Integrand size = 24, antiderivative size = 155 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (\frac {A}{a}-\frac {b B-a C}{b^2}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (6 A b^2+a (b B-8 a C)\right ) x}{35 a^2 b^2 \left (a+b x^2\right )^{5/2}}+\frac {\left (24 A b^2+a (4 b B+3 a C)\right ) x}{105 a^3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 \left (24 A b^2+a (4 b B+3 a C)\right ) x}{105 a^4 b^2 \sqrt {a+b x^2}} \] Output:
1/7*(A/a-(B*b-C*a)/b^2)*x/(b*x^2+a)^(7/2)+1/35*(6*A*b^2+a*(B*b-8*C*a))*x/a ^2/b^2/(b*x^2+a)^(5/2)+1/105*(24*A*b^2+a*(4*B*b+3*C*a))*x/a^3/b^2/(b*x^2+a )^(3/2)+2/105*(24*A*b^2+a*(4*B*b+3*C*a))*x/a^4/b^2/(b*x^2+a)^(1/2)
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {48 A b^3 x^7+8 a b^2 x^5 \left (21 A+B x^2\right )+2 a^2 b x^3 \left (105 A+14 B x^2+3 C x^4\right )+7 a^3 \left (15 A x+5 B x^3+3 C x^5\right )}{105 a^4 \left (a+b x^2\right )^{7/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(a + b*x^2)^(9/2),x]
Output:
(48*A*b^3*x^7 + 8*a*b^2*x^5*(21*A + B*x^2) + 2*a^2*b*x^3*(105*A + 14*B*x^2 + 3*C*x^4) + 7*a^3*(15*A*x + 5*B*x^3 + 3*C*x^5))/(105*a^4*(a + b*x^2)^(7/ 2))
Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1469, 2075, 362, 245, 242}
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1469 |
| \(\displaystyle \frac {\int \frac {x^2 \left (6 A b+a \left (C x^2+B\right )\right )}{\left (b x^2+a\right )^{9/2}}dx}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {x^2 \left (a C x^2+6 A b+a B\right )}{\left (b x^2+a\right )^{9/2}}dx}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {4 (a B+6 A b)}{a}+\frac {3 a C}{b}\right ) \int \frac {x^2}{\left (b x^2+a\right )^{7/2}}dx+\frac {x^3 \left (a \left (B-\frac {a C}{b}\right )+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {4 (a B+6 A b)}{a}+\frac {3 a C}{b}\right ) \left (\frac {2 b \int \frac {x^4}{\left (b x^2+a\right )^{7/2}}dx}{3 a}+\frac {x^3}{3 a \left (a+b x^2\right )^{5/2}}\right )+\frac {x^3 \left (a \left (B-\frac {a C}{b}\right )+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 b x^5}{15 a^2 \left (a+b x^2\right )^{5/2}}+\frac {x^3}{3 a \left (a+b x^2\right )^{5/2}}\right ) \left (\frac {4 (a B+6 A b)}{a}+\frac {3 a C}{b}\right )+\frac {x^3 \left (a \left (B-\frac {a C}{b}\right )+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(a + b*x^2)^(9/2),x]
Output:
(A*x)/(a*(a + b*x^2)^(7/2)) + (((6*A*b + a*(B - (a*C)/b))*x^3)/(7*a*(a + b *x^2)^(7/2)) + (((4*(6*A*b + a*B))/a + (3*a*C)/b)*(x^3/(3*a*(a + b*x^2)^(5 /2)) + (2*b*x^5)/(15*a^2*(a + b*x^2)^(5/2))))/7)/a
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[a^p*x*((d + e*x^2)^(q + 1)/d), x] + Simp[1/d Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x] - e* a^p*(2*q + 3)), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4 *p + 2*q + 1, 0]
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Time = 0.52 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (\frac {1}{5} C \,x^{4}+\frac {1}{3} x^{2} B +A \right ) a^{3}+2 \left (\frac {1}{35} C \,x^{4}+\frac {2}{15} x^{2} B +A \right ) x^{2} b \,a^{2}+\frac {8 b^{2} \left (\frac {x^{2} B}{21}+A \right ) x^{4} a}{5}+\frac {16 A \,b^{3} x^{6}}{35}\right ) x}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(83\) |
| gosper | \(\frac {x \left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 C \,a^{2} b \,x^{6}+168 a A \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 C \,a^{3} x^{4}+210 a^{2} A b \,x^{2}+35 B \,a^{3} x^{2}+105 a^{3} A \right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(100\) |
| trager | \(\frac {x \left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 C \,a^{2} b \,x^{6}+168 a A \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 C \,a^{3} x^{4}+210 a^{2} A b \,x^{2}+35 B \,a^{3} x^{2}+105 a^{3} A \right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(100\) |
| orering | \(\frac {x \left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 C \,a^{2} b \,x^{6}+168 a A \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 C \,a^{3} x^{4}+210 a^{2} A b \,x^{2}+35 B \,a^{3} x^{2}+105 a^{3} A \right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(100\) |
| default | \(A \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+C \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+B \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )\) | \(295\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
((1/5*C*x^4+1/3*x^2*B+A)*a^3+2*(1/35*C*x^4+2/15*x^2*B+A)*x^2*b*a^2+8/5*b^2 *(1/21*x^2*B+A)*x^4*a+16/35*A*b^3*x^6)/(b*x^2+a)^(7/2)*x/a^4
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (2 \, {\left (3 \, C a^{2} b + 4 \, B a b^{2} + 24 \, A b^{3}\right )} x^{7} + 7 \, {\left (3 \, C a^{3} + 4 \, B a^{2} b + 24 \, A a b^{2}\right )} x^{5} + 105 \, A a^{3} x + 35 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
1/105*(2*(3*C*a^2*b + 4*B*a*b^2 + 24*A*b^3)*x^7 + 7*(3*C*a^3 + 4*B*a^2*b + 24*A*a*b^2)*x^5 + 105*A*a^3*x + 35*(B*a^3 + 6*A*a^2*b)*x^3)*sqrt(b*x^2 + a)/(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8)
Leaf count of result is larger than twice the leaf count of optimal. 1989 vs. \(2 (150) = 300\).
Time = 38.41 (sec) , antiderivative size = 1989, normalized size of antiderivative = 12.83 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)
Output:
A*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt (1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2 )*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a ) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12* sqrt(1 + b*x**2/a)) + 175*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b* *4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(35*a** (37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525* a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b** 5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 429*a**11*b**3*x**7/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x* *2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a **(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b *x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6 *x**12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*a**(37/2)*sqrt(1 + b* x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x** 4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525...
Time = 0.04 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.46 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {16 \, A x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} + \frac {3 \, C x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, C x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, C a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, B x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
-1/4*C*x^3/((b*x^2 + a)^(7/2)*b) + 16/35*A*x/(sqrt(b*x^2 + a)*a^4) + 8/35* A*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*A*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*A*x/( (b*x^2 + a)^(7/2)*a) + 3/140*C*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*C*x/(sqrt( b*x^2 + a)*a^2*b^2) + 1/35*C*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*C*a*x/((b* x^2 + a)^(7/2)*b^2) - 1/7*B*x/((b*x^2 + a)^(7/2)*b) + 8/105*B*x/(sqrt(b*x^ 2 + a)*a^3*b) + 4/105*B*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*B*x/((b*x^2 + a )^(5/2)*a*b)
Time = 0.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {2 \, {\left (3 \, C a^{2} b^{4} + 4 \, B a b^{5} + 24 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac {7 \, {\left (3 \, C a^{3} b^{3} + 4 \, B a^{2} b^{4} + 24 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac {35 \, {\left (B a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac {105 \, A}{a}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
1/105*((x^2*(2*(3*C*a^2*b^4 + 4*B*a*b^5 + 24*A*b^6)*x^2/(a^4*b^3) + 7*(3*C *a^3*b^3 + 4*B*a^2*b^4 + 24*A*a*b^5)/(a^4*b^3)) + 35*(B*a^3*b^3 + 6*A*a^2* b^4)/(a^4*b^3))*x^2 + 105*A/a)*x/(b*x^2 + a)^(7/2)
Time = 0.55 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x\,\left (\frac {A}{7\,a}-\frac {a\,\left (\frac {B}{7\,a}-\frac {C}{7\,b}\right )}{b}\right )}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {x\,\left (\frac {C}{5\,b^2}-\frac {-C\,a^2+B\,a\,b+6\,A\,b^2}{35\,a^2\,b^2}\right )}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {x\,\left (3\,C\,a^2+4\,B\,a\,b+24\,A\,b^2\right )}{105\,a^3\,b^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x\,\left (6\,C\,a^2+8\,B\,a\,b+48\,A\,b^2\right )}{105\,a^4\,b^2\,\sqrt {b\,x^2+a}} \] Input:
int((A + B*x^2 + C*x^4)/(a + b*x^2)^(9/2),x)
Output:
(x*(A/(7*a) - (a*(B/(7*a) - C/(7*b)))/b))/(a + b*x^2)^(7/2) - (x*(C/(5*b^2 ) - (6*A*b^2 - C*a^2 + B*a*b)/(35*a^2*b^2)))/(a + b*x^2)^(5/2) + (x*(24*A* b^2 + 3*C*a^2 + 4*B*a*b))/(105*a^3*b^2*(a + b*x^2)^(3/2)) + (x*(48*A*b^2 + 6*C*a^2 + 8*B*a*b))/(105*a^4*b^2*(a + b*x^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {105 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x +245 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{3}+21 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,x^{5}+196 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{5}+6 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{7}+56 \sqrt {b \,x^{2}+a}\, b^{6} x^{7}-6 \sqrt {b}\, a^{5} c -56 \sqrt {b}\, a^{4} b^{2}-24 \sqrt {b}\, a^{4} b c \,x^{2}-224 \sqrt {b}\, a^{3} b^{3} x^{2}-36 \sqrt {b}\, a^{3} b^{2} c \,x^{4}-336 \sqrt {b}\, a^{2} b^{4} x^{4}-24 \sqrt {b}\, a^{2} b^{3} c \,x^{6}-224 \sqrt {b}\, a \,b^{5} x^{6}-6 \sqrt {b}\, a \,b^{4} c \,x^{8}-56 \sqrt {b}\, b^{6} x^{8}}{105 a^{3} b^{3} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)
Output:
(105*sqrt(a + b*x**2)*a**3*b**3*x + 245*sqrt(a + b*x**2)*a**2*b**4*x**3 + 21*sqrt(a + b*x**2)*a**2*b**3*c*x**5 + 196*sqrt(a + b*x**2)*a*b**5*x**5 + 6*sqrt(a + b*x**2)*a*b**4*c*x**7 + 56*sqrt(a + b*x**2)*b**6*x**7 - 6*sqrt( b)*a**5*c - 56*sqrt(b)*a**4*b**2 - 24*sqrt(b)*a**4*b*c*x**2 - 224*sqrt(b)* a**3*b**3*x**2 - 36*sqrt(b)*a**3*b**2*c*x**4 - 336*sqrt(b)*a**2*b**4*x**4 - 24*sqrt(b)*a**2*b**3*c*x**6 - 224*sqrt(b)*a*b**5*x**6 - 6*sqrt(b)*a*b**4 *c*x**8 - 56*sqrt(b)*b**6*x**8)/(105*a**3*b**3*(a**4 + 4*a**3*b*x**2 + 6*a **2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))