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

Optimal. Leaf size=235 \[ \frac{x^5 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^5 \left (-13 a^3 f+9 a^2 b e-5 a b^2 d+b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 \sqrt{a} b^{11/2}}-\frac{x \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 a b^5}+\frac{x^3 \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{24 a^2 b^4}+\frac{f x^5}{5 b^3} \]

[Out]

-((3*b^3*c - 15*a*b^2*d + 35*a^2*b*e - 63*a^3*f)*x)/(8*a*b^5) + ((3*b^3*c - 15*a
*b^2*d + 35*a^2*b*e - 63*a^3*f)*x^3)/(24*a^2*b^4) + (f*x^5)/(5*b^3) + ((c - (a*(
b^2*d - a*b*e + a^2*f))/b^3)*x^5)/(4*a*(a + b*x^2)^2) - ((b^3*c - 5*a*b^2*d + 9*
a^2*b*e - 13*a^3*f)*x^5)/(8*a^2*b^3*(a + b*x^2)) + ((3*b^3*c - 15*a*b^2*d + 35*a
^2*b*e - 63*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(11/2))

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Rubi [A]  time = 0.867842, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x^5 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^5 \left (-13 a^3 f+9 a^2 b e-5 a b^2 d+b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 \sqrt{a} b^{11/2}}-\frac{x \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 a b^5}+\frac{x^3 \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{24 a^2 b^4}+\frac{f x^5}{5 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-((3*b^3*c - 15*a*b^2*d + 35*a^2*b*e - 63*a^3*f)*x)/(8*a*b^5) + ((3*b^3*c - 15*a
*b^2*d + 35*a^2*b*e - 63*a^3*f)*x^3)/(24*a^2*b^4) + (f*x^5)/(5*b^3) + ((c - (a*(
b^2*d - a*b*e + a^2*f))/b^3)*x^5)/(4*a*(a + b*x^2)^2) - ((b^3*c - 5*a*b^2*d + 9*
a^2*b*e - 13*a^3*f)*x^5)/(8*a^2*b^3*(a + b*x^2)) + ((3*b^3*c - 15*a*b^2*d + 35*a
^2*b*e - 63*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(11/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.306987, size = 176, normalized size = 0.75 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 \sqrt{a} b^{11/2}}+\frac{x \left (945 a^4 f-525 a^3 b \left (e-3 f x^2\right )+a^2 b^2 \left (225 d-875 e x^2+504 f x^4\right )-a b^3 \left (45 c-375 d x^2+280 e x^4+72 f x^6\right )+b^4 x^2 \left (8 \left (15 d x^2+5 e x^4+3 f x^6\right )-75 c\right )\right )}{120 b^5 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(945*a^4*f - 525*a^3*b*(e - 3*f*x^2) + a^2*b^2*(225*d - 875*e*x^2 + 504*f*x^4
) - a*b^3*(45*c - 375*d*x^2 + 280*e*x^4 + 72*f*x^6) + b^4*x^2*(-75*c + 8*(15*d*x
^2 + 5*e*x^4 + 3*f*x^6))))/(120*b^5*(a + b*x^2)^2) + ((3*b^3*c - 15*a*b^2*d + 35
*a^2*b*e - 63*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(11/2))

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Maple [A]  time = 0.018, size = 294, normalized size = 1.3 \[{\frac{f{x}^{5}}{5\,{b}^{3}}}-{\frac{a{x}^{3}f}{{b}^{4}}}+{\frac{{x}^{3}e}{3\,{b}^{3}}}+6\,{\frac{{a}^{2}fx}{{b}^{5}}}-3\,{\frac{aex}{{b}^{4}}}+{\frac{dx}{{b}^{3}}}+{\frac{17\,{x}^{3}{a}^{3}f}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{x}^{3}{a}^{2}e}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}d}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{x}^{3}c}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,f{a}^{4}x}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,{a}^{3}ex}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}dx}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,acx}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{a}^{3}f}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}e}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ad}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*f*x^5/b^3-1/b^4*x^3*a*f+1/3/b^3*x^3*e+6/b^5*a^2*f*x-3/b^4*a*e*x+1/b^3*d*x+17
/8/b^4/(b*x^2+a)^2*x^3*a^3*f-13/8/b^3/(b*x^2+a)^2*x^3*a^2*e+9/8/b^2/(b*x^2+a)^2*
x^3*a*d-5/8/b/(b*x^2+a)^2*x^3*c+15/8/b^5/(b*x^2+a)^2*a^4*f*x-11/8/b^4/(b*x^2+a)^
2*a^3*e*x+7/8/b^3/(b*x^2+a)^2*a^2*d*x-3/8/b^2/(b*x^2+a)^2*a*c*x-63/8/b^5/(a*b)^(
1/2)*arctan(x*b/(a*b)^(1/2))*a^3*f+35/8/b^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*
a^2*e-15/8/b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a*d+3/8/b^2/(a*b)^(1/2)*arcta
n(x*b/(a*b)^(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((f*x^6 + e*x^4 + d*x^2 + c)*x^4/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24066, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (3 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 35 \, a^{4} b e - 63 \, a^{5} f +{\left (3 \, b^{5} c - 15 \, a b^{4} d + 35 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c - 15 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x^{2}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (24 \, b^{4} f x^{9} + 8 \,{\left (5 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 8 \,{\left (15 \, b^{4} d - 35 \, a b^{3} e + 63 \, a^{2} b^{2} f\right )} x^{5} - 25 \,{\left (3 \, b^{4} c - 15 \, a b^{3} d + 35 \, a^{2} b^{2} e - 63 \, a^{3} b f\right )} x^{3} - 15 \,{\left (3 \, a b^{3} c - 15 \, a^{2} b^{2} d + 35 \, a^{3} b e - 63 \, a^{4} f\right )} x\right )} \sqrt{-a b}}{240 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )} \sqrt{-a b}}, \frac{15 \,{\left (3 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 35 \, a^{4} b e - 63 \, a^{5} f +{\left (3 \, b^{5} c - 15 \, a b^{4} d + 35 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c - 15 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (24 \, b^{4} f x^{9} + 8 \,{\left (5 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 8 \,{\left (15 \, b^{4} d - 35 \, a b^{3} e + 63 \, a^{2} b^{2} f\right )} x^{5} - 25 \,{\left (3 \, b^{4} c - 15 \, a b^{3} d + 35 \, a^{2} b^{2} e - 63 \, a^{3} b f\right )} x^{3} - 15 \,{\left (3 \, a b^{3} c - 15 \, a^{2} b^{2} d + 35 \, a^{3} b e - 63 \, a^{4} f\right )} x\right )} \sqrt{a b}}{120 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 24.5198, size = 279, normalized size = 1.19 \[ \frac{\sqrt{- \frac{1}{a b^{11}}} \left (63 a^{3} f - 35 a^{2} b e + 15 a b^{2} d - 3 b^{3} c\right ) \log{\left (- a b^{5} \sqrt{- \frac{1}{a b^{11}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a b^{11}}} \left (63 a^{3} f - 35 a^{2} b e + 15 a b^{2} d - 3 b^{3} c\right ) \log{\left (a b^{5} \sqrt{- \frac{1}{a b^{11}}} + x \right )}}{16} + \frac{x^{3} \left (17 a^{3} b f - 13 a^{2} b^{2} e + 9 a b^{3} d - 5 b^{4} c\right ) + x \left (15 a^{4} f - 11 a^{3} b e + 7 a^{2} b^{2} d - 3 a b^{3} c\right )}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} + \frac{f x^{5}}{5 b^{3}} - \frac{x^{3} \left (3 a f - b e\right )}{3 b^{4}} + \frac{x \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(a*b**11))*(63*a**3*f - 35*a**2*b*e + 15*a*b**2*d - 3*b**3*c)*log(-a*b**
5*sqrt(-1/(a*b**11)) + x)/16 - sqrt(-1/(a*b**11))*(63*a**3*f - 35*a**2*b*e + 15*
a*b**2*d - 3*b**3*c)*log(a*b**5*sqrt(-1/(a*b**11)) + x)/16 + (x**3*(17*a**3*b*f
- 13*a**2*b**2*e + 9*a*b**3*d - 5*b**4*c) + x*(15*a**4*f - 11*a**3*b*e + 7*a**2*
b**2*d - 3*a*b**3*c))/(8*a**2*b**5 + 16*a*b**6*x**2 + 8*b**7*x**4) + f*x**5/(5*b
**3) - x**3*(3*a*f - b*e)/(3*b**4) + x*(6*a**2*f - 3*a*b*e + b**2*d)/b**5

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GIAC/XCAS [A]  time = 0.216108, size = 270, normalized size = 1.15 \[ \frac{{\left (3 \, b^{3} c - 15 \, a b^{2} d - 63 \, a^{3} f + 35 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{5}} - \frac{5 \, b^{4} c x^{3} - 9 \, a b^{3} d x^{3} - 17 \, a^{3} b f x^{3} + 13 \, a^{2} b^{2} x^{3} e + 3 \, a b^{3} c x - 7 \, a^{2} b^{2} d x - 15 \, a^{4} f x + 11 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} b^{5}} + \frac{3 \, b^{12} f x^{5} - 15 \, a b^{11} f x^{3} + 5 \, b^{12} x^{3} e + 15 \, b^{12} d x + 90 \, a^{2} b^{10} f x - 45 \, a b^{11} x e}{15 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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