3.289 \(\int \frac{a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac{8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac{2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{a x}{d \left (d+e x^2\right )^{11/2}} \]

[Out]

(a*x)/(d*(d + e*x^2)^(11/2)) + ((b*d + 10*a*e)*x^3)/(3*d^2*(d + e*x^2)^(11/2)) +
((3*c*d^2 + 8*e*(b*d + 10*a*e))*x^5)/(15*d^3*(d + e*x^2)^(11/2)) + (2*e*(3*c*d^
2 + 8*e*(b*d + 10*a*e))*x^7)/(35*d^4*(d + e*x^2)^(11/2)) + (8*e^2*(3*c*d^2 + 8*e
*(b*d + 10*a*e))*x^9)/(315*d^5*(d + e*x^2)^(11/2)) + (16*e^3*(3*c*d^2 + 8*e*(b*d
+ 10*a*e))*x^11)/(3465*d^6*(d + e*x^2)^(11/2))

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Rubi [A]  time = 0.437505, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac{8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac{2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{a x}{d \left (d+e x^2\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]

[Out]

(a*x)/(d*(d + e*x^2)^(11/2)) + ((b*d + 10*a*e)*x^3)/(3*d^2*(d + e*x^2)^(11/2)) +
((3*c*d^2 + 8*e*(b*d + 10*a*e))*x^5)/(15*d^3*(d + e*x^2)^(11/2)) + (2*e*(3*c*d^
2 + 8*e*(b*d + 10*a*e))*x^7)/(35*d^4*(d + e*x^2)^(11/2)) + (8*e^2*(3*c*d^2 + 8*e
*(b*d + 10*a*e))*x^9)/(315*d^5*(d + e*x^2)^(11/2)) + (16*e^3*(3*c*d^2 + 8*e*(b*d
+ 10*a*e))*x^11)/(3465*d^6*(d + e*x^2)^(11/2))

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Rubi in Sympy [A]  time = 39.3151, size = 240, normalized size = 1.14 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{11 d e^{2} \left (d + e x^{2}\right )^{\frac{11}{2}}} + \frac{x \left (10 a e^{2} + b d e - 12 c d^{2}\right )}{99 d^{2} e^{2} \left (d + e x^{2}\right )^{\frac{9}{2}}} + \frac{x \left (80 a e^{2} + 8 b d e + 3 c d^{2}\right )}{693 d^{3} e^{2} \left (d + e x^{2}\right )^{\frac{7}{2}}} + \frac{2 x \left (80 a e^{2} + 8 b d e + 3 c d^{2}\right )}{1155 d^{4} e^{2} \left (d + e x^{2}\right )^{\frac{5}{2}}} + \frac{8 x \left (80 a e^{2} + 8 b d e + 3 c d^{2}\right )}{3465 d^{5} e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{16 x \left (80 a e^{2} + 8 b d e + 3 c d^{2}\right )}{3465 d^{6} e^{2} \sqrt{d + e x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(13/2),x)

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(11*d*e**2*(d + e*x**2)**(11/2)) + x*(10*a*e**2 + b*
d*e - 12*c*d**2)/(99*d**2*e**2*(d + e*x**2)**(9/2)) + x*(80*a*e**2 + 8*b*d*e + 3
*c*d**2)/(693*d**3*e**2*(d + e*x**2)**(7/2)) + 2*x*(80*a*e**2 + 8*b*d*e + 3*c*d*
*2)/(1155*d**4*e**2*(d + e*x**2)**(5/2)) + 8*x*(80*a*e**2 + 8*b*d*e + 3*c*d**2)/
(3465*d**5*e**2*(d + e*x**2)**(3/2)) + 16*x*(80*a*e**2 + 8*b*d*e + 3*c*d**2)/(34
65*d**6*e**2*sqrt(d + e*x**2))

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Mathematica [A]  time = 0.157361, size = 167, normalized size = 0.8 \[ \frac{5 a \left (693 d^5 x+2310 d^4 e x^3+3696 d^3 e^2 x^5+3168 d^2 e^3 x^7+1408 d e^4 x^9+256 e^5 x^{11}\right )+d x^3 \left (b \left (1155 d^4+1848 d^3 e x^2+1584 d^2 e^2 x^4+704 d e^3 x^6+128 e^4 x^8\right )+3 c d x^2 \left (231 d^3+198 d^2 e x^2+88 d e^2 x^4+16 e^3 x^6\right )\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]

[Out]

(5*a*(693*d^5*x + 2310*d^4*e*x^3 + 3696*d^3*e^2*x^5 + 3168*d^2*e^3*x^7 + 1408*d*
e^4*x^9 + 256*e^5*x^11) + d*x^3*(3*c*d*x^2*(231*d^3 + 198*d^2*e*x^2 + 88*d*e^2*x
^4 + 16*e^3*x^6) + b*(1155*d^4 + 1848*d^3*e*x^2 + 1584*d^2*e^2*x^4 + 704*d*e^3*x
^6 + 128*e^4*x^8)))/(3465*d^6*(d + e*x^2)^(11/2))

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Maple [A]  time = 0.009, size = 172, normalized size = 0.8 \[{\frac{x \left ( 1280\,a{e}^{5}{x}^{10}+128\,bd{e}^{4}{x}^{10}+48\,c{d}^{2}{e}^{3}{x}^{10}+7040\,ad{e}^{4}{x}^{8}+704\,b{d}^{2}{e}^{3}{x}^{8}+264\,c{d}^{3}{e}^{2}{x}^{8}+15840\,a{d}^{2}{e}^{3}{x}^{6}+1584\,b{d}^{3}{e}^{2}{x}^{6}+594\,c{d}^{4}e{x}^{6}+18480\,a{d}^{3}{e}^{2}{x}^{4}+1848\,b{d}^{4}e{x}^{4}+693\,c{d}^{5}{x}^{4}+11550\,a{d}^{4}e{x}^{2}+1155\,b{d}^{5}{x}^{2}+3465\,a{d}^{5} \right ) }{3465\,{d}^{6}} \left ( e{x}^{2}+d \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x)

[Out]

1/3465*x*(1280*a*e^5*x^10+128*b*d*e^4*x^10+48*c*d^2*e^3*x^10+7040*a*d*e^4*x^8+70
4*b*d^2*e^3*x^8+264*c*d^3*e^2*x^8+15840*a*d^2*e^3*x^6+1584*b*d^3*e^2*x^6+594*c*d
^4*e*x^6+18480*a*d^3*e^2*x^4+1848*b*d^4*e*x^4+693*c*d^5*x^4+11550*a*d^4*e*x^2+11
55*b*d^5*x^2+3465*a*d^5)/(e*x^2+d)^(11/2)/d^6

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Maxima [A]  time = 0.739303, size = 452, normalized size = 2.15 \[ -\frac{c x^{3}}{8 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} e} + \frac{256 \, a x}{693 \, \sqrt{e x^{2} + d} d^{6}} + \frac{128 \, a x}{693 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{5}} + \frac{32 \, a x}{231 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{4}} + \frac{80 \, a x}{693 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d^{3}} + \frac{10 \, a x}{99 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} d^{2}} + \frac{a x}{11 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} d} + \frac{c x}{264 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e^{2}} + \frac{16 \, c x}{1155 \, \sqrt{e x^{2} + d} d^{4} e^{2}} + \frac{8 \, c x}{1155 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3} e^{2}} + \frac{2 \, c x}{385 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} e^{2}} + \frac{c x}{231 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d e^{2}} - \frac{3 \, c d x}{88 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} e^{2}} - \frac{b x}{11 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} e} + \frac{128 \, b x}{3465 \, \sqrt{e x^{2} + d} d^{5} e} + \frac{64 \, b x}{3465 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{4} e} + \frac{16 \, b x}{1155 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{3} e} + \frac{8 \, b x}{693 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d^{2} e} + \frac{b x}{99 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(13/2),x, algorithm="maxima")

[Out]

-1/8*c*x^3/((e*x^2 + d)^(11/2)*e) + 256/693*a*x/(sqrt(e*x^2 + d)*d^6) + 128/693*
a*x/((e*x^2 + d)^(3/2)*d^5) + 32/231*a*x/((e*x^2 + d)^(5/2)*d^4) + 80/693*a*x/((
e*x^2 + d)^(7/2)*d^3) + 10/99*a*x/((e*x^2 + d)^(9/2)*d^2) + 1/11*a*x/((e*x^2 + d
)^(11/2)*d) + 1/264*c*x/((e*x^2 + d)^(9/2)*e^2) + 16/1155*c*x/(sqrt(e*x^2 + d)*d
^4*e^2) + 8/1155*c*x/((e*x^2 + d)^(3/2)*d^3*e^2) + 2/385*c*x/((e*x^2 + d)^(5/2)*
d^2*e^2) + 1/231*c*x/((e*x^2 + d)^(7/2)*d*e^2) - 3/88*c*d*x/((e*x^2 + d)^(11/2)*
e^2) - 1/11*b*x/((e*x^2 + d)^(11/2)*e) + 128/3465*b*x/(sqrt(e*x^2 + d)*d^5*e) +
64/3465*b*x/((e*x^2 + d)^(3/2)*d^4*e) + 16/1155*b*x/((e*x^2 + d)^(5/2)*d^3*e) +
8/693*b*x/((e*x^2 + d)^(7/2)*d^2*e) + 1/99*b*x/((e*x^2 + d)^(9/2)*d*e)

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Fricas [A]  time = 0.845007, size = 302, normalized size = 1.44 \[ \frac{{\left (16 \,{\left (3 \, c d^{2} e^{3} + 8 \, b d e^{4} + 80 \, a e^{5}\right )} x^{11} + 88 \,{\left (3 \, c d^{3} e^{2} + 8 \, b d^{2} e^{3} + 80 \, a d e^{4}\right )} x^{9} + 198 \,{\left (3 \, c d^{4} e + 8 \, b d^{3} e^{2} + 80 \, a d^{2} e^{3}\right )} x^{7} + 3465 \, a d^{5} x + 231 \,{\left (3 \, c d^{5} + 8 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} x^{5} + 1155 \,{\left (b d^{5} + 10 \, a d^{4} e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{3465 \,{\left (d^{6} e^{6} x^{12} + 6 \, d^{7} e^{5} x^{10} + 15 \, d^{8} e^{4} x^{8} + 20 \, d^{9} e^{3} x^{6} + 15 \, d^{10} e^{2} x^{4} + 6 \, d^{11} e x^{2} + d^{12}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(13/2),x, algorithm="fricas")

[Out]

1/3465*(16*(3*c*d^2*e^3 + 8*b*d*e^4 + 80*a*e^5)*x^11 + 88*(3*c*d^3*e^2 + 8*b*d^2
*e^3 + 80*a*d*e^4)*x^9 + 198*(3*c*d^4*e + 8*b*d^3*e^2 + 80*a*d^2*e^3)*x^7 + 3465
*a*d^5*x + 231*(3*c*d^5 + 8*b*d^4*e + 80*a*d^3*e^2)*x^5 + 1155*(b*d^5 + 10*a*d^4
*e)*x^3)*sqrt(e*x^2 + d)/(d^6*e^6*x^12 + 6*d^7*e^5*x^10 + 15*d^8*e^4*x^8 + 20*d^
9*e^3*x^6 + 15*d^10*e^2*x^4 + 6*d^11*e*x^2 + d^12)

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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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.271024, size = 255, normalized size = 1.21 \[ \frac{{\left ({\left ({\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \,{\left (3 \, c d^{2} e^{8} + 8 \, b d e^{9} + 80 \, a e^{10}\right )} x^{2} e^{\left (-5\right )}}{d^{6}} + \frac{11 \,{\left (3 \, c d^{3} e^{7} + 8 \, b d^{2} e^{8} + 80 \, a d e^{9}\right )} e^{\left (-5\right )}}{d^{6}}\right )} + \frac{99 \,{\left (3 \, c d^{4} e^{6} + 8 \, b d^{3} e^{7} + 80 \, a d^{2} e^{8}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac{231 \,{\left (3 \, c d^{5} e^{5} + 8 \, b d^{4} e^{6} + 80 \, a d^{3} e^{7}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac{1155 \,{\left (b d^{5} e^{5} + 10 \, a d^{4} e^{6}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac{3465 \, a}{d}\right )} x}{3465 \,{\left (x^{2} e + d\right )}^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(13/2),x, algorithm="giac")

[Out]

1/3465*(((2*(4*x^2*(2*(3*c*d^2*e^8 + 8*b*d*e^9 + 80*a*e^10)*x^2*e^(-5)/d^6 + 11*
(3*c*d^3*e^7 + 8*b*d^2*e^8 + 80*a*d*e^9)*e^(-5)/d^6) + 99*(3*c*d^4*e^6 + 8*b*d^3
*e^7 + 80*a*d^2*e^8)*e^(-5)/d^6)*x^2 + 231*(3*c*d^5*e^5 + 8*b*d^4*e^6 + 80*a*d^3
*e^7)*e^(-5)/d^6)*x^2 + 1155*(b*d^5*e^5 + 10*a*d^4*e^6)*e^(-5)/d^6)*x^2 + 3465*a
/d)*x/(x^2*e + d)^(11/2)