3.167 \(\int \frac{A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=334 \[ -\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]

[Out]

-A/(7*a*x^7*(a + b*x^2)^(7/2)) + (2*A*b - a*B)/(5*a^2*x^5*(a + b*x^2)^(7/2)) - (
24*A*b^2 - a*(12*b*B - 5*a*C))/(15*a^3*x^3*(a + b*x^2)^(7/2)) + (48*A*b^3 - a*(2
4*b^2*B - 10*a*b*C + 3*a^2*D))/(3*a^4*x*(a + b*x^2)^(7/2)) + (8*b*(48*A*b^3 - a*
(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(21*a^5*(a + b*x^2)^(7/2)) + (16*b*(48*A*b^3
- a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(35*a^6*(a + b*x^2)^(5/2)) + (64*b*(48*
A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^7*(a + b*x^2)^(3/2)) + (128
*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^8*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.992973, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ -\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]

[Out]

-A/(7*a*x^7*(a + b*x^2)^(7/2)) + (2*A*b - a*B)/(5*a^2*x^5*(a + b*x^2)^(7/2)) - (
24*A*b^2 - a*(12*b*B - 5*a*C))/(15*a^3*x^3*(a + b*x^2)^(7/2)) + (48*A*b^3 - a*(2
4*b^2*B - 10*a*b*C + 3*a^2*D))/(3*a^4*x*(a + b*x^2)^(7/2)) + (8*b*(48*A*b^3 - a*
(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(21*a^5*(a + b*x^2)^(7/2)) + (16*b*(48*A*b^3
- a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(35*a^6*(a + b*x^2)^(5/2)) + (64*b*(48*
A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^7*(a + b*x^2)^(3/2)) + (128
*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^8*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 179.705, size = 354, normalized size = 1.06 \[ - \frac{D}{10 b^{2} x^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (\frac{A b^{3}}{x^{8}} - \frac{B a b^{2}}{x^{8}} + \frac{C a^{2} b}{x^{8}} - \frac{D a^{3}}{x^{8}}\right )}{7 a b^{3} \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{B b^{2} - C a b + D a^{2}}{7 a b^{3} x^{7} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{24 B b^{2} - 38 C a b + 45 D a^{2}}{70 a^{2} b^{2} x^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{24 B b^{2} - 38 C a b + 45 D a^{2}}{21 a^{3} b x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{8 \left (24 B b^{2} - 38 C a b + 45 D a^{2}\right )}{21 a^{4} x \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{16 b x \left (24 B b^{2} - 38 C a b + 45 D a^{2}\right )}{35 a^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{64 b x \left (24 B b^{2} - 38 C a b + 45 D a^{2}\right )}{105 a^{6} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{128 b x \left (24 B b^{2} - 38 C a b + 45 D a^{2}\right )}{105 a^{7} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((D*x**6+C*x**4+B*x**2+A)/x**8/(b*x**2+a)**(9/2),x)

[Out]

-D/(10*b**2*x**5*(a + b*x**2)**(5/2)) + x*(A*b**3/x**8 - B*a*b**2/x**8 + C*a**2*
b/x**8 - D*a**3/x**8)/(7*a*b**3*(a + b*x**2)**(7/2)) - (B*b**2 - C*a*b + D*a**2)
/(7*a*b**3*x**7*(a + b*x**2)**(5/2)) + (24*B*b**2 - 38*C*a*b + 45*D*a**2)/(70*a*
*2*b**2*x**5*(a + b*x**2)**(5/2)) - (24*B*b**2 - 38*C*a*b + 45*D*a**2)/(21*a**3*
b*x**3*(a + b*x**2)**(5/2)) + 8*(24*B*b**2 - 38*C*a*b + 45*D*a**2)/(21*a**4*x*(a
+ b*x**2)**(5/2)) + 16*b*x*(24*B*b**2 - 38*C*a*b + 45*D*a**2)/(35*a**5*(a + b*x
**2)**(5/2)) + 64*b*x*(24*B*b**2 - 38*C*a*b + 45*D*a**2)/(105*a**6*(a + b*x**2)*
*(3/2)) + 128*b*x*(24*B*b**2 - 38*C*a*b + 45*D*a**2)/(105*a**7*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.428251, size = 234, normalized size = 0.7 \[ \frac{-a^7 \left (15 A+21 B x^2+35 x^4 \left (C+3 D x^2\right )\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 D x^6\right )-56 a^5 b^2 x^4 \left (3 A+15 B x^2-50 C x^4+30 D x^6\right )+112 a^4 b^3 x^6 \left (15 A-60 B x^2+50 C x^4-12 D x^6\right )+128 a^3 b^4 x^8 \left (105 A-105 B x^2+35 C x^4-3 D x^6\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )-3072 a b^6 x^{12} \left (B x^2-7 A\right )+6144 A b^7 x^{14}}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]

[Out]

(6144*A*b^7*x^14 - 3072*a*b^6*x^12*(-7*A + B*x^2) + 256*a^2*b^5*x^10*(105*A - 42
*B*x^2 + 5*C*x^4) + 14*a^6*b*x^2*(3*A + 6*B*x^2 + 25*C*x^4 - 60*D*x^6) + 112*a^4
*b^3*x^6*(15*A - 60*B*x^2 + 50*C*x^4 - 12*D*x^6) + 128*a^3*b^4*x^8*(105*A - 105*
B*x^2 + 35*C*x^4 - 3*D*x^6) - 56*a^5*b^2*x^4*(3*A + 15*B*x^2 - 50*C*x^4 + 30*D*x
^6) - a^7*(15*A + 21*B*x^2 + 35*x^4*(C + 3*D*x^2)))/(105*a^8*x^7*(a + b*x^2)^(7/
2))

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Maple [A]  time = 0.012, size = 301, normalized size = 0.9 \[ -{\frac{-6144\,A{b}^{7}{x}^{14}+3072\,Ba{b}^{6}{x}^{14}-1280\,C{a}^{2}{b}^{5}{x}^{14}+384\,D{a}^{3}{b}^{4}{x}^{14}-21504\,Aa{b}^{6}{x}^{12}+10752\,B{a}^{2}{b}^{5}{x}^{12}-4480\,C{a}^{3}{b}^{4}{x}^{12}+1344\,D{a}^{4}{b}^{3}{x}^{12}-26880\,A{a}^{2}{b}^{5}{x}^{10}+13440\,B{a}^{3}{b}^{4}{x}^{10}-5600\,C{a}^{4}{b}^{3}{x}^{10}+1680\,D{a}^{5}{b}^{2}{x}^{10}-13440\,A{a}^{3}{b}^{4}{x}^{8}+6720\,B{a}^{4}{b}^{3}{x}^{8}-2800\,C{a}^{5}{b}^{2}{x}^{8}+840\,D{a}^{6}b{x}^{8}-1680\,A{a}^{4}{b}^{3}{x}^{6}+840\,B{a}^{5}{b}^{2}{x}^{6}-350\,C{a}^{6}b{x}^{6}+105\,D{a}^{7}{x}^{6}+168\,A{a}^{5}{b}^{2}{x}^{4}-84\,B{a}^{6}b{x}^{4}+35\,C{a}^{7}{x}^{4}-42\,A{a}^{6}b{x}^{2}+21\,B{a}^{7}{x}^{2}+15\,A{a}^{7}}{105\,{x}^{7}{a}^{8}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((D*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(-6144*A*b^7*x^14+3072*B*a*b^6*x^14-1280*C*a^2*b^5*x^14+384*D*a^3*b^4*x^1
4-21504*A*a*b^6*x^12+10752*B*a^2*b^5*x^12-4480*C*a^3*b^4*x^12+1344*D*a^4*b^3*x^1
2-26880*A*a^2*b^5*x^10+13440*B*a^3*b^4*x^10-5600*C*a^4*b^3*x^10+1680*D*a^5*b^2*x
^10-13440*A*a^3*b^4*x^8+6720*B*a^4*b^3*x^8-2800*C*a^5*b^2*x^8+840*D*a^6*b*x^8-16
80*A*a^4*b^3*x^6+840*B*a^5*b^2*x^6-350*C*a^6*b*x^6+105*D*a^7*x^6+168*A*a^5*b^2*x
^4-84*B*a^6*b*x^4+35*C*a^7*x^4-42*A*a^6*b*x^2+21*B*a^7*x^2+15*A*a^7)/x^7/(b*x^2+
a)^(7/2)/a^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)/((b*x^2 + a)^(9/2)*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.96954, size = 420, normalized size = 1.26 \[ -\frac{{\left (128 \,{\left (3 \, D a^{3} b^{4} - 10 \, C a^{2} b^{5} + 24 \, B a b^{6} - 48 \, A b^{7}\right )} x^{14} + 448 \,{\left (3 \, D a^{4} b^{3} - 10 \, C a^{3} b^{4} + 24 \, B a^{2} b^{5} - 48 \, A a b^{6}\right )} x^{12} + 560 \,{\left (3 \, D a^{5} b^{2} - 10 \, C a^{4} b^{3} + 24 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5}\right )} x^{10} + 280 \,{\left (3 \, D a^{6} b - 10 \, C a^{5} b^{2} + 24 \, B a^{4} b^{3} - 48 \, A a^{3} b^{4}\right )} x^{8} + 15 \, A a^{7} + 35 \,{\left (3 \, D a^{7} - 10 \, C a^{6} b + 24 \, B a^{5} b^{2} - 48 \, A a^{4} b^{3}\right )} x^{6} + 7 \,{\left (5 \, C a^{7} - 12 \, B a^{6} b + 24 \, A a^{5} b^{2}\right )} x^{4} + 21 \,{\left (B a^{7} - 2 \, A a^{6} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{8} b^{4} x^{15} + 4 \, a^{9} b^{3} x^{13} + 6 \, a^{10} b^{2} x^{11} + 4 \, a^{11} b x^{9} + a^{12} x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)/((b*x^2 + a)^(9/2)*x^8),x, algorithm="fricas")

[Out]

-1/105*(128*(3*D*a^3*b^4 - 10*C*a^2*b^5 + 24*B*a*b^6 - 48*A*b^7)*x^14 + 448*(3*D
*a^4*b^3 - 10*C*a^3*b^4 + 24*B*a^2*b^5 - 48*A*a*b^6)*x^12 + 560*(3*D*a^5*b^2 - 1
0*C*a^4*b^3 + 24*B*a^3*b^4 - 48*A*a^2*b^5)*x^10 + 280*(3*D*a^6*b - 10*C*a^5*b^2
+ 24*B*a^4*b^3 - 48*A*a^3*b^4)*x^8 + 15*A*a^7 + 35*(3*D*a^7 - 10*C*a^6*b + 24*B*
a^5*b^2 - 48*A*a^4*b^3)*x^6 + 7*(5*C*a^7 - 12*B*a^6*b + 24*A*a^5*b^2)*x^4 + 21*(
B*a^7 - 2*A*a^6*b)*x^2)*sqrt(b*x^2 + a)/(a^8*b^4*x^15 + 4*a^9*b^3*x^13 + 6*a^10*
b^2*x^11 + 4*a^11*b*x^9 + a^12*x^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x**6+C*x**4+B*x**2+A)/x**8/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.255198, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)/((b*x^2 + a)^(9/2)*x^8),x, algorithm="giac")

[Out]

Done