3.287 \(\int \frac{a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{2 e x^7 \left (4 e (6 a e+b d)+3 c d^2\right )}{105 d^4 \left (d+e x^2\right )^{7/2}}+\frac{x^5 \left (4 e (6 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac{x^3 (6 a e+b d)}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{a x}{d \left (d+e x^2\right )^{7/2}} \]

[Out]

(a*x)/(d*(d + e*x^2)^(7/2)) + ((b*d + 6*a*e)*x^3)/(3*d^2*(d + e*x^2)^(7/2)) + ((
3*c*d^2 + 4*e*(b*d + 6*a*e))*x^5)/(15*d^3*(d + e*x^2)^(7/2)) + (2*e*(3*c*d^2 + 4
*e*(b*d + 6*a*e))*x^7)/(105*d^4*(d + e*x^2)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.267988, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 e x^7 \left (4 e (6 a e+b d)+3 c d^2\right )}{105 d^4 \left (d+e x^2\right )^{7/2}}+\frac{x^5 \left (4 e (6 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac{x^3 (6 a e+b d)}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{a x}{d \left (d+e x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(9/2),x]

[Out]

(a*x)/(d*(d + e*x^2)^(7/2)) + ((b*d + 6*a*e)*x^3)/(3*d^2*(d + e*x^2)^(7/2)) + ((
3*c*d^2 + 4*e*(b*d + 6*a*e))*x^5)/(15*d^3*(d + e*x^2)^(7/2)) + (2*e*(3*c*d^2 + 4
*e*(b*d + 6*a*e))*x^7)/(105*d^4*(d + e*x^2)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.5851, size = 155, normalized size = 1.23 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{7 d e^{2} \left (d + e x^{2}\right )^{\frac{7}{2}}} + \frac{x \left (6 a e^{2} + b d e - 8 c d^{2}\right )}{35 d^{2} e^{2} \left (d + e x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (24 a e^{2} + 4 b d e + 3 c d^{2}\right )}{105 d^{3} e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{2 x \left (24 a e^{2} + 4 b d e + 3 c d^{2}\right )}{105 d^{4} e^{2} \sqrt{d + e x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(9/2),x)

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(7*d*e**2*(d + e*x**2)**(7/2)) + x*(6*a*e**2 + b*d*e
- 8*c*d**2)/(35*d**2*e**2*(d + e*x**2)**(5/2)) + x*(24*a*e**2 + 4*b*d*e + 3*c*d
**2)/(105*d**3*e**2*(d + e*x**2)**(3/2)) + 2*x*(24*a*e**2 + 4*b*d*e + 3*c*d**2)/
(105*d**4*e**2*sqrt(d + e*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.109725, size = 101, normalized size = 0.8 \[ \frac{3 a \left (35 d^3 x+70 d^2 e x^3+56 d e^2 x^5+16 e^3 x^7\right )+d x^3 \left (b \left (35 d^2+28 d e x^2+8 e^2 x^4\right )+3 c d x^2 \left (7 d+2 e x^2\right )\right )}{105 d^4 \left (d+e x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(9/2),x]

[Out]

(3*a*(35*d^3*x + 70*d^2*e*x^3 + 56*d*e^2*x^5 + 16*e^3*x^7) + d*x^3*(3*c*d*x^2*(7
*d + 2*e*x^2) + b*(35*d^2 + 28*d*e*x^2 + 8*e^2*x^4)))/(105*d^4*(d + e*x^2)^(7/2)
)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 100, normalized size = 0.8 \[{\frac{x \left ( 48\,a{e}^{3}{x}^{6}+8\,bd{e}^{2}{x}^{6}+6\,c{d}^{2}e{x}^{6}+168\,ad{e}^{2}{x}^{4}+28\,b{d}^{2}e{x}^{4}+21\,c{d}^{3}{x}^{4}+210\,a{d}^{2}e{x}^{2}+35\,b{d}^{3}{x}^{2}+105\,a{d}^{3} \right ) }{105\,{d}^{4}} \left ( e{x}^{2}+d \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/(e*x^2+d)^(9/2),x)

[Out]

1/105*x*(48*a*e^3*x^6+8*b*d*e^2*x^6+6*c*d^2*e*x^6+168*a*d*e^2*x^4+28*b*d^2*e*x^4
+21*c*d^3*x^4+210*a*d^2*e*x^2+35*b*d^3*x^2+105*a*d^3)/(e*x^2+d)^(7/2)/d^4

_______________________________________________________________________________________

Maxima [A]  time = 0.756594, size = 306, normalized size = 2.43 \[ -\frac{c x^{3}}{4 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e} + \frac{16 \, a x}{35 \, \sqrt{e x^{2} + d} d^{4}} + \frac{8 \, a x}{35 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3}} + \frac{6 \, a x}{35 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2}} + \frac{a x}{7 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d} + \frac{3 \, c x}{140 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} e^{2}} + \frac{2 \, c x}{35 \, \sqrt{e x^{2} + d} d^{2} e^{2}} + \frac{c x}{35 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d e^{2}} - \frac{3 \, c d x}{28 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e^{2}} - \frac{b x}{7 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e} + \frac{8 \, b x}{105 \, \sqrt{e x^{2} + d} d^{3} e} + \frac{4 \, b x}{105 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2} e} + \frac{b x}{35 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(9/2),x, algorithm="maxima")

[Out]

-1/4*c*x^3/((e*x^2 + d)^(7/2)*e) + 16/35*a*x/(sqrt(e*x^2 + d)*d^4) + 8/35*a*x/((
e*x^2 + d)^(3/2)*d^3) + 6/35*a*x/((e*x^2 + d)^(5/2)*d^2) + 1/7*a*x/((e*x^2 + d)^
(7/2)*d) + 3/140*c*x/((e*x^2 + d)^(5/2)*e^2) + 2/35*c*x/(sqrt(e*x^2 + d)*d^2*e^2
) + 1/35*c*x/((e*x^2 + d)^(3/2)*d*e^2) - 3/28*c*d*x/((e*x^2 + d)^(7/2)*e^2) - 1/
7*b*x/((e*x^2 + d)^(7/2)*e) + 8/105*b*x/(sqrt(e*x^2 + d)*d^3*e) + 4/105*b*x/((e*
x^2 + d)^(3/2)*d^2*e) + 1/35*b*x/((e*x^2 + d)^(5/2)*d*e)

_______________________________________________________________________________________

Fricas [A]  time = 0.402601, size = 184, normalized size = 1.46 \[ \frac{{\left (2 \,{\left (3 \, c d^{2} e + 4 \, b d e^{2} + 24 \, a e^{3}\right )} x^{7} + 7 \,{\left (3 \, c d^{3} + 4 \, b d^{2} e + 24 \, a d e^{2}\right )} x^{5} + 105 \, a d^{3} x + 35 \,{\left (b d^{3} + 6 \, a d^{2} e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{105 \,{\left (d^{4} e^{4} x^{8} + 4 \, d^{5} e^{3} x^{6} + 6 \, d^{6} e^{2} x^{4} + 4 \, d^{7} e x^{2} + d^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(9/2),x, algorithm="fricas")

[Out]

1/105*(2*(3*c*d^2*e + 4*b*d*e^2 + 24*a*e^3)*x^7 + 7*(3*c*d^3 + 4*b*d^2*e + 24*a*
d*e^2)*x^5 + 105*a*d^3*x + 35*(b*d^3 + 6*a*d^2*e)*x^3)*sqrt(e*x^2 + d)/(d^4*e^4*
x^8 + 4*d^5*e^3*x^6 + 6*d^6*e^2*x^4 + 4*d^7*e*x^2 + d^8)

_______________________________________________________________________________________

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**4+b*x**2+a)/(e*x**2+d)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269935, size = 153, normalized size = 1.21 \[ \frac{{\left ({\left (x^{2}{\left (\frac{2 \,{\left (3 \, c d^{2} e^{4} + 4 \, b d e^{5} + 24 \, a e^{6}\right )} x^{2} e^{\left (-3\right )}}{d^{4}} + \frac{7 \,{\left (3 \, c d^{3} e^{3} + 4 \, b d^{2} e^{4} + 24 \, a d e^{5}\right )} e^{\left (-3\right )}}{d^{4}}\right )} + \frac{35 \,{\left (b d^{3} e^{3} + 6 \, a d^{2} e^{4}\right )} e^{\left (-3\right )}}{d^{4}}\right )} x^{2} + \frac{105 \, a}{d}\right )} x}{105 \,{\left (x^{2} e + d\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*(2*(3*c*d^2*e^4 + 4*b*d*e^5 + 24*a*e^6)*x^2*e^(-3)/d^4 + 7*(3*c*d^3*
e^3 + 4*b*d^2*e^4 + 24*a*d*e^5)*e^(-3)/d^4) + 35*(b*d^3*e^3 + 6*a*d^2*e^4)*e^(-3
)/d^4)*x^2 + 105*a/d)*x/(x^2*e + d)^(7/2)