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

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]

-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*(48*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 = 0.670315, 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 \[ \frac{x \left (80 A b^2-3 a (8 b B-a C)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{8 b^2 x^7 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac{x^3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{3 a^4 \left (a+b x^2\right )^{7/2}}-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]

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*(48*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 in Sympy [A]  time = 163.22, size = 270, normalized size = 1.12 \[ - \frac{D}{6 b^{2} x \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (\frac{A b^{3}}{x^{4}} - \frac{B a b^{2}}{x^{4}} + \frac{C a^{2} b}{x^{4}} - \frac{D a^{3}}{x^{4}}\right )}{7 a b^{3} \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{B b^{2} - C a b + D a^{2}}{3 a b^{3} x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{16 B b^{2} - 22 C a b + 23 D a^{2}}{6 a^{2} b^{2} x \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (16 B b^{2} - 22 C a b + 23 D a^{2}\right )}{5 a^{3} b \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{4 x \left (16 B b^{2} - 22 C a b + 23 D a^{2}\right )}{15 a^{4} b \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{8 x \left (16 B b^{2} - 22 C a b + 23 D a^{2}\right )}{15 a^{5} b \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-D/(6*b**2*x*(a + b*x**2)**(5/2)) + x*(A*b**3/x**4 - B*a*b**2/x**4 + C*a**2*b/x*
*4 - D*a**3/x**4)/(7*a*b**3*(a + b*x**2)**(7/2)) - (B*b**2 - C*a*b + D*a**2)/(3*
a*b**3*x**3*(a + b*x**2)**(5/2)) + (16*B*b**2 - 22*C*a*b + 23*D*a**2)/(6*a**2*b*
*2*x*(a + b*x**2)**(5/2)) + x*(16*B*b**2 - 22*C*a*b + 23*D*a**2)/(5*a**3*b*(a +
b*x**2)**(5/2)) + 4*x*(16*B*b**2 - 22*C*a*b + 23*D*a**2)/(15*a**4*b*(a + b*x**2)
**(3/2)) + 8*x*(16*B*b**2 - 22*C*a*b + 23*D*a**2)/(15*a**5*b*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.211512, 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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Maple [A]  time = 0.012, size = 205, normalized size = 0.9 \[ -{\frac{-1280\,A{b}^{5}{x}^{10}+384\,Ba{b}^{4}{x}^{10}-48\,C{a}^{2}{b}^{3}{x}^{10}-8\,D{a}^{3}{b}^{2}{x}^{10}-4480\,Aa{b}^{4}{x}^{8}+1344\,B{a}^{2}{b}^{3}{x}^{8}-168\,C{a}^{3}{b}^{2}{x}^{8}-28\,D{a}^{4}b{x}^{8}-5600\,A{a}^{2}{b}^{3}{x}^{6}+1680\,B{a}^{3}{b}^{2}{x}^{6}-210\,C{a}^{4}b{x}^{6}-35\,D{a}^{5}{x}^{6}-2800\,A{a}^{3}{b}^{2}{x}^{4}+840\,B{a}^{4}b{x}^{4}-105\,C{a}^{5}{x}^{4}-350\,A{a}^{4}b{x}^{2}+105\,B{a}^{5}{x}^{2}+35\,A{a}^{5}}{105\,{x}^{3}{a}^{6}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

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^10-448
0*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]  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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.680248, size = 304, normalized size = 1.26 \[ \frac{{\left (8 \,{\left (D a^{3} b^{2} + 6 \, C a^{2} b^{3} - 48 \, B a b^{4} + 160 \, A b^{5}\right )} x^{10} + 28 \,{\left (D a^{4} b + 6 \, C a^{3} b^{2} - 48 \, B a^{2} b^{3} + 160 \, A a b^{4}\right )} x^{8} + 35 \,{\left (D a^{5} + 6 \, C a^{4} b - 48 \, B a^{3} b^{2} + 160 \, A a^{2} b^{3}\right )} x^{6} - 35 \, A a^{5} + 35 \,{\left (3 \, C a^{5} - 24 \, B a^{4} b + 80 \, A a^{3} b^{2}\right )} x^{4} - 35 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22656, size = 471, normalized size = 1.95 \[ \frac{{\left ({\left (x^{2}{\left (\frac{{\left (8 \, D a^{15} b^{5} + 48 \, C a^{14} b^{6} - 279 \, B a^{13} b^{7} + 790 \, A a^{12} b^{8}\right )} x^{2}}{a^{18} b^{3}} + \frac{7 \,{\left (4 \, D a^{16} b^{4} + 24 \, C a^{15} b^{5} - 132 \, B a^{14} b^{6} + 365 \, A a^{13} b^{7}\right )}}{a^{18} b^{3}}\right )} + \frac{35 \,{\left (D a^{17} b^{3} + 6 \, C a^{16} b^{4} - 30 \, B a^{15} b^{5} + 80 \, A a^{14} b^{6}\right )}}{a^{18} b^{3}}\right )} x^{2} + \frac{105 \,{\left (C a^{17} b^{3} - 4 \, B a^{16} b^{4} + 10 \, A a^{15} b^{5}\right )}}{a^{18} b^{3}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a \sqrt{b} - 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} \sqrt{b} + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{3}{2}} + 3 \, B a^{3} \sqrt{b} - 14 \, A a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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