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

[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))

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Rubi [A]  time = 0.321803, antiderivative size = 242, normalized size of antiderivative = 1., 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 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^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*}

Mathematica [A]  time = 0.129439, size = 165, normalized size = 0.68 \[ \frac{8 a^3 b^2 x^4 \left (350 A-210 B x^2+21 C x^4+D x^6\right )+16 a^2 b^3 x^6 \left (350 A-84 B x^2+3 C x^4\right )+14 a^4 b x^2 \left (25 A-60 B x^2+15 C x^4+2 D x^6\right )-35 a^5 \left (A+3 B x^2-3 C x^4-D x^6\right )+128 a b^4 x^8 \left (35 A-3 B x^2\right )+1280 A b^5 x^{10}}{105 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]

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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Maple [A]  time = 0.007, size = 205, normalized size = 0.9 \begin{align*} -{\frac{-1280\,A{b}^{5}{x}^{10}+384\,Ba{b}^{4}{x}^{10}-48\,C{a}^{2}{b}^{3}{x}^{10}-8\,D{a}^{3}{b}^{2}{x}^{10}-4480\,Aa{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\,{x}^{3}{a}^{6}} \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^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)/x^3/(b*x^2+a)^(7/2)/a^6

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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^4/(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^4/(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**4/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21572, size = 471, normalized size = 1.95 \begin{align*} \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}} \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^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)