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

Optimal. Leaf size=281 \[ -\frac{16 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^7 \sqrt{a+b x^2}}-\frac{8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac{2 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \]

[Out]

-A/(5*a*x^5*(a + b*x^2)^(7/2)) + (12*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(7/2)) - (24*A*b^2 - a*(10*b*B - 3*a
*C))/(3*a^3*x*(a + b*x^2)^(7/2)) - ((192*A*b^3 - a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(21*a^4*(a + b*x^2)^(7/
2)) - (2*(192*A*b^3 - a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(35*a^5*(a + b*x^2)^(5/2)) - (8*(192*A*b^3 - a*(80
*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(105*a^6*(a + b*x^2)^(3/2)) - (16*(192*A*b^3 - a*(80*b^2*B - 24*a*b*C + 3*a^2
*D))*x)/(105*a^7*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.428996, antiderivative size = 275, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1803, 12, 192, 191} \[ -\frac{16 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{105 a^7 \sqrt{a+b x^2}}-\frac{8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac{2 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(5*a*x^5*(a + b*x^2)^(7/2)) + (12*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(7/2)) - (24*A*b^2 - a*(10*b*B - 3*a
*C))/(3*a^3*x*(a + b*x^2)^(7/2)) - ((192*A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(21*a^4*(a + b*x^2)^(7/2
)) - (2*(192*A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(35*a^5*(a + b*x^2)^(5/2)) - (8*(192*A*b^3 - a*(80*b
^2*B - 24*a*b*C + 3*a^2*D))*x)/(105*a^6*(a + b*x^2)^(3/2)) - (16*(192*A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3*D
)*x)/(105*a^7*Sqrt[a + b*x^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 12

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

Rule 192

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

Rule 191

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

Rubi steps

\begin{align*} \int \frac{A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{12 A b-5 a \left (B+C x^2+D x^4\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{5 a}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{10 b (12 A b-5 a B)-3 a \left (-5 a C-5 a D x^2\right )}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{15 a^2}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{8 b \left (120 A b^2-50 a b B+15 a^2 C\right )-15 a^3 D}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{\left (2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^4}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{\left (8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^5}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac{\left (16 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^6}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac{\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac{16 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^7 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.146291, size = 202, normalized size = 0.72 \[ \frac{16 a^3 b^3 x^6 \left (-420 A+350 B x^2-84 C x^4+3 D x^6\right )+56 a^4 b^2 x^4 \left (-15 A+50 B x^2-30 C x^4+3 D x^6\right )-128 a^2 b^4 x^8 \left (105 A-35 B x^2+3 C x^4\right )+14 a^5 b x^2 \left (6 A+25 B x^2-60 C x^4+15 D x^6\right )-7 a^6 \left (3 A+5 x^2 \left (B+3 C x^2-3 D x^4\right )\right )+256 a b^5 x^{10} \left (5 B x^2-42 A\right )-3072 A b^6 x^{12}}{105 a^7 x^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3072*A*b^6*x^12 + 256*a*b^5*x^10*(-42*A + 5*B*x^2) - 128*a^2*b^4*x^8*(105*A - 35*B*x^2 + 3*C*x^4) + 16*a^3*b
^3*x^6*(-420*A + 350*B*x^2 - 84*C*x^4 + 3*D*x^6) + 56*a^4*b^2*x^4*(-15*A + 50*B*x^2 - 30*C*x^4 + 3*D*x^6) + 14
*a^5*b*x^2*(6*A + 25*B*x^2 - 60*C*x^4 + 15*D*x^6) - 7*a^6*(3*A + 5*x^2*(B + 3*C*x^2 - 3*D*x^4)))/(105*a^7*x^5*
(a + b*x^2)^(7/2))

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Maple [A]  time = 0.007, size = 253, normalized size = 0.9 \begin{align*} -{\frac{3072\,A{b}^{6}{x}^{12}-1280\,Ba{b}^{5}{x}^{12}+384\,C{a}^{2}{b}^{4}{x}^{12}-48\,D{a}^{3}{b}^{3}{x}^{12}+10752\,Aa{b}^{5}{x}^{10}-4480\,B{a}^{2}{b}^{4}{x}^{10}+1344\,C{a}^{3}{b}^{3}{x}^{10}-168\,D{a}^{4}{b}^{2}{x}^{10}+13440\,A{a}^{2}{b}^{4}{x}^{8}-5600\,B{a}^{3}{b}^{3}{x}^{8}+1680\,C{a}^{4}{b}^{2}{x}^{8}-210\,D{a}^{5}b{x}^{8}+6720\,A{a}^{3}{b}^{3}{x}^{6}-2800\,B{a}^{4}{b}^{2}{x}^{6}+840\,C{a}^{5}b{x}^{6}-105\,D{a}^{6}{x}^{6}+840\,A{a}^{4}{b}^{2}{x}^{4}-350\,B{a}^{5}b{x}^{4}+105\,C{a}^{6}{x}^{4}-84\,A{a}^{5}b{x}^{2}+35\,B{a}^{6}{x}^{2}+21\,A{a}^{6}}{105\,{x}^{5}{a}^{7}} \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^6/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(3072*A*b^6*x^12-1280*B*a*b^5*x^12+384*C*a^2*b^4*x^12-48*D*a^3*b^3*x^12+10752*A*a*b^5*x^10-4480*B*a^2*b
^4*x^10+1344*C*a^3*b^3*x^10-168*D*a^4*b^2*x^10+13440*A*a^2*b^4*x^8-5600*B*a^3*b^3*x^8+1680*C*a^4*b^2*x^8-210*D
*a^5*b*x^8+6720*A*a^3*b^3*x^6-2800*B*a^4*b^2*x^6+840*C*a^5*b*x^6-105*D*a^6*x^6+840*A*a^4*b^2*x^4-350*B*a^5*b*x
^4+105*C*a^6*x^4-84*A*a^5*b*x^2+35*B*a^6*x^2+21*A*a^6)/x^5/(b*x^2+a)^(7/2)/a^7

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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^6/(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^6/(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**6/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.25856, size = 799, normalized size = 2.84 \begin{align*} \frac{{\left ({\left (x^{2}{\left (\frac{{\left (48 \, D a^{18} b^{6} - 279 \, C a^{17} b^{7} + 790 \, B a^{16} b^{8} - 1686 \, A a^{15} b^{9}\right )} x^{2}}{a^{22} b^{3}} + \frac{7 \,{\left (24 \, D a^{19} b^{5} - 132 \, C a^{18} b^{6} + 365 \, B a^{17} b^{7} - 768 \, A a^{16} b^{8}\right )}}{a^{22} b^{3}}\right )} + \frac{35 \,{\left (6 \, D a^{20} b^{4} - 30 \, C a^{19} b^{5} + 80 \, B a^{18} b^{6} - 165 \, A a^{17} b^{7}\right )}}{a^{22} b^{3}}\right )} x^{2} + \frac{105 \,{\left (D a^{21} b^{3} - 4 \, C a^{20} b^{4} + 10 \, B a^{19} b^{5} - 20 \, A a^{18} b^{6}\right )}}{a^{22} b^{3}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} C a^{2} \sqrt{b} - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} + 150 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} C a^{3} \sqrt{b} + 270 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} - 720 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{5}{2}} + 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} C a^{4} \sqrt{b} - 430 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} + 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} C a^{5} \sqrt{b} + 290 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} - 840 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{5}{2}} + 15 \, C a^{6} \sqrt{b} - 70 \, B a^{5} b^{\frac{3}{2}} + 198 \, A a^{4} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \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^6/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*((48*D*a^18*b^6 - 279*C*a^17*b^7 + 790*B*a^16*b^8 - 1686*A*a^15*b^9)*x^2/(a^22*b^3) + 7*(24*D*a^19
*b^5 - 132*C*a^18*b^6 + 365*B*a^17*b^7 - 768*A*a^16*b^8)/(a^22*b^3)) + 35*(6*D*a^20*b^4 - 30*C*a^19*b^5 + 80*B
*a^18*b^6 - 165*A*a^17*b^7)/(a^22*b^3))*x^2 + 105*(D*a^21*b^3 - 4*C*a^20*b^4 + 10*B*a^19*b^5 - 20*A*a^18*b^6)/
(a^22*b^3))*x/(b*x^2 + a)^(7/2) + 2/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^2*sqrt(b) - 60*(sqrt(b)*x - sqr
t(b*x^2 + a))^8*B*a*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))
^6*C*a^3*sqrt(b) + 270*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a
*b^(5/2) + 90*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^4*sqrt(b) - 430*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2
) + 1260*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^5*sqrt(b) + 29
0*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) - 840*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(5/2) + 15*C*a^6
*sqrt(b) - 70*B*a^5*b^(3/2) + 198*A*a^4*b^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^6)