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

Optimal. Leaf size=127 \[ \frac{8 x (a C+6 A b)}{105 a^4 b \sqrt{a+b x^2}}+\frac{4 x (a C+6 A b)}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac{x (a C+6 A b)}{35 a^2 b \left (a+b x^2\right )^{5/2}}-\frac{a B-x (A b-a C)}{7 a b \left (a+b x^2\right )^{7/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.153129, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{8 x (a C+6 A b)}{105 a^4 b \sqrt{a+b x^2}}+\frac{4 x (a C+6 A b)}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac{x (a C+6 A b)}{35 a^2 b \left (a+b x^2\right )^{5/2}}-\frac{a B-x (A b-a C)}{7 a b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.633, size = 110, normalized size = 0.87 \[ - \frac{B a - x \left (A b - C a\right )}{7 a b \left (a + b x^{2}\right )^{\frac{7}{2}}} + \frac{x \left (6 A b + C a\right )}{35 a^{2} b \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{4 x \left (6 A b + C a\right )}{105 a^{3} b \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{8 x \left (6 A b + C a\right )}{105 a^{4} b \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0919634, size = 92, normalized size = 0.72 \[ \frac{-15 a^4 B+35 a^3 b x \left (3 A+C x^2\right )+14 a^2 b^2 x^3 \left (15 A+2 C x^2\right )+8 a b^3 x^5 \left (21 A+C x^2\right )+48 A b^4 x^7}{105 a^4 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 96, normalized size = 0.8 \[{\frac{48\,A{b}^{4}{x}^{7}+8\,Ca{b}^{3}{x}^{7}+168\,Aa{b}^{3}{x}^{5}+28\,C{a}^{2}{b}^{2}{x}^{5}+210\,A{a}^{2}{b}^{2}{x}^{3}+35\,C{a}^{3}b{x}^{3}+105\,Ax{a}^{3}b-15\,B{a}^{4}}{105\,{a}^{4}b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(48*A*b^4*x^7+8*C*a*b^3*x^7+168*A*a*b^3*x^5+28*C*a^2*b^2*x^5+210*A*a^2*b^2
*x^3+35*C*a^3*b*x^3+105*A*a^3*b*x-15*B*a^4)/(b*x^2+a)^(7/2)/a^4/b

_______________________________________________________________________________________

Maxima [A]  time = 1.48432, size = 207, normalized size = 1.63 \[ \frac{16 \, A x}{35 \, \sqrt{b x^{2} + a} a^{4}} + \frac{8 \, A x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{6 \, A x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2}} + \frac{A x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a} - \frac{C x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{8 \, C x}{105 \, \sqrt{b x^{2} + a} a^{3} b} + \frac{4 \, C x}{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b} + \frac{C x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b} - \frac{B}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.342734, size = 185, normalized size = 1.46 \[ \frac{{\left (8 \,{\left (C a b^{3} + 6 \, A b^{4}\right )} x^{7} + 105 \, A a^{3} b x + 28 \,{\left (C a^{2} b^{2} + 6 \, A a b^{3}\right )} x^{5} - 15 \, B a^{4} + 35 \,{\left (C a^{3} b + 6 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{4} b^{5} x^{8} + 4 \, a^{5} b^{4} x^{6} + 6 \, a^{6} b^{3} x^{4} + 4 \, a^{7} b^{2} x^{2} + a^{8} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22074, size = 151, normalized size = 1.19 \[ \frac{{\left ({\left (4 \, x^{2}{\left (\frac{2 \,{\left (C a b^{5} + 6 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac{7 \,{\left (C a^{2} b^{4} + 6 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac{35 \,{\left (C a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac{105 \, A}{a}\right )} x - \frac{15 \, B}{b}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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