3.280 \(\int \frac{x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^3} \, dx\)

Optimal. Leaf size=187 \[ \frac{d x \left (11 c d^2-e (7 b d-3 a e)\right )}{8 e^5 \left (d+e x^2\right )}+\frac{x \left (13 c d^2-e (7 b d-3 a e)\right )}{2 e^5}-\frac{x^3 \left (15 c d^2-e (7 b d-3 a e)\right )}{12 d e^4}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 a e^2-35 b d e+63 c d^2\right )}{8 e^{11/2}}+\frac{x^7 \left (a+\frac{d (c d-b e)}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{c x^5}{5 e^3} \]

[Out]

((13*c*d^2 - e*(7*b*d - 3*a*e))*x)/(2*e^5) - ((15*c*d^2 - e*(7*b*d - 3*a*e))*x^3
)/(12*d*e^4) + (c*x^5)/(5*e^3) + ((a + (d*(c*d - b*e))/e^2)*x^7)/(4*d*(d + e*x^2
)^2) + (d*(11*c*d^2 - e*(7*b*d - 3*a*e))*x)/(8*e^5*(d + e*x^2)) - (Sqrt[d]*(63*c
*d^2 - 35*b*d*e + 15*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*e^(11/2))

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Rubi [A]  time = 0.599985, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{d x \left (11 c d^2-e (7 b d-3 a e)\right )}{8 e^5 \left (d+e x^2\right )}+\frac{x \left (13 c d^2-e (7 b d-3 a e)\right )}{2 e^5}-\frac{x^3 \left (15 c d^2-e (7 b d-3 a e)\right )}{12 d e^4}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 a e^2-35 b d e+63 c d^2\right )}{8 e^{11/2}}+\frac{x^7 \left (a+\frac{d (c d-b e)}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{c x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(a + b*x^2 + c*x^4))/(d + e*x^2)^3,x]

[Out]

((13*c*d^2 - e*(7*b*d - 3*a*e))*x)/(2*e^5) - ((15*c*d^2 - e*(7*b*d - 3*a*e))*x^3
)/(12*d*e^4) + (c*x^5)/(5*e^3) + ((a + (d*(c*d - b*e))/e^2)*x^7)/(4*d*(d + e*x^2
)^2) + (d*(11*c*d^2 - e*(7*b*d - 3*a*e))*x)/(8*e^5*(d + e*x^2)) - (Sqrt[d]*(63*c
*d^2 - 35*b*d*e + 15*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*e^(11/2))

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Rubi in Sympy [A]  time = 129.758, size = 165, normalized size = 0.88 \[ \frac{c x^{5}}{5 e^{3}} - \frac{\sqrt{d} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 e^{\frac{11}{2}}} - \frac{d^{2} x \left (a e^{2} - b d e + c d^{2}\right )}{4 e^{5} \left (d + e x^{2}\right )^{2}} + \frac{d x \left (9 a e^{2} - 13 b d e + 17 c d^{2}\right )}{8 e^{5} \left (d + e x^{2}\right )} + \frac{x^{3} \left (b e - 3 c d\right )}{3 e^{4}} + \frac{x \left (a e^{2} - 3 b d e + 6 c d^{2}\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(c*x**4+b*x**2+a)/(e*x**2+d)**3,x)

[Out]

c*x**5/(5*e**3) - sqrt(d)*(15*a*e**2 - 35*b*d*e + 63*c*d**2)*atan(sqrt(e)*x/sqrt
(d))/(8*e**(11/2)) - d**2*x*(a*e**2 - b*d*e + c*d**2)/(4*e**5*(d + e*x**2)**2) +
d*x*(9*a*e**2 - 13*b*d*e + 17*c*d**2)/(8*e**5*(d + e*x**2)) + x**3*(b*e - 3*c*d
)/(3*e**4) + x*(a*e**2 - 3*b*d*e + 6*c*d**2)/e**5

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Mathematica [A]  time = 0.190401, size = 170, normalized size = 0.91 \[ \frac{x \left (d e (9 a e-13 b d)+17 c d^3\right )}{8 e^5 \left (d+e x^2\right )}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 e (3 a e-7 b d)+63 c d^2\right )}{8 e^{11/2}}+\frac{x \left (e (a e-3 b d)+6 c d^2\right )}{e^5}-\frac{x \left (d^2 e (a e-b d)+c d^4\right )}{4 e^5 \left (d+e x^2\right )^2}+\frac{x^3 (b e-3 c d)}{3 e^4}+\frac{c x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(a + b*x^2 + c*x^4))/(d + e*x^2)^3,x]

[Out]

((6*c*d^2 + e*(-3*b*d + a*e))*x)/e^5 + ((-3*c*d + b*e)*x^3)/(3*e^4) + (c*x^5)/(5
*e^3) - ((c*d^4 + d^2*e*(-(b*d) + a*e))*x)/(4*e^5*(d + e*x^2)^2) + ((17*c*d^3 +
d*e*(-13*b*d + 9*a*e))*x)/(8*e^5*(d + e*x^2)) - (Sqrt[d]*(63*c*d^2 + 5*e*(-7*b*d
+ 3*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*e^(11/2))

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Maple [A]  time = 0.018, size = 239, normalized size = 1.3 \[{\frac{c{x}^{5}}{5\,{e}^{3}}}+{\frac{b{x}^{3}}{3\,{e}^{3}}}-{\frac{cd{x}^{3}}{{e}^{4}}}+{\frac{ax}{{e}^{3}}}-3\,{\frac{bxd}{{e}^{4}}}+6\,{\frac{c{d}^{2}x}{{e}^{5}}}+{\frac{9\,d{x}^{3}a}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{13\,{d}^{2}{x}^{3}b}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{17\,{d}^{3}{x}^{3}c}{8\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,a{d}^{2}x}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{11\,b{d}^{3}x}{8\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{15\,c{d}^{4}x}{8\,{e}^{5} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{15\,ad}{8\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,b{d}^{2}}{8\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{63\,c{d}^{3}}{8\,{e}^{5}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(c*x^4+b*x^2+a)/(e*x^2+d)^3,x)

[Out]

1/5*c*x^5/e^3+1/3/e^3*x^3*b-1/e^4*c*d*x^3+1/e^3*a*x-3/e^4*x*b*d+6/e^5*c*d^2*x+9/
8*d/e^2/(e*x^2+d)^2*x^3*a-13/8*d^2/e^3/(e*x^2+d)^2*x^3*b+17/8*d^3/e^4/(e*x^2+d)^
2*x^3*c+7/8*d^2/e^3/(e*x^2+d)^2*a*x-11/8*d^3/e^4/(e*x^2+d)^2*b*x+15/8*d^4/e^5/(e
*x^2+d)^2*c*x-15/8*d/e^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a+35/8*d^2/e^4/(d*e
)^(1/2)*arctan(x*e/(d*e)^(1/2))*b-63/8*d^3/e^5/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2
))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^6/(e*x^2 + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267524, size = 1, normalized size = 0.01 \[ \left [\frac{48 \, c e^{4} x^{9} - 16 \,{\left (9 \, c d e^{3} - 5 \, b e^{4}\right )} x^{7} + 16 \,{\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{5} + 50 \,{\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{3} + 15 \,{\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2} +{\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{4} + 2 \,{\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{2}\right )} \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 30 \,{\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2}\right )} x}{240 \,{\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}}, \frac{24 \, c e^{4} x^{9} - 8 \,{\left (9 \, c d e^{3} - 5 \, b e^{4}\right )} x^{7} + 8 \,{\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{5} + 25 \,{\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{3} - 15 \,{\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2} +{\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{4} + 2 \,{\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{2}\right )} \sqrt{\frac{d}{e}} \arctan \left (\frac{x}{\sqrt{\frac{d}{e}}}\right ) + 15 \,{\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2}\right )} x}{120 \,{\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^6/(e*x^2 + d)^3,x, algorithm="fricas")

[Out]

[1/240*(48*c*e^4*x^9 - 16*(9*c*d*e^3 - 5*b*e^4)*x^7 + 16*(63*c*d^2*e^2 - 35*b*d*
e^3 + 15*a*e^4)*x^5 + 50*(63*c*d^3*e - 35*b*d^2*e^2 + 15*a*d*e^3)*x^3 + 15*(63*c
*d^4 - 35*b*d^3*e + 15*a*d^2*e^2 + (63*c*d^2*e^2 - 35*b*d*e^3 + 15*a*e^4)*x^4 +
2*(63*c*d^3*e - 35*b*d^2*e^2 + 15*a*d*e^3)*x^2)*sqrt(-d/e)*log((e*x^2 - 2*e*x*sq
rt(-d/e) - d)/(e*x^2 + d)) + 30*(63*c*d^4 - 35*b*d^3*e + 15*a*d^2*e^2)*x)/(e^7*x
^4 + 2*d*e^6*x^2 + d^2*e^5), 1/120*(24*c*e^4*x^9 - 8*(9*c*d*e^3 - 5*b*e^4)*x^7 +
8*(63*c*d^2*e^2 - 35*b*d*e^3 + 15*a*e^4)*x^5 + 25*(63*c*d^3*e - 35*b*d^2*e^2 +
15*a*d*e^3)*x^3 - 15*(63*c*d^4 - 35*b*d^3*e + 15*a*d^2*e^2 + (63*c*d^2*e^2 - 35*
b*d*e^3 + 15*a*e^4)*x^4 + 2*(63*c*d^3*e - 35*b*d^2*e^2 + 15*a*d*e^3)*x^2)*sqrt(d
/e)*arctan(x/sqrt(d/e)) + 15*(63*c*d^4 - 35*b*d^3*e + 15*a*d^2*e^2)*x)/(e^7*x^4
+ 2*d*e^6*x^2 + d^2*e^5)]

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Sympy [A]  time = 15.8907, size = 233, normalized size = 1.25 \[ \frac{c x^{5}}{5 e^{3}} + \frac{\sqrt{- \frac{d}{e^{11}}} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \log{\left (- e^{5} \sqrt{- \frac{d}{e^{11}}} + x \right )}}{16} - \frac{\sqrt{- \frac{d}{e^{11}}} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \log{\left (e^{5} \sqrt{- \frac{d}{e^{11}}} + x \right )}}{16} + \frac{x^{3} \left (9 a d e^{3} - 13 b d^{2} e^{2} + 17 c d^{3} e\right ) + x \left (7 a d^{2} e^{2} - 11 b d^{3} e + 15 c d^{4}\right )}{8 d^{2} e^{5} + 16 d e^{6} x^{2} + 8 e^{7} x^{4}} + \frac{x^{3} \left (b e - 3 c d\right )}{3 e^{4}} + \frac{x \left (a e^{2} - 3 b d e + 6 c d^{2}\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(c*x**4+b*x**2+a)/(e*x**2+d)**3,x)

[Out]

c*x**5/(5*e**3) + sqrt(-d/e**11)*(15*a*e**2 - 35*b*d*e + 63*c*d**2)*log(-e**5*sq
rt(-d/e**11) + x)/16 - sqrt(-d/e**11)*(15*a*e**2 - 35*b*d*e + 63*c*d**2)*log(e**
5*sqrt(-d/e**11) + x)/16 + (x**3*(9*a*d*e**3 - 13*b*d**2*e**2 + 17*c*d**3*e) + x
*(7*a*d**2*e**2 - 11*b*d**3*e + 15*c*d**4))/(8*d**2*e**5 + 16*d*e**6*x**2 + 8*e*
*7*x**4) + x**3*(b*e - 3*c*d)/(3*e**4) + x*(a*e**2 - 3*b*d*e + 6*c*d**2)/e**5

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GIAC/XCAS [A]  time = 0.27167, size = 216, normalized size = 1.16 \[ -\frac{{\left (63 \, c d^{3} - 35 \, b d^{2} e + 15 \, a d e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{11}{2}\right )}}{8 \, \sqrt{d}} + \frac{1}{15} \,{\left (3 \, c x^{5} e^{12} - 15 \, c d x^{3} e^{11} + 5 \, b x^{3} e^{12} + 90 \, c d^{2} x e^{10} - 45 \, b d x e^{11} + 15 \, a x e^{12}\right )} e^{\left (-15\right )} + \frac{{\left (17 \, c d^{3} x^{3} e - 13 \, b d^{2} x^{3} e^{2} + 15 \, c d^{4} x + 9 \, a d x^{3} e^{3} - 11 \, b d^{3} x e + 7 \, a d^{2} x e^{2}\right )} e^{\left (-5\right )}}{8 \,{\left (x^{2} e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^6/(e*x^2 + d)^3,x, algorithm="giac")

[Out]

-1/8*(63*c*d^3 - 35*b*d^2*e + 15*a*d*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-11/2)/sq
rt(d) + 1/15*(3*c*x^5*e^12 - 15*c*d*x^3*e^11 + 5*b*x^3*e^12 + 90*c*d^2*x*e^10 -
45*b*d*x*e^11 + 15*a*x*e^12)*e^(-15) + 1/8*(17*c*d^3*x^3*e - 13*b*d^2*x^3*e^2 +
15*c*d^4*x + 9*a*d*x^3*e^3 - 11*b*d^3*x*e + 7*a*d^2*x*e^2)*e^(-5)/(x^2*e + d)^2