3.51 \(\int \frac{x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.358691, 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 \[ \frac{2 B x}{35 a^2 b^2 \sqrt{a+b x^2}}-\frac{2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 29.0823, size = 124, normalized size = 0.89 \[ \frac{2 B x}{35 a^{2} b^{2} \sqrt{a + b x^{2}}} - \frac{x^{3} \left (B a - x \left (A b - C a\right )\right )}{7 a b \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{x^{2} \left (3 A b - 3 B b x + 4 C a\right )}{35 a b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{6 A b + 6 B b x + 8 C a}{105 a b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.0960272, size = 84, normalized size = 0.6 \[ \frac{-8 a^4 C-2 a^3 b \left (3 A+14 C x^2\right )-7 a^2 b^2 x^2 \left (3 A+5 C x^2\right )+21 a b^3 B x^5+6 b^4 B x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 85, normalized size = 0.6 \[ -{\frac{-6\,B{b}^{4}{x}^{7}-21\,B{x}^{5}a{b}^{3}+35\,C{x}^{4}{a}^{2}{b}^{2}+21\,A{a}^{2}{b}^{2}{x}^{2}+28\,C{a}^{3}b{x}^{2}+6\,A{a}^{3}b+8\,C{a}^{4}}{105\,{a}^{2}{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)

[Out]

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

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Maxima [A]  time = 1.38504, size = 242, normalized size = 1.74 \[ -\frac{C x^{4}}{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{B x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{4 \, C a x^{2}}{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{A x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{3 \, B x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, B x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{B x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, B a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, C a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} - \frac{2 \, A a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^3/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.285538, size = 177, normalized size = 1.27 \[ \frac{{\left (6 \, B b^{4} x^{7} + 21 \, B a b^{3} x^{5} - 35 \, C a^{2} b^{2} x^{4} - 8 \, C a^{4} - 6 \, A a^{3} b - 7 \,{\left (4 \, C a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{2} b^{7} x^{8} + 4 \, a^{3} b^{6} x^{6} + 6 \, a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^3/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(6*B*b^4*x^7 + 21*B*a*b^3*x^5 - 35*C*a^2*b^2*x^4 - 8*C*a^4 - 6*A*a^3*b - 7
*(4*C*a^3*b + 3*A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/(a^2*b^7*x^8 + 4*a^3*b^6*x^6 + 6
*a^4*b^5*x^4 + 4*a^5*b^4*x^2 + a^6*b^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(x**3*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22317, size = 128, normalized size = 0.92 \[ \frac{{\left ({\left (3 \,{\left (\frac{2 \, B b x^{2}}{a^{2}} + \frac{7 \, B}{a}\right )} x - \frac{35 \, C}{b}\right )} x^{2} - \frac{7 \,{\left (4 \, C a^{4} b + 3 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}\right )} x^{2} - \frac{2 \,{\left (4 \, C a^{5} + 3 \, A a^{4} b\right )}}{a^{3} b^{3}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^3/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

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