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

Optimal. Leaf size=139 \[ \frac{2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt{a+b x^2}}+\frac{x (3 a C+4 A b)}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac{x (3 a C+4 A b)+2 a B}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

[Out]

-(x^2*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (2*a*B + (4*A*b + 3*a*C
)*x)/(35*a*b^2*(a + b*x^2)^(5/2)) + ((4*A*b + 3*a*C)*x)/(105*a^2*b^2*(a + b*x^2)
^(3/2)) + (2*(4*A*b + 3*a*C)*x)/(105*a^3*b^2*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.297266, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt{a+b x^2}}+\frac{x (3 a C+4 A b)}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac{x (3 a C+4 A b)+2 a B}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(x^2*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (2*a*B + (4*A*b + 3*a*C
)*x)/(35*a*b^2*(a + b*x^2)^(5/2)) + ((4*A*b + 3*a*C)*x)/(105*a^2*b^2*(a + b*x^2)
^(3/2)) + (2*(4*A*b + 3*a*C)*x)/(105*a^3*b^2*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 23.5693, size = 126, normalized size = 0.91 \[ - \frac{x^{2} \left (B a - x \left (A b - C a\right )\right )}{7 a b \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{2 B a + x \left (4 A b + 3 C a\right )}{35 a b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (4 A b + 3 C a\right )}{105 a^{2} b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{2 x \left (4 A b + 3 C a\right )}{105 a^{3} b^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*(B*a - x*(A*b - C*a))/(7*a*b*(a + b*x**2)**(7/2)) - (2*B*a + x*(4*A*b + 3*
C*a))/(35*a*b**2*(a + b*x**2)**(5/2)) + x*(4*A*b + 3*C*a)/(105*a**2*b**2*(a + b*
x**2)**(3/2)) + 2*x*(4*A*b + 3*C*a)/(105*a**3*b**2*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.107597, size = 87, normalized size = 0.63 \[ \frac{-6 a^4 B-21 a^3 b B x^2+7 a^2 b^2 x^3 \left (5 A+3 C x^2\right )+2 a b^3 x^5 \left (14 A+3 C x^2\right )+8 A b^4 x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*a^4*B - 21*a^3*b*B*x^2 + 8*A*b^4*x^7 + 7*a^2*b^2*x^3*(5*A + 3*C*x^2) + 2*a*b
^3*x^5*(14*A + 3*C*x^2))/(105*a^3*b^2*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.009, size = 88, normalized size = 0.6 \[{\frac{8\,A{b}^{4}{x}^{7}+6\,Ca{b}^{3}{x}^{7}+28\,Aa{b}^{3}{x}^{5}+21\,C{a}^{2}{b}^{2}{x}^{5}+35\,A{x}^{3}{a}^{2}{b}^{2}-21\,B{x}^{2}{a}^{3}b-6\,B{a}^{4}}{105\,{a}^{3}{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(8*A*b^4*x^7+6*C*a*b^3*x^7+28*A*a*b^3*x^5+21*C*a^2*b^2*x^5+35*A*a^2*b^2*x^
3-21*B*a^3*b*x^2-6*B*a^4)/(b*x^2+a)^(7/2)/a^3/b^2

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Maxima [A]  time = 1.36151, size = 266, normalized size = 1.91 \[ -\frac{C x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{B x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{3 \, C x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, C x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{C x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, C a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{A x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{8 \, A x}{105 \, \sqrt{b x^{2} + a} a^{3} b} + \frac{4 \, A x}{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b} + \frac{A x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b} - \frac{2 \, B a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*C*x^3/((b*x^2 + a)^(7/2)*b) - 1/5*B*x^2/((b*x^2 + a)^(7/2)*b) + 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*A*x/((b*x^2 + a)^(7/
2)*b) + 8/105*A*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*A*x/((b*x^2 + a)^(3/2)*a^2*b)
+ 1/35*A*x/((b*x^2 + a)^(5/2)*a*b) - 2/35*B*a/((b*x^2 + a)^(7/2)*b^2)

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Fricas [A]  time = 0.3116, size = 181, normalized size = 1.3 \[ \frac{{\left (35 \, A a^{2} b^{2} x^{3} + 2 \,{\left (3 \, C a b^{3} + 4 \, A b^{4}\right )} x^{7} - 21 \, B a^{3} b x^{2} + 7 \,{\left (3 \, C a^{2} b^{2} + 4 \, A a b^{3}\right )} x^{5} - 6 \, B a^{4}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{3} b^{6} x^{8} + 4 \, a^{4} b^{5} x^{6} + 6 \, a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(35*A*a^2*b^2*x^3 + 2*(3*C*a*b^3 + 4*A*b^4)*x^7 - 21*B*a^3*b*x^2 + 7*(3*C*
a^2*b^2 + 4*A*a*b^3)*x^5 - 6*B*a^4)*sqrt(b*x^2 + a)/(a^3*b^6*x^8 + 4*a^4*b^5*x^6
+ 6*a^5*b^4*x^4 + 4*a^6*b^3*x^2 + a^7*b^2)

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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(x**2*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219227, size = 127, normalized size = 0.91 \[ \frac{{\left ({\left (x^{2}{\left (\frac{2 \,{\left (3 \, C a b^{4} + 4 \, A b^{5}\right )} x^{2}}{a^{3} b^{3}} + \frac{7 \,{\left (3 \, C a^{2} b^{3} + 4 \, A a b^{4}\right )}}{a^{3} b^{3}}\right )} + \frac{35 \, A}{a}\right )} x - \frac{21 \, B}{b}\right )} x^{2} - \frac{6 \, B a}{b^{2}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(((x^2*(2*(3*C*a*b^4 + 4*A*b^5)*x^2/(a^3*b^3) + 7*(3*C*a^2*b^3 + 4*A*a*b^4
)/(a^3*b^3)) + 35*A/a)*x - 21*B/b)*x^2 - 6*B*a/b^2)/(b*x^2 + a)^(7/2)