {b d \left (x^{2}+\frac {a}{b}\right )^{2}+\left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}}{3 b^{3} \left (-a d +b c \right )^{2} \left (x^{2}+\frac {a}{b}\right )}+\frac {2 a \sqrt {b d \left (x^{2}+\frac {a}{b}\right )^{2}+\left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}}{b^{3} \left (-a d +b c \right ) \left (x^{2}+\frac {a}{b}\right )}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(285\) |
| default | \(\frac {\left (-8 \sqrt {b d}\, a^{2} b \,d^{2} x^{4}+12 \sqrt {b d}\, a \,b^{2} c d \,x^{4}+3 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,d^{2} x^{2}-6 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c d \,x^{2}+3 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{2}-6 \sqrt {b d}\, a^{3} d^{2} x^{2}+2 \sqrt {b d}\, a^{2} b c d \,x^{2}+12 \sqrt {b d}\, a \,b^{2} c^{2} x^{2}+3 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{2}-6 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c d +3 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2}-6 \sqrt {b d}\, a^{3} c d +10 \sqrt {b d}\, a^{2} b \,c^{2}\right ) \sqrt {d \,x^{2}+c}}{6 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {b d}\, \left (a d -b c \right )^{2} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right )}\) | \(609\) |
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/2/b^2*ln((1 /2*a*d+1/2*b*c+b*d*x^2)/(b*d)^(1/2)+(b*d*x^4+(a*d+b*c)*x^2+a*c)^(1/2))/(b* d)^(1/2)-1/3*a^2/b^4/(-a*d+b*c)/(x^2+a/b)^2*(b*d*(x^2+a/b)^2+(-a*d+b*c)*(x ^2+a/b))^(1/2)+2/3*a^2/b^3*d/(-a*d+b*c)^2/(x^2+a/b)*(b*d*(x^2+a/b)^2+(-a*d +b*c)*(x^2+a/b))^(1/2)+2*a/b^3/(-a*d+b*c)/(x^2+a/b)*(b*d*(x^2+a/b)^2+(-a*d +b*c)*(x^2+a/b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (113) = 226\).
Time = 0.37 (sec) , antiderivative size = 706, normalized size of antiderivative = 5.15 \[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) + 4 \, {\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \, {\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{12 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} + {\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{4} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \, {\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{6 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} + {\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{4} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x^{2}\right )}}\right ] \]
[1/12*(3*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a ^2*b^2*d^2)*x^4 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2)*sqrt(b*d) *log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2) *x^2 + 4*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d) ) + 4*(5*a^2*b^2*c*d - 3*a^3*b*d^2 + 2*(3*a*b^3*c*d - 2*a^2*b^2*d^2)*x^2)* sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^ 3*d^3 + (b^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*x^4 + 2*(a*b^6*c^2*d - 2 *a^2*b^5*c*d^2 + a^3*b^4*d^3)*x^2), -1/6*(3*(a^2*b^2*c^2 - 2*a^3*b*c*d + a ^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^4 + 2*(a*b^3*c^2 - 2*a^2* b^2*c*d + a^3*b*d^2)*x^2)*sqrt(-b*d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sq rt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d)/(b^2*d^2*x^4 + a*b*c*d + (b^2*c*d + a*b*d^2)*x^2)) - 2*(5*a^2*b^2*c*d - 3*a^3*b*d^2 + 2*(3*a*b^3*c*d - 2*a^ 2*b^2*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^2*b^5*c^2*d - 2*a^3*b^ 4*c*d^2 + a^4*b^3*d^3 + (b^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*x^4 + 2* (a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*x^2)]
\[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{5}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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. 312 vs. \(2 (113) = 226\).
Time = 0.37 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.28 \[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {\log \left ({\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{2 \, \sqrt {b d} b {\left | b \right |}} + \frac {4 \, {\left (3 \, a b^{4} c^{2} d - 5 \, a^{2} b^{3} c d^{2} + 2 \, a^{3} b^{2} d^{3} - 6 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{2} c d + 3 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b d^{2} + 3 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a d\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{3} \sqrt {b d} {\left | b \right |}} \]
-1/2*log((sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d ))^2)/(sqrt(b*d)*b*abs(b)) + 4/3*(3*a*b^4*c^2*d - 5*a^2*b^3*c*d^2 + 2*a^3* b^2*d^3 - 6*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a* b*d))^2*a*b^2*c*d + 3*(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*d^2 + 3*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a*d)/((b^2*c - a*b*d - (sqrt(b*x^2 + a)*sqrt(b *d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)^3*sqrt(b*d)*abs(b))
Timed out. \[ \int \frac {x^5}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^5}{{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {d\,x^2+c}} \,d x \]