3.165 \(\int \frac {A+B x^2+C x^4+D x^6}{x^4 (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=242 \[ \frac {x \left (80 A b^2-3 a (8 b B-a C)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {8 b^2 x^7 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{7/2}}+\frac {4 b x^5 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac {x^3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{3 a^4 \left (a+b x^2\right )^{7/2}}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]

[Out]

-1/3*A/a/x^3/(b*x^2+a)^(7/2)+1/3*(10*A*b-3*B*a)/a^2/x/(b*x^2+a)^(7/2)+1/3*(80*A*b^2-3*a*(8*B*b-C*a))*x/a^3/(b*
x^2+a)^(7/2)+1/3*(160*A*b^3-a*(48*B*b^2-6*C*a*b-D*a^2))*x^3/a^4/(b*x^2+a)^(7/2)+4/15*b*(160*A*b^3-a*(48*B*b^2-
6*C*a*b-D*a^2))*x^5/a^5/(b*x^2+a)^(7/2)+8/105*b^2*(160*A*b^3-a*(48*B*b^2-6*C*a*b-D*a^2))*x^7/a^6/(b*x^2+a)^(7/
2)

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Rubi [A]  time = 0.32, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1803, 1813, 12, 271, 264} \[ \frac {8 b^2 x^7 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{7/2}}+\frac {4 b x^5 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac {x^3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac {x \left (80 A b^2-3 a (8 b B-a C)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(a + b*x^2)^(9/2)),x]

[Out]

-A/(3*a*x^3*(a + b*x^2)^(7/2)) + (10*A*b - 3*a*B)/(3*a^2*x*(a + b*x^2)^(7/2)) + ((80*A*b^2 - 3*a*(8*b*B - a*C)
)*x)/(3*a^3*(a + b*x^2)^(7/2)) + ((160*A*b^3 - a*(48*b^2*B - 6*a*b*C - a^2*D))*x^3)/(3*a^4*(a + b*x^2)^(7/2))
+ (4*b*(160*A*b^3 - a*(48*b^2*B - 6*a*b*C - a^2*D))*x^5)/(15*a^5*(a + b*x^2)^(7/2)) + (8*b^2*(160*A*b^3 - a*(4
8*b^2*B - 6*a*b*C - a^2*D))*x^7)/(105*a^6*(a + b*x^2)^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 1803

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[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] + Dist[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]

Rule 1813

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient[Pq - Coef
f[Pq, x, 0], x^2, x]}, Simp[(A*x*(a + b*x^2)^(p + 1))/a, x] + Dist[1/a, Int[x^2*(a + b*x^2)^p*(a*Q - A*b*(2*p
+ 3)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && ILtQ[p + 1/2, 0] && LtQ[Expon[Pq, x] + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {10 A b-3 a \left (B+C x^2+D x^4\right )}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a}\\ &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {8 b (10 A b-3 a B)-a \left (-3 a C-3 a D x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^2}\\ &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {\left (6 b \left (80 A b^2-24 a b B+3 a^2 C\right )+3 a^3 D\right ) x^2}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) \int \frac {x^2}{\left (a+b x^2\right )^{9/2}} \, dx}{a^3}\\ &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right )\right ) \int \frac {x^4}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^4}\\ &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac {4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^5}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac {\left (8 b^2 \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right )\right ) \int \frac {x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^5}\\ &=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac {4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^5}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac {8 b^2 \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^7}{105 a^6 \left (a+b x^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 165, normalized size = 0.68 \[ \frac {-35 a^5 \left (A+3 B x^2-3 C x^4-D x^6\right )+14 a^4 b x^2 \left (25 A-60 B x^2+15 C x^4+2 D x^6\right )+8 a^3 b^2 x^4 \left (350 A-210 B x^2+21 C x^4+D x^6\right )+16 a^2 b^3 x^6 \left (350 A-84 B x^2+3 C x^4\right )+128 a b^4 x^8 \left (35 A-3 B x^2\right )+1280 A b^5 x^{10}}{105 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(a + b*x^2)^(9/2)),x]

[Out]

(1280*A*b^5*x^10 + 128*a*b^4*x^8*(35*A - 3*B*x^2) + 16*a^2*b^3*x^6*(350*A - 84*B*x^2 + 3*C*x^4) - 35*a^5*(A +
3*B*x^2 - 3*C*x^4 - D*x^6) + 8*a^3*b^2*x^4*(350*A - 210*B*x^2 + 21*C*x^4 + D*x^6) + 14*a^4*b*x^2*(25*A - 60*B*
x^2 + 15*C*x^4 + 2*D*x^6))/(105*a^6*x^3*(a + b*x^2)^(7/2))

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fricas [A]  time = 1.00, size = 225, normalized size = 0.93 \[ \frac {{\left (8 \, {\left (D a^{3} b^{2} + 6 \, C a^{2} b^{3} - 48 \, B a b^{4} + 160 \, A b^{5}\right )} x^{10} + 28 \, {\left (D a^{4} b + 6 \, C a^{3} b^{2} - 48 \, B a^{2} b^{3} + 160 \, A a b^{4}\right )} x^{8} + 35 \, {\left (D a^{5} + 6 \, C a^{4} b - 48 \, B a^{3} b^{2} + 160 \, A a^{2} b^{3}\right )} x^{6} - 35 \, A a^{5} + 35 \, {\left (3 \, C a^{5} - 24 \, B a^{4} b + 80 \, A a^{3} b^{2}\right )} x^{4} - 35 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(8*(D*a^3*b^2 + 6*C*a^2*b^3 - 48*B*a*b^4 + 160*A*b^5)*x^10 + 28*(D*a^4*b + 6*C*a^3*b^2 - 48*B*a^2*b^3 +
160*A*a*b^4)*x^8 + 35*(D*a^5 + 6*C*a^4*b - 48*B*a^3*b^2 + 160*A*a^2*b^3)*x^6 - 35*A*a^5 + 35*(3*C*a^5 - 24*B*a
^4*b + 80*A*a^3*b^2)*x^4 - 35*(3*B*a^5 - 10*A*a^4*b)*x^2)*sqrt(b*x^2 + a)/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^
8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)

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giac [A]  time = 0.56, size = 349, normalized size = 1.44 \[ \frac {{\left ({\left (x^{2} {\left (\frac {{\left (8 \, D a^{15} b^{5} + 48 \, C a^{14} b^{6} - 279 \, B a^{13} b^{7} + 790 \, A a^{12} b^{8}\right )} x^{2}}{a^{18} b^{3}} + \frac {7 \, {\left (4 \, D a^{16} b^{4} + 24 \, C a^{15} b^{5} - 132 \, B a^{14} b^{6} + 365 \, A a^{13} b^{7}\right )}}{a^{18} b^{3}}\right )} + \frac {35 \, {\left (D a^{17} b^{3} + 6 \, C a^{16} b^{4} - 30 \, B a^{15} b^{5} + 80 \, A a^{14} b^{6}\right )}}{a^{18} b^{3}}\right )} x^{2} + \frac {105 \, {\left (C a^{17} b^{3} - 4 \, B a^{16} b^{4} + 10 \, A a^{15} b^{5}\right )}}{a^{18} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} + 3 \, B a^{3} \sqrt {b} - 14 \, A a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*((8*D*a^15*b^5 + 48*C*a^14*b^6 - 279*B*a^13*b^7 + 790*A*a^12*b^8)*x^2/(a^18*b^3) + 7*(4*D*a^16*b^4
+ 24*C*a^15*b^5 - 132*B*a^14*b^6 + 365*A*a^13*b^7)/(a^18*b^3)) + 35*(D*a^17*b^3 + 6*C*a^16*b^4 - 30*B*a^15*b^
5 + 80*A*a^14*b^6)/(a^18*b^3))*x^2 + 105*(C*a^17*b^3 - 4*B*a^16*b^4 + 10*A*a^15*b^5)/(a^18*b^3))*x/(b*x^2 + a)
^(7/2) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a*sqrt(b) - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) - 6
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b) + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2) + 3*B*a^3*sqrt
(b) - 14*A*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^5)

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maple [A]  time = 0.01, size = 205, normalized size = 0.85 \[ -\frac {-1280 A \,b^{5} x^{10}+384 B a \,b^{4} x^{10}-48 C \,a^{2} b^{3} x^{10}-8 D a^{3} b^{2} x^{10}-4480 A a \,b^{4} x^{8}+1344 B \,a^{2} b^{3} x^{8}-168 C \,a^{3} b^{2} x^{8}-28 D a^{4} b \,x^{8}-5600 A \,a^{2} b^{3} x^{6}+1680 B \,a^{3} b^{2} x^{6}-210 C \,a^{4} b \,x^{6}-35 D a^{5} x^{6}-2800 A \,a^{3} b^{2} x^{4}+840 B \,a^{4} b \,x^{4}-105 C \,a^{5} x^{4}-350 A \,a^{4} b \,x^{2}+105 B \,a^{5} x^{2}+35 A \,a^{5}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(-1280*A*b^5*x^10+384*B*a*b^4*x^10-48*C*a^2*b^3*x^10-8*D*a^3*b^2*x^10-4480*A*a*b^4*x^8+1344*B*a^2*b^3*x
^8-168*C*a^3*b^2*x^8-28*D*a^4*b*x^8-5600*A*a^2*b^3*x^6+1680*B*a^3*b^2*x^6-210*C*a^4*b*x^6-35*D*a^5*x^6-2800*A*
a^3*b^2*x^4+840*B*a^4*b*x^4-105*C*a^5*x^4-350*A*a^4*b*x^2+105*B*a^5*x^2+35*A*a^5)/(b*x^2+a)^(7/2)/x^3/a^6

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maxima [A]  time = 1.47, size = 337, normalized size = 1.39 \[ \frac {16 \, C x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {D x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, D x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, D x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {128 \, B b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, B b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, B b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, B b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, A b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, A b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, A b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, A b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {B}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, A b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

16/35*C*x/(sqrt(b*x^2 + a)*a^4) + 8/35*C*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*C*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*C*
x/((b*x^2 + a)^(7/2)*a) - 1/7*D*x/((b*x^2 + a)^(7/2)*b) + 8/105*D*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*D*x/((b*x^
2 + a)^(3/2)*a^2*b) + 1/35*D*x/((b*x^2 + a)^(5/2)*a*b) - 128/35*B*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*B*b*x/((b*
x^2 + a)^(3/2)*a^4) - 48/35*B*b*x/((b*x^2 + a)^(5/2)*a^3) - 8/7*B*b*x/((b*x^2 + a)^(7/2)*a^2) + 256/21*A*b^2*x
/(sqrt(b*x^2 + a)*a^6) + 128/21*A*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7*A*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21
*A*b^2*x/((b*x^2 + a)^(7/2)*a^3) - B/((b*x^2 + a)^(7/2)*a*x) + 10/3*A*b/((b*x^2 + a)^(7/2)*a^2*x) - 1/3*A/((b*
x^2 + a)^(7/2)*a*x^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^4\,{\left (b\,x^2+a\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2 + C*x^4 + x^6*D)/(x^4*(a + b*x^2)^(9/2)),x)

[Out]

int((A + B*x^2 + C*x^4 + x^6*D)/(x^4*(a + b*x^2)^(9/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**6+C*x**4+B*x**2+A)/x**4/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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