3.288 \(\int \frac{a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx\)

Optimal. Leaf size=165 \[ \frac{8 e^2 x^9 \left (2 e (8 a e+b d)+c d^2\right )}{315 d^5 \left (d+e x^2\right )^{9/2}}+\frac{4 e x^7 \left (2 e (8 a e+b d)+c d^2\right )}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac{x^5 \left (2 e (8 a e+b d)+c d^2\right )}{5 d^3 \left (d+e x^2\right )^{9/2}}+\frac{x^3 (8 a e+b d)}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac{a x}{d \left (d+e x^2\right )^{9/2}} \]

[Out]

(a*x)/(d*(d + e*x^2)^(9/2)) + ((b*d + 8*a*e)*x^3)/(3*d^2*(d + e*x^2)^(9/2)) + ((
c*d^2 + 2*e*(b*d + 8*a*e))*x^5)/(5*d^3*(d + e*x^2)^(9/2)) + (4*e*(c*d^2 + 2*e*(b
*d + 8*a*e))*x^7)/(35*d^4*(d + e*x^2)^(9/2)) + (8*e^2*(c*d^2 + 2*e*(b*d + 8*a*e)
)*x^9)/(315*d^5*(d + e*x^2)^(9/2))

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Rubi [A]  time = 0.390302, antiderivative size = 164, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x^5 \left (\frac{2 e (8 a e+b d)}{d^2}+c\right )}{5 d \left (d+e x^2\right )^{9/2}}+\frac{8 e^2 x^9 \left (2 e (8 a e+b d)+c d^2\right )}{315 d^5 \left (d+e x^2\right )^{9/2}}+\frac{4 e x^7 \left (2 e (8 a e+b d)+c d^2\right )}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac{x^3 (8 a e+b d)}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac{a x}{d \left (d+e x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(11/2),x]

[Out]

(a*x)/(d*(d + e*x^2)^(9/2)) + ((b*d + 8*a*e)*x^3)/(3*d^2*(d + e*x^2)^(9/2)) + ((
c + (2*e*(b*d + 8*a*e))/d^2)*x^5)/(5*d*(d + e*x^2)^(9/2)) + (4*e*(c*d^2 + 2*e*(b
*d + 8*a*e))*x^7)/(35*d^4*(d + e*x^2)^(9/2)) + (8*e^2*(c*d^2 + 2*e*(b*d + 8*a*e)
)*x^9)/(315*d^5*(d + e*x^2)^(9/2))

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Rubi in Sympy [A]  time = 35.1959, size = 192, normalized size = 1.16 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{9 d e^{2} \left (d + e x^{2}\right )^{\frac{9}{2}}} + \frac{x \left (8 a e^{2} + b d e - 10 c d^{2}\right )}{63 d^{2} e^{2} \left (d + e x^{2}\right )^{\frac{7}{2}}} + \frac{x \left (16 a e^{2} + 2 b d e + c d^{2}\right )}{105 d^{3} e^{2} \left (d + e x^{2}\right )^{\frac{5}{2}}} + \frac{4 x \left (16 a e^{2} + 2 b d e + c d^{2}\right )}{315 d^{4} e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{8 x \left (16 a e^{2} + 2 b d e + c d^{2}\right )}{315 d^{5} e^{2} \sqrt{d + e x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(11/2),x)

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(9*d*e**2*(d + e*x**2)**(9/2)) + x*(8*a*e**2 + b*d*e
- 10*c*d**2)/(63*d**2*e**2*(d + e*x**2)**(7/2)) + x*(16*a*e**2 + 2*b*d*e + c*d*
*2)/(105*d**3*e**2*(d + e*x**2)**(5/2)) + 4*x*(16*a*e**2 + 2*b*d*e + c*d**2)/(31
5*d**4*e**2*(d + e*x**2)**(3/2)) + 8*x*(16*a*e**2 + 2*b*d*e + c*d**2)/(315*d**5*
e**2*sqrt(d + e*x**2))

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Mathematica [A]  time = 0.149372, size = 132, normalized size = 0.8 \[ \frac{a \left (315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right )+d x^3 \left (b \left (105 d^3+126 d^2 e x^2+72 d e^2 x^4+16 e^3 x^6\right )+c d x^2 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(11/2),x]

[Out]

(a*(315*d^4*x + 840*d^3*e*x^3 + 1008*d^2*e^2*x^5 + 576*d*e^3*x^7 + 128*e^4*x^9)
+ d*x^3*(c*d*x^2*(63*d^2 + 36*d*e*x^2 + 8*e^2*x^4) + b*(105*d^3 + 126*d^2*e*x^2
+ 72*d*e^2*x^4 + 16*e^3*x^6)))/(315*d^5*(d + e*x^2)^(9/2))

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Maple [A]  time = 0.008, size = 136, normalized size = 0.8 \[{\frac{x \left ( 128\,a{e}^{4}{x}^{8}+16\,bd{e}^{3}{x}^{8}+8\,c{d}^{2}{e}^{2}{x}^{8}+576\,ad{e}^{3}{x}^{6}+72\,b{d}^{2}{e}^{2}{x}^{6}+36\,c{d}^{3}e{x}^{6}+1008\,a{d}^{2}{e}^{2}{x}^{4}+126\,b{d}^{3}e{x}^{4}+63\,c{d}^{4}{x}^{4}+840\,a{d}^{3}e{x}^{2}+105\,b{d}^{4}{x}^{2}+315\,a{d}^{4} \right ) }{315\,{d}^{5}} \left ( e{x}^{2}+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/(e*x^2+d)^(11/2),x)

[Out]

1/315*x*(128*a*e^4*x^8+16*b*d*e^3*x^8+8*c*d^2*e^2*x^8+576*a*d*e^3*x^6+72*b*d^2*e
^2*x^6+36*c*d^3*e*x^6+1008*a*d^2*e^2*x^4+126*b*d^3*e*x^4+63*c*d^4*x^4+840*a*d^3*
e*x^2+105*b*d^4*x^2+315*a*d^4)/(e*x^2+d)^(9/2)/d^5

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Maxima [A]  time = 0.746915, size = 379, normalized size = 2.3 \[ -\frac{c x^{3}}{6 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e} + \frac{128 \, a x}{315 \, \sqrt{e x^{2} + d} d^{5}} + \frac{64 \, a x}{315 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{4}} + \frac{16 \, a x}{105 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{3}} + \frac{8 \, a x}{63 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d^{2}} + \frac{a x}{9 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} d} + \frac{c x}{126 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e^{2}} + \frac{8 \, c x}{315 \, \sqrt{e x^{2} + d} d^{3} e^{2}} + \frac{4 \, c x}{315 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2} e^{2}} + \frac{c x}{105 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d e^{2}} - \frac{c d x}{18 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e^{2}} - \frac{b x}{9 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e} + \frac{16 \, b x}{315 \, \sqrt{e x^{2} + d} d^{4} e} + \frac{8 \, b x}{315 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3} e} + \frac{2 \, b x}{105 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} e} + \frac{b x}{63 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(11/2),x, algorithm="maxima")

[Out]

-1/6*c*x^3/((e*x^2 + d)^(9/2)*e) + 128/315*a*x/(sqrt(e*x^2 + d)*d^5) + 64/315*a*
x/((e*x^2 + d)^(3/2)*d^4) + 16/105*a*x/((e*x^2 + d)^(5/2)*d^3) + 8/63*a*x/((e*x^
2 + d)^(7/2)*d^2) + 1/9*a*x/((e*x^2 + d)^(9/2)*d) + 1/126*c*x/((e*x^2 + d)^(7/2)
*e^2) + 8/315*c*x/(sqrt(e*x^2 + d)*d^3*e^2) + 4/315*c*x/((e*x^2 + d)^(3/2)*d^2*e
^2) + 1/105*c*x/((e*x^2 + d)^(5/2)*d*e^2) - 1/18*c*d*x/((e*x^2 + d)^(9/2)*e^2) -
1/9*b*x/((e*x^2 + d)^(9/2)*e) + 16/315*b*x/(sqrt(e*x^2 + d)*d^4*e) + 8/315*b*x/
((e*x^2 + d)^(3/2)*d^3*e) + 2/105*b*x/((e*x^2 + d)^(5/2)*d^2*e) + 1/63*b*x/((e*x
^2 + d)^(7/2)*d*e)

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Fricas [A]  time = 0.555679, size = 239, normalized size = 1.45 \[ \frac{{\left (8 \,{\left (c d^{2} e^{2} + 2 \, b d e^{3} + 16 \, a e^{4}\right )} x^{9} + 36 \,{\left (c d^{3} e + 2 \, b d^{2} e^{2} + 16 \, a d e^{3}\right )} x^{7} + 315 \, a d^{4} x + 63 \,{\left (c d^{4} + 2 \, b d^{3} e + 16 \, a d^{2} e^{2}\right )} x^{5} + 105 \,{\left (b d^{4} + 8 \, a d^{3} e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{315 \,{\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(11/2),x, algorithm="fricas")

[Out]

1/315*(8*(c*d^2*e^2 + 2*b*d*e^3 + 16*a*e^4)*x^9 + 36*(c*d^3*e + 2*b*d^2*e^2 + 16
*a*d*e^3)*x^7 + 315*a*d^4*x + 63*(c*d^4 + 2*b*d^3*e + 16*a*d^2*e^2)*x^5 + 105*(b
*d^4 + 8*a*d^3*e)*x^3)*sqrt(e*x^2 + d)/(d^5*e^5*x^10 + 5*d^6*e^4*x^8 + 10*d^7*e^
3*x^6 + 10*d^8*e^2*x^4 + 5*d^9*e*x^2 + d^10)

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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((c*x**4+b*x**2+a)/(e*x**2+d)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.271193, size = 200, normalized size = 1.21 \[ \frac{{\left ({\left ({\left (4 \, x^{2}{\left (\frac{2 \,{\left (c d^{2} e^{6} + 2 \, b d e^{7} + 16 \, a e^{8}\right )} x^{2} e^{\left (-4\right )}}{d^{5}} + \frac{9 \,{\left (c d^{3} e^{5} + 2 \, b d^{2} e^{6} + 16 \, a d e^{7}\right )} e^{\left (-4\right )}}{d^{5}}\right )} + \frac{63 \,{\left (c d^{4} e^{4} + 2 \, b d^{3} e^{5} + 16 \, a d^{2} e^{6}\right )} e^{\left (-4\right )}}{d^{5}}\right )} x^{2} + \frac{105 \,{\left (b d^{4} e^{4} + 8 \, a d^{3} e^{5}\right )} e^{\left (-4\right )}}{d^{5}}\right )} x^{2} + \frac{315 \, a}{d}\right )} x}{315 \,{\left (x^{2} e + d\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(11/2),x, algorithm="giac")

[Out]

1/315*(((4*x^2*(2*(c*d^2*e^6 + 2*b*d*e^7 + 16*a*e^8)*x^2*e^(-4)/d^5 + 9*(c*d^3*e
^5 + 2*b*d^2*e^6 + 16*a*d*e^7)*e^(-4)/d^5) + 63*(c*d^4*e^4 + 2*b*d^3*e^5 + 16*a*
d^2*e^6)*e^(-4)/d^5)*x^2 + 105*(b*d^4*e^4 + 8*a*d^3*e^5)*e^(-4)/d^5)*x^2 + 315*a
/d)*x/(x^2*e + d)^(9/2)