Integrand size = 26, antiderivative size = 338 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {(9 b c+11 a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}}{10 b^2 d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}} \] Output:
1/256*(-a*d+b*c)^2*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^ 2+c)^(1/2)/b^2/d^5-1/384*(-a*d+b*c)*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x ^2+a)^(3/2)*(d*x^2+c)^(1/2)/b^2/d^4+1/480*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2 )*(b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)/b^2/d^3-1/80*(11*a*d+9*b*c)*(b*x^2+a)^(7 /2)*(d*x^2+c)^(1/2)/b^2/d^2+1/10*(b*x^2+a)^(9/2)*(d*x^2+c)^(1/2)/b^2/d-1/2 56*(-a*d+b*c)^3*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*arctanh(d^(1/2)*(b*x^2+a )^(1/2)/b^(1/2)/(d*x^2+c)^(1/2))/b^(5/2)/d^(11/2)
Time = 3.86 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.80 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-\frac {24 (3 b c+a d) \left (a+b x^2\right )^4}{b d}+64 x^2 \left (a+b x^2\right )^4+\frac {5 (b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (\frac {2 d \left (a+b x^2\right )}{b c-a d}-\frac {4 d^2 \left (a+b x^2\right )^2}{3 (b c-a d)^2}+\frac {16 d^3 \left (a+b x^2\right )^3}{15 (b c-a d)^3}-\frac {2 \sqrt {d} \sqrt {a+b x^2} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}}\right )}{4 b d^5}\right )}{640 b d \sqrt {a+b x^2}} \] Input:
Integrate[(x^5*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]
Output:
(Sqrt[c + d*x^2]*((-24*(3*b*c + a*d)*(a + b*x^2)^4)/(b*d) + 64*x^2*(a + b* x^2)^4 + (5*(b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*((2*d*(a + b*x^2))/(b*c - a*d) - (4*d^2*(a + b*x^2)^2)/(3*(b*c - a*d)^2) + (16*d^3*( a + b*x^2)^3)/(15*(b*c - a*d)^3) - (2*Sqrt[d]*Sqrt[a + b*x^2]*ArcSinh[(Sqr t[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^ 2))/(b*c - a*d)])))/(4*b*d^5)))/(640*b*d*Sqrt[a + b*x^2])
Time = 0.35 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {354, 101, 27, 90, 60, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \left (b x^2+a\right )^{5/2}}{\sqrt {d x^2+c}}dx^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\left (b x^2+a\right )^{5/2} \left (3 (3 b c+a d) x^2+2 a c\right )}{2 \sqrt {d x^2+c}}dx^2}{5 b d}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\int \frac {\left (b x^2+a\right )^{5/2} \left (3 (3 b c+a d) x^2+2 a c\right )}{\sqrt {d x^2+c}}dx^2}{10 b d}\right )\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \int \frac {\left (b x^2+a\right )^{5/2}}{\sqrt {d x^2+c}}dx^2}{8 b d}}{10 b d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 d}-\frac {5 (b c-a d) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c}}dx^2}{6 d}\right )}{8 b d}}{10 b d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}dx^2}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx^2}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\right )\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {(b c-a d) \int \frac {1}{b-d x^4}d\frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{3 d}-\frac {5 (b c-a d) \left (\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\right )\) |
Input:
Int[(x^5*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]
Output:
((x^2*(a + b*x^2)^(7/2)*Sqrt[c + d*x^2])/(5*b*d) - ((3*(3*b*c + a*d)*(a + b*x^2)^(7/2)*Sqrt[c + d*x^2])/(4*b*d) - ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2* d^2)*(((a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(3*d) - (5*(b*c - a*d)*(((a + b* x^2)^(3/2)*Sqrt[c + d*x^2])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x^2]*Sqrt[ c + d*x^2])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sq rt[c + d*x^2])])/(Sqrt[b]*d^(3/2))))/(4*d)))/(6*d)))/(8*b*d))/(10*b*d))/2
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_)^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 = 0.64 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(-\frac {\left (-384 b^{4} x^{8} d^{4}-1008 a \,b^{3} d^{4} x^{6}+432 b^{4} c \,d^{3} x^{6}-744 a^{2} b^{2} d^{4} x^{4}+1184 a \,b^{3} c \,d^{3} x^{4}-504 b^{4} c^{2} d^{2} x^{4}-30 a^{3} b \,d^{4} x^{2}+962 a^{2} b^{2} c \,d^{3} x^{2}-1498 a \,b^{3} c^{2} d^{2} x^{2}+630 b^{4} c^{3} d \,x^{2}+45 d^{4} a^{4}+90 a^{3} b c \,d^{3}-1564 a^{2} b^{2} c^{2} d^{2}+2310 a \,b^{3} c^{3} d -945 c^{4} b^{4}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3840 b^{2} d^{5}}+\frac {\left (3 a^{5} d^{5}+5 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}-150 a^{2} b^{3} c^{3} d^{2}+175 a \,b^{4} c^{4} d -63 c^{5} b^{5}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{512 b^{2} d^{5} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(368\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (768 b^{4} d^{4} x^{8} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+2016 a \,b^{3} d^{4} x^{6} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}-864 b^{4} c \,d^{3} x^{6} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+1488 a^{2} b^{2} d^{4} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}-2368 a \,b^{3} c \,d^{3} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+1008 b^{4} c^{2} d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+60 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x^{2}-1924 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x^{2}+2996 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x^{2}-1260 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d \,x^{2}+45 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} d^{5}+75 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b c \,d^{4}+450 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{2} c^{2} d^{3}-2250 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{3} d^{2}+2625 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{4} c^{4} d -945 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{5} c^{5}-90 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a^{4} d^{4}-180 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}+3128 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-4620 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d +1890 \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{7680 b^{2} d^{5} \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \sqrt {b d}}\) | \(900\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {15 \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 ) a^{3} c^{2}}{256 d^{2} \sqrt {b d}}-\frac {481 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{2} c}{1920 d^{2}}-\frac {63 b^{3} \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 ) c^{5}}{512 d^{5} \sqrt {b d}}+\frac {21 b \,x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{80 d}-\frac {9 b^{2} x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{80 d^{2}}+\frac {21 b^{2} x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{2}}{160 d^{3}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{3}}{128 b d}-\frac {21 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c^{3}}{128 d^{4}}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{3} c}{128 b \,d^{2}}-\frac {77 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a \,c^{3}}{128 d^{4}}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{4}}{256 b^{2} d}+\frac {63 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{4}}{256 d^{5}}+\frac {3 \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 ) a^{5}}{512 b^{2} \sqrt {b d}}+\frac {b^{2} x^{8} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{10 d}+\frac {31 x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2}}{160 d}+\frac {391 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2} c^{2}}{960 d^{3}}+\frac {175 b^{2} \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 ) c^{4} a}{512 d^{4} \sqrt {b d}}+\frac {749 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a \,c^{2}}{1920 d^{3}}+\frac {5 \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 ) a^{4} c}{512 b d \sqrt {b d}}-\frac {75 b \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 ) a^{2} c^{3}}{256 d^{3} \sqrt {b d}}-\frac {37 b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a c}{120 d^{2}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(932\) |
Input:
int(x^5*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/3840/b^2*(-384*b^4*d^4*x^8-1008*a*b^3*d^4*x^6+432*b^4*c*d^3*x^6-744*a^2 *b^2*d^4*x^4+1184*a*b^3*c*d^3*x^4-504*b^4*c^2*d^2*x^4-30*a^3*b*d^4*x^2+962 *a^2*b^2*c*d^3*x^2-1498*a*b^3*c^2*d^2*x^2+630*b^4*c^3*d*x^2+45*a^4*d^4+90* a^3*b*c*d^3-1564*a^2*b^2*c^2*d^2+2310*a*b^3*c^3*d-945*b^4*c^4)*(b*x^2+a)^( 1/2)*(d*x^2+c)^(1/2)/d^5+1/512/b^2*(3*a^5*d^5+5*a^4*b*c*d^4+30*a^3*b^2*c^2 *d^3-150*a^2*b^3*c^3*d^2+175*a*b^4*c^4*d-63*b^5*c^5)/d^5*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)*((b *x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.19 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.17 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{8} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{6} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{4} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15360 \, b^{3} d^{6}}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{8} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{6} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{4} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{7680 \, b^{3} d^{6}}\right ] \] Input:
integrate(x^5*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
[-1/15360*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3 *b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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*(384*b^5*d^5*x^8 + 945*b^5*c^4*d - 2310*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c *d^4 - 45*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^6 + 8*(63*b^5*c^2* d^3 - 148*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^4 - 2*(315*b^5*c^3*d^2 - 749*a*b ^4*c^2*d^3 + 481*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x^2)*sqrt(b*x^2 + a)*sqrt (d*x^2 + c))/(b^3*d^6), 1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2 *b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*sqrt(-b*d)* arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(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*(384*b^5*d^5*x^ 8 + 945*b^5*c^4*d - 2310*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2 *c*d^4 - 45*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^6 + 8*(63*b^5*c^ 2*d^3 - 148*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^4 - 2*(315*b^5*c^3*d^2 - 749*a *b^4*c^2*d^3 + 481*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x^2)*sqrt(b*x^2 + a)*sq rt(d*x^2 + c))/(b^3*d^6)]
\[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{5} \left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**5*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**5*(a + b*x**2)**(5/2)/sqrt(c + d*x**2), x)
Exception generated. \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^5*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
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
Time = 0.15 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.18 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (6 \, {\left (b x^{2} + a\right )} {\left (\frac {8 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac {63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac {5 \, {\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} + \frac {15 \, {\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} + \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{5}}\right )} b}{3840 \, {\left | b \right |}} \] Input:
integrate(x^5*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
1/3840*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a)*(4*(b*x^2 + a)*(6*(b*x^2 + a)*(8*(b*x^2 + a)/(b^3*d) - (9*b^7*c*d^7 + 1 1*a*b^6*d^8)/(b^9*d^9)) + (63*b^8*c^2*d^6 + 14*a*b^7*c*d^7 + 3*a^2*b^6*d^8 )/(b^9*d^9)) - 5*(63*b^9*c^3*d^5 - 49*a*b^8*c^2*d^6 - 11*a^2*b^7*c*d^7 - 3 *a^3*b^6*d^8)/(b^9*d^9)) + 15*(63*b^10*c^4*d^4 - 112*a*b^9*c^3*d^5 + 38*a^ 2*b^8*c^2*d^6 + 8*a^3*b^7*c*d^7 + 3*a^4*b^6*d^8)/(b^9*d^9)) + 15*(63*b^5*c ^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b* c*d^4 - 3*a^5*d^5)*log(abs(-sqrt(b*x^2 + a)*sqrt(b*d) + sqrt(b^2*c + (b*x^ 2 + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^5))*b/abs(b)
Timed out. \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^5\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \] Input:
int((x^5*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2),x)
Output:
int((x^5*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2), x)
Time = 0.37 (sec) , antiderivative size = 689, normalized size of antiderivative = 2.04 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {-45 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{4} b \,d^{5}-90 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c \,d^{4}+30 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d^{5} x^{2}+1564 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{2} d^{3}-962 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,d^{4} x^{2}+744 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d^{5} x^{4}-2310 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{3} d^{2}+1498 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{2} d^{3} x^{2}-1184 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,d^{4} x^{4}+1008 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{4} d^{5} x^{6}+945 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{5} c^{4} d -630 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{5} c^{3} d^{2} x^{2}+504 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{5} c^{2} d^{3} x^{4}-432 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{5} c \,d^{4} x^{6}+384 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{5} d^{5} x^{8}-45 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{5} d^{5}-75 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{4} b c \,d^{4}-450 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} b^{2} c^{2} d^{3}+2250 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} b^{3} c^{3} d^{2}-2625 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a \,b^{4} c^{4} d +945 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{5} c^{5}}{3840 b^{3} d^{6}} \] Input:
int(x^5*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)
Output:
( - 45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**4*b*d**5 - 90*sqrt(c + d*x**2) *sqrt(a + b*x**2)*a**3*b**2*c*d**4 + 30*sqrt(c + d*x**2)*sqrt(a + b*x**2)* a**3*b**2*d**5*x**2 + 1564*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c** 2*d**3 - 962*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*c*d**4*x**2 + 744 *sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**3*d**5*x**4 - 2310*sqrt(c + d*x **2)*sqrt(a + b*x**2)*a*b**4*c**3*d**2 + 1498*sqrt(c + d*x**2)*sqrt(a + b* x**2)*a*b**4*c**2*d**3*x**2 - 1184*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b** 4*c*d**4*x**4 + 1008*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**4*d**5*x**6 + 945*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**5*c**4*d - 630*sqrt(c + d*x**2)*s qrt(a + b*x**2)*b**5*c**3*d**2*x**2 + 504*sqrt(c + d*x**2)*sqrt(a + b*x**2 )*b**5*c**2*d**3*x**4 - 432*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**5*c*d**4* x**6 + 384*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**5*d**5*x**8 - 45*sqrt(d)*s qrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a** 5*d**5 - 75*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sq rt(c + d*x**2)*b)*a**4*b*c*d**4 - 450*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt( a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a**3*b**2*c**2*d**3 + 2250*sqr t(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)* b)*a**2*b**3*c**3*d**2 - 2625*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x* *2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a*b**4*c**4*d + 945*sqrt(d)*sqrt(b)*lo g( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*b**5*c**5...