Integrand size = 21, antiderivative size = 255 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \]
1/3*(-a*d+b*c)*x*(d*x^2+c)^3/a/b/(b*x^2+a)^(3/2)+1/8*d^2*(35*a^2*d^2-80*a* b*c*d+48*b^2*c^2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(9/2)+1/3*(-a*d+b*c )*(7*a*d+2*b*c)*x*(d*x^2+c)^2/a^2/b^2/(b*x^2+a)^(1/2)-1/24*d*(105*a^3*d^3- 170*a^2*b*c*d^2+40*a*b^2*c^2*d+16*b^3*c^3)*x*(b*x^2+a)^(1/2)/a^2/b^4-1/12* d*(-35*a^2*d^2+24*a*b*c*d+8*b^2*c^2)*x*(d*x^2+c)*(b*x^2+a)^(1/2)/a^2/b^3
Time = 0.48 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (-105 a^5 d^4+16 b^5 c^4 x^2+20 a^4 b d^3 \left (12 c-7 d x^2\right )+8 a b^4 c^3 \left (3 c+4 d x^2\right )+a^3 b^2 d^2 \left (-144 c^2+320 c d x^2-21 d^2 x^4\right )+6 a^2 b^3 d^2 x^2 \left (-32 c^2+8 c d x^2+d^2 x^4\right )\right )}{24 a^2 b^4 \left (a+b x^2\right )^{3/2}}-\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{9/2}} \]
(x*(-105*a^5*d^4 + 16*b^5*c^4*x^2 + 20*a^4*b*d^3*(12*c - 7*d*x^2) + 8*a*b^ 4*c^3*(3*c + 4*d*x^2) + a^3*b^2*d^2*(-144*c^2 + 320*c*d*x^2 - 21*d^2*x^4) + 6*a^2*b^3*d^2*x^2*(-32*c^2 + 8*c*d*x^2 + d^2*x^4)))/(24*a^2*b^4*(a + b*x ^2)^(3/2)) - (d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8*b^(9/2))
Time = 0.43 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {315, 401, 27, 403, 299, 224, 219}
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\int \frac {\left (d x^2+c\right )^2 \left (c (2 b c+a d)-d (4 b c-7 a d) x^2\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 \left (\frac {2 b c^2}{a}-\frac {7 a d^2}{b}+5 c d\right )}{\sqrt {a+b x^2}}-\frac {\int \frac {d \left (d x^2+c\right ) \left (\left (8 b^2 c^2+24 a b d c-35 a^2 d^2\right ) x^2+a c (4 b c-7 a d)\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 \left (\frac {2 b c^2}{a}-\frac {7 a d^2}{b}+5 c d\right )}{\sqrt {a+b x^2}}-\frac {d \int \frac {\left (d x^2+c\right ) \left (\left (8 b^2 c^2+24 a b d c-35 a^2 d^2\right ) x^2+a c (4 b c-7 a d)\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 \left (\frac {2 b c^2}{a}-\frac {7 a d^2}{b}+5 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {\int \frac {\left (16 b^3 c^3+40 a b^2 d c^2-170 a^2 b d^2 c+105 a^3 d^3\right ) x^2+a c \left (8 b^2 c^2-52 a b d c+35 a^2 d^2\right )}{\sqrt {b x^2+a}}dx}{4 b}+\frac {x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 \left (\frac {2 b c^2}{a}-\frac {7 a d^2}{b}+5 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {\frac {x \sqrt {a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{2 b}-\frac {3 a^2 d \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}}{4 b}+\frac {x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 \left (\frac {2 b c^2}{a}-\frac {7 a d^2}{b}+5 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {\frac {x \sqrt {a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{2 b}-\frac {3 a^2 d \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}}{4 b}+\frac {x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 \left (\frac {2 b c^2}{a}-\frac {7 a d^2}{b}+5 c d\right )}{\sqrt {a+b x^2}}-\frac {d \left (\frac {x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{4 b}+\frac {\frac {x \sqrt {a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{2 b}-\frac {3 a^2 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right )}{2 b^{3/2}}}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
((b*c - a*d)*x*(c + d*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) + ((((2*b*c^2)/a + 5*c*d - (7*a*d^2)/b)*x*(c + d*x^2)^2)/Sqrt[a + b*x^2] - (d*(((8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*b) + (((16*b^3 *c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/ (2*b) - (3*a^2*d*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b]*x )/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(4*b)))/(a*b))/(3*a*b)
3.1.89.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Time = 2.55 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(\frac {\frac {35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} d^{2} \left (a^{2} d^{2}-\frac {16}{7} a b c d +\frac {48}{35} b^{2} c^{2}\right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}-\frac {35 x \left (\frac {48 d^{2} a^{3} \left (\frac {7}{48} d^{2} x^{4}-\frac {20}{9} c d \,x^{2}+c^{2}\right ) b^{\frac {5}{2}}}{35}+\frac {64 x^{2} \left (-\frac {1}{32} d^{2} x^{4}-\frac {1}{4} c d \,x^{2}+c^{2}\right ) d^{2} a^{2} b^{\frac {7}{2}}}{35}-\frac {16 \left (-\frac {7 d \,x^{2}}{12}+c \right ) d^{3} a^{4} b^{\frac {3}{2}}}{7}-\frac {8 \left (\frac {4 d \,x^{2}}{3}+c \right ) c^{3} a \,b^{\frac {9}{2}}}{35}+\sqrt {b}\, a^{5} d^{4}-\frac {16 b^{\frac {11}{2}} c^{4} x^{2}}{105}\right )}{8}}{a^{2} b^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(198\) |
| default | \(c^{4} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{4} \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )+4 c \,d^{3} \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+6 c^{2} d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+4 c^{3} d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(360\) |
| risch | \(-\frac {d^{3} x \left (-2 b d \,x^{2}+11 a d -16 b c \right ) \sqrt {b \,x^{2}+a}}{8 b^{4}}+\frac {\frac {d^{2} \left (35 a^{2} d^{2}-80 a b c d +48 b^{2} c^{2}\right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{b a}-\frac {2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{b a}-\frac {2 \left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {2 \left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )}}{8 b^{4}}\) | \(675\) |
35/8*((b*x^2+a)^(3/2)*d^2*(a^2*d^2-16/7*a*b*c*d+48/35*b^2*c^2)*a^2*arctanh ((b*x^2+a)^(1/2)/x/b^(1/2))-x*(48/35*d^2*a^3*(7/48*d^2*x^4-20/9*c*d*x^2+c^ 2)*b^(5/2)+64/35*x^2*(-1/32*d^2*x^4-1/4*c*d*x^2+c^2)*d^2*a^2*b^(7/2)-16/7* (-7/12*d*x^2+c)*d^3*a^4*b^(3/2)-8/35*(4/3*d*x^2+c)*c^3*a*b^(9/2)+b^(1/2)*a ^5*d^4-16/105*b^(11/2)*c^4*x^2))/b^(9/2)/(b*x^2+a)^(3/2)/a^2
Time = 0.40 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.68 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, a^{2} b^{4} d^{4} x^{7} + 3 \, {\left (16 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 4 \, {\left (4 \, b^{6} c^{4} + 8 \, a b^{5} c^{3} d - 48 \, a^{2} b^{4} c^{2} d^{2} + 80 \, a^{3} b^{3} c d^{3} - 35 \, a^{4} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, a b^{5} c^{4} - 48 \, a^{3} b^{3} c^{2} d^{2} + 80 \, a^{4} b^{2} c d^{3} - 35 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac {3 \, {\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, a^{2} b^{4} d^{4} x^{7} + 3 \, {\left (16 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 4 \, {\left (4 \, b^{6} c^{4} + 8 \, a b^{5} c^{3} d - 48 \, a^{2} b^{4} c^{2} d^{2} + 80 \, a^{3} b^{3} c d^{3} - 35 \, a^{4} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, a b^{5} c^{4} - 48 \, a^{3} b^{3} c^{2} d^{2} + 80 \, a^{4} b^{2} c d^{3} - 35 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{24 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \]
[1/48*(3*(48*a^4*b^2*c^2*d^2 - 80*a^5*b*c*d^3 + 35*a^6*d^4 + (48*a^2*b^4*c ^2*d^2 - 80*a^3*b^3*c*d^3 + 35*a^4*b^2*d^4)*x^4 + 2*(48*a^3*b^3*c^2*d^2 - 80*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(6*a^2*b^4*d^4*x^7 + 3*(16*a^2*b^4*c*d^3 - 7*a^3*b ^3*d^4)*x^5 + 4*(4*b^6*c^4 + 8*a*b^5*c^3*d - 48*a^2*b^4*c^2*d^2 + 80*a^3*b ^3*c*d^3 - 35*a^4*b^2*d^4)*x^3 + 3*(8*a*b^5*c^4 - 48*a^3*b^3*c^2*d^2 + 80* a^4*b^2*c*d^3 - 35*a^5*b*d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^7*x^4 + 2*a^3*b^6 *x^2 + a^4*b^5), -1/24*(3*(48*a^4*b^2*c^2*d^2 - 80*a^5*b*c*d^3 + 35*a^6*d^ 4 + (48*a^2*b^4*c^2*d^2 - 80*a^3*b^3*c*d^3 + 35*a^4*b^2*d^4)*x^4 + 2*(48*a ^3*b^3*c^2*d^2 - 80*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*x^2)*sqrt(-b)*arctan(sqr t(-b)*x/sqrt(b*x^2 + a)) - (6*a^2*b^4*d^4*x^7 + 3*(16*a^2*b^4*c*d^3 - 7*a^ 3*b^3*d^4)*x^5 + 4*(4*b^6*c^4 + 8*a*b^5*c^3*d - 48*a^2*b^4*c^2*d^2 + 80*a^ 3*b^3*c*d^3 - 35*a^4*b^2*d^4)*x^3 + 3*(8*a*b^5*c^4 - 48*a^3*b^3*c^2*d^2 + 80*a^4*b^2*c*d^3 - 35*a^5*b*d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^7*x^4 + 2*a^3* b^6*x^2 + a^4*b^5)]
\[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.54 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^{4} x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, c d^{3} x^{5}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, a d^{4} x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - 2 \, c^{2} d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {10 \, a c d^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b} - \frac {35 \, a^{2} d^{4} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{24 \, b^{2}} + \frac {2 \, c^{4} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{4} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {4 \, c^{3} d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {4 \, c^{3} d x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {2 \, c^{2} d^{2} x}{\sqrt {b x^{2} + a} b^{2}} + \frac {10 \, a c d^{3} x}{3 \, \sqrt {b x^{2} + a} b^{3}} - \frac {35 \, a^{2} d^{4} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {6 \, c^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {10 \, a c d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {7}{2}}} + \frac {35 \, a^{2} d^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} \]
1/4*d^4*x^7/((b*x^2 + a)^(3/2)*b) + 2*c*d^3*x^5/((b*x^2 + a)^(3/2)*b) - 7/ 8*a*d^4*x^5/((b*x^2 + a)^(3/2)*b^2) - 2*c^2*d^2*x*(3*x^2/((b*x^2 + a)^(3/2 )*b) + 2*a/((b*x^2 + a)^(3/2)*b^2)) + 10/3*a*c*d^3*x*(3*x^2/((b*x^2 + a)^( 3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b - 35/24*a^2*d^4*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 2/3*c^4*x/(sqrt(b*x^2 + a)*a^2) + 1/3*c^4*x/((b*x^2 + a)^(3/2)*a) - 4/3*c^3*d*x/((b*x^2 + a)^(3/2 )*b) + 4/3*c^3*d*x/(sqrt(b*x^2 + a)*a*b) - 2*c^2*d^2*x/(sqrt(b*x^2 + a)*b^ 2) + 10/3*a*c*d^3*x/(sqrt(b*x^2 + a)*b^3) - 35/24*a^2*d^4*x/(sqrt(b*x^2 + a)*b^4) + 6*c^2*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 10*a*c*d^3*arcsinh(b* x/sqrt(a*b))/b^(7/2) + 35/8*a^2*d^4*arcsinh(b*x/sqrt(a*b))/b^(9/2)
Time = 0.30 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, d^{4} x^{2}}{b} + \frac {16 \, a^{2} b^{6} c d^{3} - 7 \, a^{3} b^{5} d^{4}}{a^{2} b^{7}}\right )} x^{2} + \frac {4 \, {\left (4 \, b^{8} c^{4} + 8 \, a b^{7} c^{3} d - 48 \, a^{2} b^{6} c^{2} d^{2} + 80 \, a^{3} b^{5} c d^{3} - 35 \, a^{4} b^{4} d^{4}\right )}}{a^{2} b^{7}}\right )} x^{2} + \frac {3 \, {\left (8 \, a b^{7} c^{4} - 48 \, a^{3} b^{5} c^{2} d^{2} + 80 \, a^{4} b^{4} c d^{3} - 35 \, a^{5} b^{3} d^{4}\right )}}{a^{2} b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{2} c^{2} d^{2} - 80 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {9}{2}}} \]
1/24*((3*(2*d^4*x^2/b + (16*a^2*b^6*c*d^3 - 7*a^3*b^5*d^4)/(a^2*b^7))*x^2 + 4*(4*b^8*c^4 + 8*a*b^7*c^3*d - 48*a^2*b^6*c^2*d^2 + 80*a^3*b^5*c*d^3 - 3 5*a^4*b^4*d^4)/(a^2*b^7))*x^2 + 3*(8*a*b^7*c^4 - 48*a^3*b^5*c^2*d^2 + 80*a ^4*b^4*c*d^3 - 35*a^5*b^3*d^4)/(a^2*b^7))*x/(b*x^2 + a)^(3/2) - 1/8*(48*b^ 2*c^2*d^2 - 80*a*b*c*d^3 + 35*a^2*d^4)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a )))/b^(9/2)
Timed out. \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^4}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]