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

Optimal. Leaf size=185 \[ -\frac{2 b x^7 \left (4 b \left (48 A b^2-a (a C+6 b B)\right )-3 a^3 D\right )}{105 a^5 \left (a+b x^2\right )^{7/2}}-\frac{x^5 \left (4 b \left (48 A b^2-a (a C+6 b B)\right )-3 a^3 D\right )}{15 a^4 \left (a+b x^2\right )^{7/2}}-\frac{x^3 \left (48 A b^2-a (a C+6 b B)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac{x (8 A b-a B)}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{A}{a x \left (a+b x^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.24635, antiderivative size = 179, normalized size of antiderivative = 0.97, number of steps used = 6, 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} \[ -\frac{2 b x^7 \left (-3 a^3 D-4 a b (a C+6 b B)+192 A b^3\right )}{105 a^5 \left (a+b x^2\right )^{7/2}}-\frac{x^5 \left (-3 a^3 D-4 a b (a C+6 b B)+192 A b^3\right )}{15 a^4 \left (a+b x^2\right )^{7/2}}-\frac{x^3 \left (48 A b^2-a (a C+6 b B)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac{x (8 A b-a B)}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{A}{a x \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(a*x*(a + b*x^2)^(7/2))) - ((8*A*b - a*B)*x)/(a^2*(a + b*x^2)^(7/2)) - ((48*A*b^2 - a*(6*b*B + a*C))*x^3)/
(3*a^3*(a + b*x^2)^(7/2)) - ((192*A*b^3 - 4*a*b*(6*b*B + a*C) - 3*a^3*D)*x^5)/(15*a^4*(a + b*x^2)^(7/2)) - (2*
b*(192*A*b^3 - 4*a*b*(6*b*B + a*C) - 3*a^3*D)*x^7)/(105*a^5*(a + b*x^2)^(7/2))

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]

Rule 12

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

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 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]

Rubi steps

\begin{align*} \int \frac{A+B x^2+C x^4+D x^6}{x^2 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{a x \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{8 A b-a \left (B+C x^2+D x^4\right )}{\left (a+b x^2\right )^{9/2}} \, dx}{a}\\ &=-\frac{A}{a x \left (a+b x^2\right )^{7/2}}-\frac{(8 A b-a B) x}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^2 \left (6 b (8 A b-a B)+a \left (-a C-a D x^2\right )\right )}{\left (a+b x^2\right )^{9/2}} \, dx}{a^2}\\ &=-\frac{A}{a x \left (a+b x^2\right )^{7/2}}-\frac{(8 A b-a B) x}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{\left (48 A b^2-a (6 b B+a C)\right ) x^3}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{\left (4 b \left (48 A b^2-6 a b B-a^2 C\right )-3 a^3 D\right ) x^4}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac{A}{a x \left (a+b x^2\right )^{7/2}}-\frac{(8 A b-a B) x}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{\left (48 A b^2-a (6 b B+a C)\right ) x^3}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-4 a b (6 b B+a C)-3 a^3 D\right ) \int \frac{x^4}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac{A}{a x \left (a+b x^2\right )^{7/2}}-\frac{(8 A b-a B) x}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{\left (48 A b^2-a (6 b B+a C)\right ) x^3}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-4 a b (6 b B+a C)-3 a^3 D\right ) x^5}{15 a^4 \left (a+b x^2\right )^{7/2}}-\frac{\left (2 b \left (192 A b^3-4 a b (6 b B+a C)-3 a^3 D\right )\right ) \int \frac{x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^4}\\ &=-\frac{A}{a x \left (a+b x^2\right )^{7/2}}-\frac{(8 A b-a B) x}{a^2 \left (a+b x^2\right )^{7/2}}-\frac{\left (48 A b^2-a (6 b B+a C)\right ) x^3}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-4 a b (6 b B+a C)-3 a^3 D\right ) x^5}{15 a^4 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (192 A b^3-4 a b (6 b B+a C)-3 a^3 D\right ) x^7}{105 a^5 \left (a+b x^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.152244, size = 133, normalized size = 0.72 \[ \frac{8 a^2 b^2 x^4 \left (-210 A+21 B x^2+C x^4\right )+2 a^3 b x^2 \left (-420 A+105 B x^2+14 C x^4+3 D x^6\right )-7 a^4 \left (15 A-15 B x^2-5 C x^4-3 D x^6\right )+48 a b^3 x^6 \left (B x^2-28 A\right )-384 A b^4 x^8}{105 a^5 x \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-384*A*b^4*x^8 + 48*a*b^3*x^6*(-28*A + B*x^2) + 8*a^2*b^2*x^4*(-210*A + 21*B*x^2 + C*x^4) - 7*a^4*(15*A - 15*
B*x^2 - 5*C*x^4 - 3*D*x^6) + 2*a^3*b*x^2*(-420*A + 105*B*x^2 + 14*C*x^4 + 3*D*x^6))/(105*a^5*x*(a + b*x^2)^(7/
2))

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Maple [A]  time = 0.006, size = 157, normalized size = 0.9 \begin{align*} -{\frac{384\,A{b}^{4}{x}^{8}-48\,Ba{b}^{3}{x}^{8}-8\,C{a}^{2}{b}^{2}{x}^{8}-6\,D{a}^{3}b{x}^{8}+1344\,Aa{b}^{3}{x}^{6}-168\,B{a}^{2}{b}^{2}{x}^{6}-28\,C{a}^{3}b{x}^{6}-21\,D{a}^{4}{x}^{6}+1680\,A{a}^{2}{b}^{2}{x}^{4}-210\,B{a}^{3}b{x}^{4}-35\,C{a}^{4}{x}^{4}+840\,A{a}^{3}b{x}^{2}-105\,B{a}^{4}{x}^{2}+105\,A{a}^{4}}{105\,x{a}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(384*A*b^4*x^8-48*B*a*b^3*x^8-8*C*a^2*b^2*x^8-6*D*a^3*b*x^8+1344*A*a*b^3*x^6-168*B*a^2*b^2*x^6-28*C*a^3
*b*x^6-21*D*a^4*x^6+1680*A*a^2*b^2*x^4-210*B*a^3*b*x^4-35*C*a^4*x^4+840*A*a^3*b*x^2-105*B*a^4*x^2+105*A*a^4)/x
/(b*x^2+a)^(7/2)/a^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.21629, size = 285, normalized size = 1.54 \begin{align*} \frac{{\left ({\left (x^{2}{\left (\frac{{\left (6 \, D a^{12} b^{4} + 8 \, C a^{11} b^{5} + 48 \, B a^{10} b^{6} - 279 \, A a^{9} b^{7}\right )} x^{2}}{a^{14} b^{3}} + \frac{7 \,{\left (3 \, D a^{13} b^{3} + 4 \, C a^{12} b^{4} + 24 \, B a^{11} b^{5} - 132 \, A a^{10} b^{6}\right )}}{a^{14} b^{3}}\right )} + \frac{35 \,{\left (C a^{13} b^{3} + 6 \, B a^{12} b^{4} - 30 \, A a^{11} b^{5}\right )}}{a^{14} b^{3}}\right )} x^{2} + \frac{105 \,{\left (B a^{13} b^{3} - 4 \, A a^{12} b^{4}\right )}}{a^{14} b^{3}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \, A \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*((x^2*((6*D*a^12*b^4 + 8*C*a^11*b^5 + 48*B*a^10*b^6 - 279*A*a^9*b^7)*x^2/(a^14*b^3) + 7*(3*D*a^13*b^3 +
4*C*a^12*b^4 + 24*B*a^11*b^5 - 132*A*a^10*b^6)/(a^14*b^3)) + 35*(C*a^13*b^3 + 6*B*a^12*b^4 - 30*A*a^11*b^5)/(a
^14*b^3))*x^2 + 105*(B*a^13*b^3 - 4*A*a^12*b^4)/(a^14*b^3))*x/(b*x^2 + a)^(7/2) + 2*A*sqrt(b)/(((sqrt(b)*x - s
qrt(b*x^2 + a))^2 - a)*a^4)