3.286 \(\int \frac{a+b x^2+c x^4}{\left (d+e x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{x^5 \left (2 e (4 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{5/2}}+\frac{x^3 (4 a e+b d)}{3 d^2 \left (d+e x^2\right )^{5/2}}+\frac{a x}{d \left (d+e x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.192974, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x^5 \left (2 e (4 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{5/2}}+\frac{x^3 (4 a e+b d)}{3 d^2 \left (d+e x^2\right )^{5/2}}+\frac{a x}{d \left (d+e x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.7167, size = 119, normalized size = 1.38 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{5 d e^{2} \left (d + e x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (4 a e^{2} + b d e - c d^{2} + 5 c d e x^{2}\right )}{15 d^{2} e^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{2 x \left (4 a e^{2} + b d e - c d^{2}\right )}{15 d^{3} e^{2} \sqrt{d + e x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(7/2),x)

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(5*d*e**2*(d + e*x**2)**(5/2)) + x*(4*a*e**2 + b*d*e
- c*d**2 + 5*c*d*e*x**2)/(15*d**2*e**2*(d + e*x**2)**(3/2)) + 2*x*(4*a*e**2 + b
*d*e - c*d**2)/(15*d**3*e**2*sqrt(d + e*x**2))

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Mathematica [A]  time = 0.0850195, size = 67, normalized size = 0.78 \[ \frac{a \left (15 d^2 x+20 d e x^3+8 e^2 x^5\right )+d x^3 \left (5 b d+2 b e x^2+3 c d x^2\right )}{15 d^3 \left (d+e x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(7/2),x]

[Out]

(d*x^3*(5*b*d + 3*c*d*x^2 + 2*b*e*x^2) + a*(15*d^2*x + 20*d*e*x^3 + 8*e^2*x^5))/
(15*d^3*(d + e*x^2)^(5/2))

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Maple [A]  time = 0.006, size = 66, normalized size = 0.8 \[{\frac{x \left ( 8\,a{e}^{2}{x}^{4}+2\,bde{x}^{4}+3\,c{d}^{2}{x}^{4}+20\,ade{x}^{2}+5\,b{d}^{2}{x}^{2}+15\,a{d}^{2} \right ) }{15\,{d}^{3}} \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/(e*x^2+d)^(7/2),x)

[Out]

1/15*x*(8*a*e^2*x^4+2*b*d*e*x^4+3*c*d^2*x^4+20*a*d*e*x^2+5*b*d^2*x^2+15*a*d^2)/(
e*x^2+d)^(5/2)/d^3

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Maxima [A]  time = 0.764345, size = 234, normalized size = 2.72 \[ -\frac{c x^{3}}{2 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} e} + \frac{8 \, a x}{15 \, \sqrt{e x^{2} + d} d^{3}} + \frac{4 \, a x}{15 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2}} + \frac{a x}{5 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d} + \frac{c x}{10 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} e^{2}} + \frac{c x}{5 \, \sqrt{e x^{2} + d} d e^{2}} - \frac{3 \, c d x}{10 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} e^{2}} - \frac{b x}{5 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} e} + \frac{2 \, b x}{15 \, \sqrt{e x^{2} + d} d^{2} e} + \frac{b x}{15 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(7/2),x, algorithm="maxima")

[Out]

-1/2*c*x^3/((e*x^2 + d)^(5/2)*e) + 8/15*a*x/(sqrt(e*x^2 + d)*d^3) + 4/15*a*x/((e
*x^2 + d)^(3/2)*d^2) + 1/5*a*x/((e*x^2 + d)^(5/2)*d) + 1/10*c*x/((e*x^2 + d)^(3/
2)*e^2) + 1/5*c*x/(sqrt(e*x^2 + d)*d*e^2) - 3/10*c*d*x/((e*x^2 + d)^(5/2)*e^2) -
1/5*b*x/((e*x^2 + d)^(5/2)*e) + 2/15*b*x/(sqrt(e*x^2 + d)*d^2*e) + 1/15*b*x/((e
*x^2 + d)^(3/2)*d*e)

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Fricas [A]  time = 0.308491, size = 126, normalized size = 1.47 \[ \frac{{\left ({\left (3 \, c d^{2} + 2 \, b d e + 8 \, a e^{2}\right )} x^{5} + 15 \, a d^{2} x + 5 \,{\left (b d^{2} + 4 \, a d e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{15 \,{\left (d^{3} e^{3} x^{6} + 3 \, d^{4} e^{2} x^{4} + 3 \, d^{5} e x^{2} + d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(7/2),x, algorithm="fricas")

[Out]

1/15*((3*c*d^2 + 2*b*d*e + 8*a*e^2)*x^5 + 15*a*d^2*x + 5*(b*d^2 + 4*a*d*e)*x^3)*
sqrt(e*x^2 + d)/(d^3*e^3*x^6 + 3*d^4*e^2*x^4 + 3*d^5*e*x^2 + d^6)

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Sympy [A]  time = 160.57, size = 639, normalized size = 7.43 \[ a \left (\frac{15 d^{5} x}{15 d^{\frac{17}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{15}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{13}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}} + 15 d^{\frac{11}{2}} e^{3} x^{6} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{35 d^{4} e x^{3}}{15 d^{\frac{17}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{15}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{13}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}} + 15 d^{\frac{11}{2}} e^{3} x^{6} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{28 d^{3} e^{2} x^{5}}{15 d^{\frac{17}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{15}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{13}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}} + 15 d^{\frac{11}{2}} e^{3} x^{6} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{8 d^{2} e^{3} x^{7}}{15 d^{\frac{17}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{15}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 45 d^{\frac{13}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}} + 15 d^{\frac{11}{2}} e^{3} x^{6} \sqrt{1 + \frac{e x^{2}}{d}}}\right ) + b \left (\frac{5 d x^{3}}{15 d^{\frac{9}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 30 d^{\frac{7}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 15 d^{\frac{5}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{2 e x^{5}}{15 d^{\frac{9}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 30 d^{\frac{7}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 15 d^{\frac{5}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}}}\right ) + \frac{c x^{5}}{5 d^{\frac{7}{2}} \sqrt{1 + \frac{e x^{2}}{d}} + 10 d^{\frac{5}{2}} e x^{2} \sqrt{1 + \frac{e x^{2}}{d}} + 5 d^{\frac{3}{2}} e^{2} x^{4} \sqrt{1 + \frac{e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(7/2),x)

[Out]

a*(15*d**5*x/(15*d**(17/2)*sqrt(1 + e*x**2/d) + 45*d**(15/2)*e*x**2*sqrt(1 + e*x
**2/d) + 45*d**(13/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 15*d**(11/2)*e**3*x**6*sqrt
(1 + e*x**2/d)) + 35*d**4*e*x**3/(15*d**(17/2)*sqrt(1 + e*x**2/d) + 45*d**(15/2)
*e*x**2*sqrt(1 + e*x**2/d) + 45*d**(13/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 15*d**(
11/2)*e**3*x**6*sqrt(1 + e*x**2/d)) + 28*d**3*e**2*x**5/(15*d**(17/2)*sqrt(1 + e
*x**2/d) + 45*d**(15/2)*e*x**2*sqrt(1 + e*x**2/d) + 45*d**(13/2)*e**2*x**4*sqrt(
1 + e*x**2/d) + 15*d**(11/2)*e**3*x**6*sqrt(1 + e*x**2/d)) + 8*d**2*e**3*x**7/(1
5*d**(17/2)*sqrt(1 + e*x**2/d) + 45*d**(15/2)*e*x**2*sqrt(1 + e*x**2/d) + 45*d**
(13/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 15*d**(11/2)*e**3*x**6*sqrt(1 + e*x**2/d))
) + b*(5*d*x**3/(15*d**(9/2)*sqrt(1 + e*x**2/d) + 30*d**(7/2)*e*x**2*sqrt(1 + e*
x**2/d) + 15*d**(5/2)*e**2*x**4*sqrt(1 + e*x**2/d)) + 2*e*x**5/(15*d**(9/2)*sqrt
(1 + e*x**2/d) + 30*d**(7/2)*e*x**2*sqrt(1 + e*x**2/d) + 15*d**(5/2)*e**2*x**4*s
qrt(1 + e*x**2/d))) + c*x**5/(5*d**(7/2)*sqrt(1 + e*x**2/d) + 10*d**(5/2)*e*x**2
*sqrt(1 + e*x**2/d) + 5*d**(3/2)*e**2*x**4*sqrt(1 + e*x**2/d))

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GIAC/XCAS [A]  time = 0.270656, size = 101, normalized size = 1.17 \[ \frac{{\left (x^{2}{\left (\frac{{\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3} + 8 \, a e^{4}\right )} x^{2} e^{\left (-2\right )}}{d^{3}} + \frac{5 \,{\left (b d^{2} e^{2} + 4 \, a d e^{3}\right )} e^{\left (-2\right )}}{d^{3}}\right )} + \frac{15 \, a}{d}\right )} x}{15 \,{\left (x^{2} e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(e*x^2 + d)^(7/2),x, algorithm="giac")

[Out]

1/15*(x^2*((3*c*d^2*e^2 + 2*b*d*e^3 + 8*a*e^4)*x^2*e^(-2)/d^3 + 5*(b*d^2*e^2 + 4
*a*d*e^3)*e^(-2)/d^3) + 15*a/d)*x/(x^2*e + d)^(5/2)