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

Optimal. Leaf size=134 \[ \frac{x^5 \left (a (3 a C+4 b B)+24 A b^2\right )}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac{x^3 (a B+6 A b)}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^7 \left (a \left (15 a^2 D+6 a b C+8 b^2 B\right )+48 A b^3\right )}{105 a^4 \left (a+b x^2\right )^{7/2}}+\frac{A x}{a \left (a+b x^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.404128, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{x^5 \left (a (3 a C+4 b B)+24 A b^2\right )}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac{x^3 (a B+6 A b)}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^7 \left (a \left (15 a^2 D+6 a b C+8 b^2 B\right )+48 A b^3\right )}{105 a^4 \left (a+b x^2\right )^{7/2}}+\frac{A x}{a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 82.8628, size = 206, normalized size = 1.54 \[ \frac{x \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{7 a b^{3} \left (a + b x^{2}\right )^{\frac{7}{2}}} + \frac{x \left (6 A b^{3} + B a b^{2} - 8 C a^{2} b + 15 D a^{3}\right )}{35 a^{2} b^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (24 A b^{3} + 4 B a b^{2} + 3 C a^{2} b - 10 D a^{3} + 35 D a^{2} b x^{2}\right )}{105 a^{3} b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{2 x \left (24 A b^{3} + 4 B a b^{2} + 3 C a^{2} b - 10 D a^{3}\right )}{105 a^{4} b^{3} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(A*b**3 - B*a*b**2 + C*a**2*b - D*a**3)/(7*a*b**3*(a + b*x**2)**(7/2)) + x*(6*
A*b**3 + B*a*b**2 - 8*C*a**2*b + 15*D*a**3)/(35*a**2*b**3*(a + b*x**2)**(5/2)) +
x*(24*A*b**3 + 4*B*a*b**2 + 3*C*a**2*b - 10*D*a**3 + 35*D*a**2*b*x**2)/(105*a**
3*b**3*(a + b*x**2)**(3/2)) + 2*x*(24*A*b**3 + 4*B*a*b**2 + 3*C*a**2*b - 10*D*a*
*3)/(105*a**4*b**3*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.13307, size = 98, normalized size = 0.73 \[ \frac{a^3 \left (105 A x+35 B x^3+21 C x^5+15 D x^7\right )+2 a^2 b x^3 \left (105 A+14 B x^2+3 C x^4\right )+8 a b^2 x^5 \left (21 A+B x^2\right )+48 A b^3 x^7}{105 a^4 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(48*A*b^3*x^7 + 8*a*b^2*x^5*(21*A + B*x^2) + 2*a^2*b*x^3*(105*A + 14*B*x^2 + 3*C
*x^4) + a^3*(105*A*x + 35*B*x^3 + 21*C*x^5 + 15*D*x^7))/(105*a^4*(a + b*x^2)^(7/
2))

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Maple [A]  time = 0.008, size = 109, normalized size = 0.8 \[{\frac{x \left ( 48\,A{b}^{3}{x}^{6}+8\,Ba{b}^{2}{x}^{6}+6\,{a}^{2}bC{x}^{6}+15\,D{a}^{3}{x}^{6}+168\,aA{b}^{2}{x}^{4}+28\,B{a}^{2}b{x}^{4}+21\,{a}^{3}C{x}^{4}+210\,{a}^{2}Ab{x}^{2}+35\,{a}^{3}B{x}^{2}+105\,A{a}^{3} \right ) }{105\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*x*(48*A*b^3*x^6+8*B*a*b^2*x^6+6*C*a^2*b*x^6+15*D*a^3*x^6+168*A*a*b^2*x^4+2
8*B*a^2*b*x^4+21*C*a^3*x^4+210*A*a^2*b*x^2+35*B*a^3*x^2+105*A*a^3)/(b*x^2+a)^(7/
2)/a^4

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Maxima [A]  time = 1.37042, size = 452, normalized size = 3.37 \[ -\frac{D x^{5}}{2 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{5 \, D a x^{3}}{8 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{C x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{16 \, A x}{35 \, \sqrt{b x^{2} + a} a^{4}} + \frac{8 \, A x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{6 \, A x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2}} + \frac{A x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a} + \frac{D x}{14 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{3}} + \frac{D x}{7 \, \sqrt{b x^{2} + a} a b^{3}} + \frac{3 \, D a x}{56 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{3}} - \frac{15 \, D a^{2} x}{56 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} + \frac{3 \, C x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, C x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{C x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, C a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{B x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{8 \, B x}{105 \, \sqrt{b x^{2} + a} a^{3} b} + \frac{4 \, B x}{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b} + \frac{B x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b} \]

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, algorithm="maxima")

[Out]

-1/2*D*x^5/((b*x^2 + a)^(7/2)*b) - 5/8*D*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1/4*C*x
^3/((b*x^2 + a)^(7/2)*b) + 16/35*A*x/(sqrt(b*x^2 + a)*a^4) + 8/35*A*x/((b*x^2 +
a)^(3/2)*a^3) + 6/35*A*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*A*x/((b*x^2 + a)^(7/2)*a)
+ 1/14*D*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*D*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*D*a
*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*D*a^2*x/((b*x^2 + a)^(7/2)*b^3) + 3/140*C*x/(
(b*x^2 + a)^(5/2)*b^2) + 2/35*C*x/(sqrt(b*x^2 + a)*a^2*b^2) + 1/35*C*x/((b*x^2 +
a)^(3/2)*a*b^2) - 3/28*C*a*x/((b*x^2 + a)^(7/2)*b^2) - 1/7*B*x/((b*x^2 + a)^(7/
2)*b) + 8/105*B*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*B*x/((b*x^2 + a)^(3/2)*a^2*b)
+ 1/35*B*x/((b*x^2 + a)^(5/2)*a*b)

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Fricas [A]  time = 0.358683, size = 190, normalized size = 1.42 \[ \frac{{\left ({\left (15 \, D a^{3} + 6 \, C a^{2} b + 8 \, B a b^{2} + 48 \, A b^{3}\right )} x^{7} + 7 \,{\left (3 \, C a^{3} + 4 \, B a^{2} b + 24 \, A a b^{2}\right )} x^{5} + 105 \, A a^{3} x + 35 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x^{3}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \]

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, algorithm="fricas")

[Out]

1/105*((15*D*a^3 + 6*C*a^2*b + 8*B*a*b^2 + 48*A*b^3)*x^7 + 7*(3*C*a^3 + 4*B*a^2*
b + 24*A*a*b^2)*x^5 + 105*A*a^3*x + 35*(B*a^3 + 6*A*a^2*b)*x^3)*sqrt(b*x^2 + a)/
(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22427, size = 177, normalized size = 1.32 \[ \frac{{\left ({\left (x^{2}{\left (\frac{{\left (15 \, D a^{3} b^{3} + 6 \, C a^{2} b^{4} + 8 \, B a b^{5} + 48 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac{7 \,{\left (3 \, C a^{3} b^{3} + 4 \, B a^{2} b^{4} + 24 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac{35 \,{\left (B a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac{105 \, A}{a}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]

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, algorithm="giac")

[Out]

1/105*((x^2*((15*D*a^3*b^3 + 6*C*a^2*b^4 + 8*B*a*b^5 + 48*A*b^6)*x^2/(a^4*b^3) +
7*(3*C*a^3*b^3 + 4*B*a^2*b^4 + 24*A*a*b^5)/(a^4*b^3)) + 35*(B*a^3*b^3 + 6*A*a^2
*b^4)/(a^4*b^3))*x^2 + 105*A/a)*x/(b*x^2 + a)^(7/2)