Integrand size = 32, antiderivative size = 235 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {b^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x}{a^6 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{9 a^2 x^9}+\frac {(17 A b-9 a B) \sqrt {a+b x^2}}{63 a^3 x^7}-\frac {\left (55 A b^2-39 a b B+21 a^2 C\right ) \sqrt {a+b x^2}}{105 a^4 x^5}+\frac {\left (325 A b^3-3 a \left (87 b^2 B-63 a b C+35 a^2 D\right )\right ) \sqrt {a+b x^2}}{315 a^5 x^3}-\frac {b \left (965 A b^3-837 a b^2 B+693 a^2 b C-525 a^3 D\right ) \sqrt {a+b x^2}}{315 a^6 x} \] Output:
-b^2*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*x/a^6/(b*x^2+a)^(1/2)-1/9*A*(b*x^2+a)^( 1/2)/a^2/x^9+1/63*(17*A*b-9*B*a)*(b*x^2+a)^(1/2)/a^3/x^7-1/105*(55*A*b^2-3 9*B*a*b+21*C*a^2)*(b*x^2+a)^(1/2)/a^4/x^5+1/315*(325*A*b^3-3*a*(87*B*b^2-6 3*C*a*b+35*D*a^2))*(b*x^2+a)^(1/2)/a^5/x^3-1/315*b*(965*A*b^3-837*B*a*b^2+ 693*C*a^2*b-525*D*a^3)*(b*x^2+a)^(1/2)/a^6/x
Time = 0.68 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\frac {-1280 A b^5 x^{10}+128 a b^4 x^8 \left (-5 A+9 B x^2\right )+16 a^2 b^3 x^6 \left (10 A+36 B x^2-63 C x^4\right )-8 a^3 b^2 x^4 \left (10 A+18 B x^2+63 C x^4-105 D x^6\right )-a^5 \left (35 A+45 B x^2+63 C x^4+105 D x^6\right )+2 a^4 b x^2 \left (25 A+36 B x^2+63 C x^4+210 D x^6\right )}{315 a^6 x^9 \sqrt {a+b x^2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(3/2)),x]
Output:
(-1280*A*b^5*x^10 + 128*a*b^4*x^8*(-5*A + 9*B*x^2) + 16*a^2*b^3*x^6*(10*A + 36*B*x^2 - 63*C*x^4) - 8*a^3*b^2*x^4*(10*A + 18*B*x^2 + 63*C*x^4 - 105*D *x^6) - a^5*(35*A + 45*B*x^2 + 63*C*x^4 + 105*D*x^6) + 2*a^4*b*x^2*(25*A + 36*B*x^2 + 63*C*x^4 + 210*D*x^6))/(315*a^6*x^9*Sqrt[a + b*x^2])
Time = 0.55 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2334, 2089, 1588, 359, 245, 245, 208}
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {10 A b-9 a \left (D x^4+C x^2+B\right )}{x^8 \left (b x^2+a\right )^{3/2}}dx}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-9 a D x^4-9 a C x^2+10 A b-9 a B}{x^8 \left (b x^2+a\right )^{3/2}}dx}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {63 D x^2 a^2+63 C a^2-72 b B a+80 A b^2}{x^6 \left (b x^2+a\right )^{3/2}}dx}{7 a}-\frac {10 A b-9 a B}{7 a x^7 \sqrt {a+b x^2}}}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-105 a^3 D-18 a b (8 b B-7 a C)+160 A b^3\right ) \int \frac {1}{x^4 \left (b x^2+a\right )^{3/2}}dx}{5 a}-\frac {63 a^2 C-72 a b B+80 A b^2}{5 a x^5 \sqrt {a+b x^2}}}{7 a}-\frac {10 A b-9 a B}{7 a x^7 \sqrt {a+b x^2}}}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-105 a^3 D-18 a b (8 b B-7 a C)+160 A b^3\right ) \left (-\frac {4 b \int \frac {1}{x^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {1}{3 a x^3 \sqrt {a+b x^2}}\right )}{5 a}-\frac {63 a^2 C-72 a b B+80 A b^2}{5 a x^5 \sqrt {a+b x^2}}}{7 a}-\frac {10 A b-9 a B}{7 a x^7 \sqrt {a+b x^2}}}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-105 a^3 D-18 a b (8 b B-7 a C)+160 A b^3\right ) \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{a}-\frac {1}{a x \sqrt {a+b x^2}}\right )}{3 a}-\frac {1}{3 a x^3 \sqrt {a+b x^2}}\right )}{5 a}-\frac {63 a^2 C-72 a b B+80 A b^2}{5 a x^5 \sqrt {a+b x^2}}}{7 a}-\frac {10 A b-9 a B}{7 a x^7 \sqrt {a+b x^2}}}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {-\frac {-\frac {63 a^2 C-72 a b B+80 A b^2}{5 a x^5 \sqrt {a+b x^2}}-\frac {3 \left (-\frac {4 b \left (-\frac {2 b x}{a^2 \sqrt {a+b x^2}}-\frac {1}{a x \sqrt {a+b x^2}}\right )}{3 a}-\frac {1}{3 a x^3 \sqrt {a+b x^2}}\right ) \left (-105 a^3 D-18 a b (8 b B-7 a C)+160 A b^3\right )}{5 a}}{7 a}-\frac {10 A b-9 a B}{7 a x^7 \sqrt {a+b x^2}}}{9 a}-\frac {A}{9 a x^9 \sqrt {a+b x^2}}\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(3/2)),x]
Output:
-1/9*A/(a*x^9*Sqrt[a + b*x^2]) - (-1/7*(10*A*b - 9*a*B)/(a*x^7*Sqrt[a + b* x^2]) - (-1/5*(80*A*b^2 - 72*a*b*B + 63*a^2*C)/(a*x^5*Sqrt[a + b*x^2]) - ( 3*(160*A*b^3 - 18*a*b*(8*b*B - 7*a*C) - 105*a^3*D)*(-1/3*1/(a*x^3*Sqrt[a + b*x^2]) - (4*b*(-(1/(a*x*Sqrt[a + b*x^2])) - (2*b*x)/(a^2*Sqrt[a + b*x^2] )))/(3*a)))/(5*a))/(7*a))/(9*a)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_)^(q_.), x_Symbol] :> Int[(f*x)^m*Expa ndToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && Binomi alQ[z, x] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ [u, x])
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coef f[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A* x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[ x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
Time = 0.58 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {\left (-105 D x^{6}-63 C \,x^{4}-45 x^{2} B -35 A \right ) a^{5}+50 \left (\frac {42}{5} D x^{6}+\frac {63}{25} C \,x^{4}+\frac {36}{25} x^{2} B +A \right ) x^{2} b \,a^{4}-80 \left (-\frac {21}{2} D x^{6}+\frac {63}{10} C \,x^{4}+\frac {9}{5} x^{2} B +A \right ) x^{4} b^{2} a^{3}+160 \left (-\frac {63}{10} C \,x^{4}+\frac {18}{5} x^{2} B +A \right ) x^{6} b^{3} a^{2}-640 \left (-\frac {9 x^{2} B}{5}+A \right ) x^{8} b^{4} a -1280 A \,b^{5} x^{10}}{315 \sqrt {b \,x^{2}+a}\, x^{9} a^{6}}\) | \(156\) |
| gosper | \(-\frac {1280 A \,b^{5} x^{10}-1152 B a \,b^{4} x^{10}+1008 C \,a^{2} b^{3} x^{10}-840 D a^{3} b^{2} x^{10}+640 a A \,b^{4} x^{8}-576 B \,a^{2} b^{3} x^{8}+504 C \,a^{3} b^{2} x^{8}-420 D a^{4} b \,x^{8}-160 a^{2} A \,b^{3} x^{6}+144 B \,a^{3} b^{2} x^{6}-126 C \,a^{4} b \,x^{6}+105 D a^{5} x^{6}+80 a^{3} A \,b^{2} x^{4}-72 B \,a^{4} b \,x^{4}+63 C \,a^{5} x^{4}-50 a^{4} A b \,x^{2}+45 B \,a^{5} x^{2}+35 a^{5} A}{315 x^{9} \sqrt {b \,x^{2}+a}\, a^{6}}\) | \(205\) |
| trager | \(-\frac {1280 A \,b^{5} x^{10}-1152 B a \,b^{4} x^{10}+1008 C \,a^{2} b^{3} x^{10}-840 D a^{3} b^{2} x^{10}+640 a A \,b^{4} x^{8}-576 B \,a^{2} b^{3} x^{8}+504 C \,a^{3} b^{2} x^{8}-420 D a^{4} b \,x^{8}-160 a^{2} A \,b^{3} x^{6}+144 B \,a^{3} b^{2} x^{6}-126 C \,a^{4} b \,x^{6}+105 D a^{5} x^{6}+80 a^{3} A \,b^{2} x^{4}-72 B \,a^{4} b \,x^{4}+63 C \,a^{5} x^{4}-50 a^{4} A b \,x^{2}+45 B \,a^{5} x^{2}+35 a^{5} A}{315 x^{9} \sqrt {b \,x^{2}+a}\, a^{6}}\) | \(205\) |
| orering | \(-\frac {1280 A \,b^{5} x^{10}-1152 B a \,b^{4} x^{10}+1008 C \,a^{2} b^{3} x^{10}-840 D a^{3} b^{2} x^{10}+640 a A \,b^{4} x^{8}-576 B \,a^{2} b^{3} x^{8}+504 C \,a^{3} b^{2} x^{8}-420 D a^{4} b \,x^{8}-160 a^{2} A \,b^{3} x^{6}+144 B \,a^{3} b^{2} x^{6}-126 C \,a^{4} b \,x^{6}+105 D a^{5} x^{6}+80 a^{3} A \,b^{2} x^{4}-72 B \,a^{4} b \,x^{4}+63 C \,a^{5} x^{4}-50 a^{4} A b \,x^{2}+45 B \,a^{5} x^{2}+35 a^{5} A}{315 x^{9} \sqrt {b \,x^{2}+a}\, a^{6}}\) | \(205\) |
| default | \(A \left (-\frac {1}{9 a \,x^{9} \sqrt {b \,x^{2}+a}}-\frac {10 b \left (-\frac {1}{7 a \,x^{7} \sqrt {b \,x^{2}+a}}-\frac {8 b \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )+B \left (-\frac {1}{7 a \,x^{7} \sqrt {b \,x^{2}+a}}-\frac {8 b \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )+C \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )+D \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )\) | \(386\) |
Input:
int((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/315*((-105*D*x^6-63*C*x^4-45*B*x^2-35*A)*a^5+50*(42/5*D*x^6+63/25*C*x^4+ 36/25*x^2*B+A)*x^2*b*a^4-80*(-21/2*D*x^6+63/10*C*x^4+9/5*x^2*B+A)*x^4*b^2* a^3+160*(-63/10*C*x^4+18/5*x^2*B+A)*x^6*b^3*a^2-640*(-9/5*x^2*B+A)*x^8*b^4 *a-1280*A*b^5*x^10)/(b*x^2+a)^(1/2)/x^9/a^6
Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (8 \, {\left (105 \, D a^{3} b^{2} - 126 \, C a^{2} b^{3} + 144 \, B a b^{4} - 160 \, A b^{5}\right )} x^{10} + 4 \, {\left (105 \, D a^{4} b - 126 \, C a^{3} b^{2} + 144 \, B a^{2} b^{3} - 160 \, A a b^{4}\right )} x^{8} - {\left (105 \, D a^{5} - 126 \, C a^{4} b + 144 \, B a^{3} b^{2} - 160 \, A a^{2} b^{3}\right )} x^{6} - 35 \, A a^{5} - {\left (63 \, C a^{5} - 72 \, B a^{4} b + 80 \, A a^{3} b^{2}\right )} x^{4} - 5 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(3/2),x, algorithm="fricas" )
Output:
1/315*(8*(105*D*a^3*b^2 - 126*C*a^2*b^3 + 144*B*a*b^4 - 160*A*b^5)*x^10 + 4*(105*D*a^4*b - 126*C*a^3*b^2 + 144*B*a^2*b^3 - 160*A*a*b^4)*x^8 - (105*D *a^5 - 126*C*a^4*b + 144*B*a^3*b^2 - 160*A*a^2*b^3)*x^6 - 35*A*a^5 - (63*C *a^5 - 72*B*a^4*b + 80*A*a^3*b^2)*x^4 - 5*(9*B*a^5 - 10*A*a^4*b)*x^2)*sqrt (b*x^2 + a)/(a^6*b*x^11 + a^7*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 2134 vs. \(2 (228) = 456\).
Time = 17.37 (sec) , antiderivative size = 2134, normalized size of antiderivative = 9.08 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/x**10/(b*x**2+a)**(3/2),x)
Output:
A*(-7*a**9*b**(51/2)*sqrt(a/(b*x**2) + 1)/(63*a**11*b**25*x**8 + 315*a**10 *b**26*x**10 + 630*a**9*b**27*x**12 + 630*a**8*b**28*x**14 + 315*a**7*b**2 9*x**16 + 63*a**6*b**30*x**18) - 18*a**8*b**(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(63*a**11*b**25*x**8 + 315*a**10*b**26*x**10 + 630*a**9*b**27*x**12 + 6 30*a**8*b**28*x**14 + 315*a**7*b**29*x**16 + 63*a**6*b**30*x**18) - 18*a** 7*b**(55/2)*x**4*sqrt(a/(b*x**2) + 1)/(63*a**11*b**25*x**8 + 315*a**10*b** 26*x**10 + 630*a**9*b**27*x**12 + 630*a**8*b**28*x**14 + 315*a**7*b**29*x* *16 + 63*a**6*b**30*x**18) - 63*a**5*b**(59/2)*x**8*sqrt(a/(b*x**2) + 1)/( 63*a**11*b**25*x**8 + 315*a**10*b**26*x**10 + 630*a**9*b**27*x**12 + 630*a **8*b**28*x**14 + 315*a**7*b**29*x**16 + 63*a**6*b**30*x**18) - 630*a**4*b **(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(63*a**11*b**25*x**8 + 315*a**10*b**26 *x**10 + 630*a**9*b**27*x**12 + 630*a**8*b**28*x**14 + 315*a**7*b**29*x**1 6 + 63*a**6*b**30*x**18) - 1680*a**3*b**(63/2)*x**12*sqrt(a/(b*x**2) + 1)/ (63*a**11*b**25*x**8 + 315*a**10*b**26*x**10 + 630*a**9*b**27*x**12 + 630* a**8*b**28*x**14 + 315*a**7*b**29*x**16 + 63*a**6*b**30*x**18) - 2016*a**2 *b**(65/2)*x**14*sqrt(a/(b*x**2) + 1)/(63*a**11*b**25*x**8 + 315*a**10*b** 26*x**10 + 630*a**9*b**27*x**12 + 630*a**8*b**28*x**14 + 315*a**7*b**29*x* *16 + 63*a**6*b**30*x**18) - 1152*a*b**(67/2)*x**16*sqrt(a/(b*x**2) + 1)/( 63*a**11*b**25*x**8 + 315*a**10*b**26*x**10 + 630*a**9*b**27*x**12 + 630*a **8*b**28*x**14 + 315*a**7*b**29*x**16 + 63*a**6*b**30*x**18) - 256*b**...
Time = 0.04 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.49 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\frac {8 \, D b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {16 \, C b^{3} x}{5 \, \sqrt {b x^{2} + a} a^{4}} + \frac {128 \, B b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {256 \, A b^{5} x}{63 \, \sqrt {b x^{2} + a} a^{6}} + \frac {4 \, D b}{3 \, \sqrt {b x^{2} + a} a^{2} x} - \frac {8 \, C b^{2}}{5 \, \sqrt {b x^{2} + a} a^{3} x} + \frac {64 \, B b^{3}}{35 \, \sqrt {b x^{2} + a} a^{4} x} - \frac {128 \, A b^{4}}{63 \, \sqrt {b x^{2} + a} a^{5} x} - \frac {D}{3 \, \sqrt {b x^{2} + a} a x^{3}} + \frac {2 \, C b}{5 \, \sqrt {b x^{2} + a} a^{2} x^{3}} - \frac {16 \, B b^{2}}{35 \, \sqrt {b x^{2} + a} a^{3} x^{3}} + \frac {32 \, A b^{3}}{63 \, \sqrt {b x^{2} + a} a^{4} x^{3}} - \frac {C}{5 \, \sqrt {b x^{2} + a} a x^{5}} + \frac {8 \, B b}{35 \, \sqrt {b x^{2} + a} a^{2} x^{5}} - \frac {16 \, A b^{2}}{63 \, \sqrt {b x^{2} + a} a^{3} x^{5}} - \frac {B}{7 \, \sqrt {b x^{2} + a} a x^{7}} + \frac {10 \, A b}{63 \, \sqrt {b x^{2} + a} a^{2} x^{7}} - \frac {A}{9 \, \sqrt {b x^{2} + a} a x^{9}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(3/2),x, algorithm="maxima" )
Output:
8/3*D*b^2*x/(sqrt(b*x^2 + a)*a^3) - 16/5*C*b^3*x/(sqrt(b*x^2 + a)*a^4) + 1 28/35*B*b^4*x/(sqrt(b*x^2 + a)*a^5) - 256/63*A*b^5*x/(sqrt(b*x^2 + a)*a^6) + 4/3*D*b/(sqrt(b*x^2 + a)*a^2*x) - 8/5*C*b^2/(sqrt(b*x^2 + a)*a^3*x) + 6 4/35*B*b^3/(sqrt(b*x^2 + a)*a^4*x) - 128/63*A*b^4/(sqrt(b*x^2 + a)*a^5*x) - 1/3*D/(sqrt(b*x^2 + a)*a*x^3) + 2/5*C*b/(sqrt(b*x^2 + a)*a^2*x^3) - 16/3 5*B*b^2/(sqrt(b*x^2 + a)*a^3*x^3) + 32/63*A*b^3/(sqrt(b*x^2 + a)*a^4*x^3) - 1/5*C/(sqrt(b*x^2 + a)*a*x^5) + 8/35*B*b/(sqrt(b*x^2 + a)*a^2*x^5) - 16/ 63*A*b^2/(sqrt(b*x^2 + a)*a^3*x^5) - 1/7*B/(sqrt(b*x^2 + a)*a*x^7) + 10/63 *A*b/(sqrt(b*x^2 + a)*a^2*x^7) - 1/9*A/(sqrt(b*x^2 + a)*a*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (213) = 426\).
Time = 0.16 (sec) , antiderivative size = 1001, normalized size of antiderivative = 4.26 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
(D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 - A*b^5)*x/(sqrt(b*x^2 + a)*a^6) - 2/315* (315*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a^3*b^(3/2) - 315*(sqrt(b)*x - sqr t(b*x^2 + a))^16*C*a^2*b^(5/2) + 315*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*a* b^(7/2) - 315*(sqrt(b)*x - sqrt(b*x^2 + a))^16*A*b^(9/2) - 3150*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^4*b^(3/2) + 3150*(sqrt(b)*x - sqrt(b*x^2 + a))^ 14*C*a^3*b^(5/2) - 3150*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^2*b^(7/2) + 3 150*(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a*b^(9/2) + 12810*(sqrt(b)*x - sqrt (b*x^2 + a))^12*D*a^5*b^(3/2) - 14490*(sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a ^4*b^(5/2) + 14490*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^3*b^(7/2) - 14490* (sqrt(b)*x - sqrt(b*x^2 + a))^12*A*a^2*b^(9/2) - 28350*(sqrt(b)*x - sqrt(b *x^2 + a))^10*D*a^6*b^(3/2) + 35910*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^5 *b^(5/2) - 40950*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^4*b^(7/2) + 40950*(s qrt(b)*x - sqrt(b*x^2 + a))^10*A*a^3*b^(9/2) + 37800*(sqrt(b)*x - sqrt(b*x ^2 + a))^8*D*a^7*b^(3/2) - 51408*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^6*b^( 5/2) + 64512*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^5*b^(7/2) - 80640*(sqrt(b )*x - sqrt(b*x^2 + a))^8*A*a^4*b^(9/2) - 31290*(sqrt(b)*x - sqrt(b*x^2 + a ))^6*D*a^8*b^(3/2) + 43722*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^7*b^(5/2) - 55818*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^6*b^(7/2) + 66570*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^5*b^(9/2) + 15750*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D *a^9*b^(3/2) - 21798*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^8*b^(5/2) + 26...
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^{10}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(a + b*x^2)^(3/2)),x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(a + b*x^2)^(3/2)), x)
Time = 0.18 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.36 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{3/2}} \, dx=\frac {-35 \sqrt {b \,x^{2}+a}\, a^{5}+5 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-63 \sqrt {b \,x^{2}+a}\, a^{4} c \,x^{4}-105 \sqrt {b \,x^{2}+a}\, a^{4} d \,x^{6}-8 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}+126 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{6}+420 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{8}+16 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}-504 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{8}+840 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{10}-64 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-1008 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{10}-128 \sqrt {b \,x^{2}+a}\, b^{5} x^{10}-840 \sqrt {b}\, a^{3} b d \,x^{9}+1008 \sqrt {b}\, a^{2} b^{2} c \,x^{9}-840 \sqrt {b}\, a^{2} b^{2} d \,x^{11}+128 \sqrt {b}\, a \,b^{4} x^{9}+1008 \sqrt {b}\, a \,b^{3} c \,x^{11}+128 \sqrt {b}\, b^{5} x^{11}}{315 a^{5} x^{9} \left (b \,x^{2}+a \right )} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(3/2),x)
Output:
( - 35*sqrt(a + b*x**2)*a**5 + 5*sqrt(a + b*x**2)*a**4*b*x**2 - 63*sqrt(a + b*x**2)*a**4*c*x**4 - 105*sqrt(a + b*x**2)*a**4*d*x**6 - 8*sqrt(a + b*x* *2)*a**3*b**2*x**4 + 126*sqrt(a + b*x**2)*a**3*b*c*x**6 + 420*sqrt(a + b*x **2)*a**3*b*d*x**8 + 16*sqrt(a + b*x**2)*a**2*b**3*x**6 - 504*sqrt(a + b*x **2)*a**2*b**2*c*x**8 + 840*sqrt(a + b*x**2)*a**2*b**2*d*x**10 - 64*sqrt(a + b*x**2)*a*b**4*x**8 - 1008*sqrt(a + b*x**2)*a*b**3*c*x**10 - 128*sqrt(a + b*x**2)*b**5*x**10 - 840*sqrt(b)*a**3*b*d*x**9 + 1008*sqrt(b)*a**2*b**2 *c*x**9 - 840*sqrt(b)*a**2*b**2*d*x**11 + 128*sqrt(b)*a*b**4*x**9 + 1008*s qrt(b)*a*b**3*c*x**11 + 128*sqrt(b)*b**5*x**11)/(315*a**5*x**9*(a + b*x**2 ))