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

Optimal. Leaf size=193 \[ \frac{x^3 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac{x^3 \left (11 a^3 f-7 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}-\frac{x \left (35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^2 b^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^{3/2} b^{9/2}}+\frac{f x^3}{3 b^3} \]

[Out]

-((b^3*c + 3*a*b^2*d - 15*a^2*b*e + 35*a^3*f)*x)/(8*a^2*b^4) + (f*x^3)/(3*b^3) +
((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^3)/(4*a*(a + b*x^2)^2) + ((b^3*c + 3*a
*b^2*d - 7*a^2*b*e + 11*a^3*f)*x^3)/(8*a^2*b^3*(a + b*x^2)) + ((b^3*c + 3*a*b^2*
d - 15*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(9/2))

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Rubi [A]  time = 0.739767, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x^3 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac{x^3 \left (11 a^3 f-7 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}-\frac{x \left (35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^2 b^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^{3/2} b^{9/2}}+\frac{f x^3}{3 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-((b^3*c + 3*a*b^2*d - 15*a^2*b*e + 35*a^3*f)*x)/(8*a^2*b^4) + (f*x^3)/(3*b^3) +
((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^3)/(4*a*(a + b*x^2)^2) + ((b^3*c + 3*a
*b^2*d - 7*a^2*b*e + 11*a^3*f)*x^3)/(8*a^2*b^3*(a + b*x^2)) + ((b^3*c + 3*a*b^2*
d - 15*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(9/2))

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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**2*(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 = 0.254623, size = 156, normalized size = 0.81 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right )}{8 a^{3/2} b^{9/2}}+\frac{x \left (-105 a^4 f+5 a^3 b \left (9 e-35 f x^2\right )+a^2 b^2 \left (-9 d+75 e x^2-56 f x^4\right )+a b^3 \left (-3 c-15 d x^2+24 e x^4+8 f x^6\right )+3 b^4 c x^2\right )}{24 a b^4 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-105*a^4*f + 3*b^4*c*x^2 + 5*a^3*b*(9*e - 35*f*x^2) + a^2*b^2*(-9*d + 75*e*x
^2 - 56*f*x^4) + a*b^3*(-3*c - 15*d*x^2 + 24*e*x^4 + 8*f*x^6)))/(24*a*b^4*(a + b
*x^2)^2) + ((b^3*c + 3*a*b^2*d - 15*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[
a]])/(8*a^(3/2)*b^(9/2))

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Maple [A]  time = 0.016, size = 259, normalized size = 1.3 \[{\frac{f{x}^{3}}{3\,{b}^{3}}}-3\,{\frac{afx}{{b}^{4}}}+{\frac{ex}{{b}^{3}}}-{\frac{13\,{x}^{3}{a}^{2}f}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}e}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{x}^{3}d}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{x}^{3}c}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{11\,{a}^{3}fx}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}ex}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,adx}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{cx}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{2}f}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ae}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,d}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{c}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236339, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a^{2} b^{3} c + 3 \, a^{3} b^{2} d - 15 \, a^{4} b e + 35 \, a^{5} f +{\left (b^{5} c + 3 \, a b^{4} d - 15 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (a b^{4} c + 3 \, a^{2} b^{3} d - 15 \, a^{3} b^{2} e + 35 \, a^{4} b f\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (8 \, a b^{3} f x^{7} + 8 \,{\left (3 \, a b^{3} e - 7 \, a^{2} b^{2} f\right )} x^{5} +{\left (3 \, b^{4} c - 15 \, a b^{3} d + 75 \, a^{2} b^{2} e - 175 \, a^{3} b f\right )} x^{3} - 3 \,{\left (a b^{3} c + 3 \, a^{2} b^{2} d - 15 \, a^{3} b e + 35 \, a^{4} f\right )} x\right )} \sqrt{-a b}}{48 \,{\left (a b^{6} x^{4} + 2 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \sqrt{-a b}}, \frac{3 \,{\left (a^{2} b^{3} c + 3 \, a^{3} b^{2} d - 15 \, a^{4} b e + 35 \, a^{5} f +{\left (b^{5} c + 3 \, a b^{4} d - 15 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (a b^{4} c + 3 \, a^{2} b^{3} d - 15 \, a^{3} b^{2} e + 35 \, a^{4} b f\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (8 \, a b^{3} f x^{7} + 8 \,{\left (3 \, a b^{3} e - 7 \, a^{2} b^{2} f\right )} x^{5} +{\left (3 \, b^{4} c - 15 \, a b^{3} d + 75 \, a^{2} b^{2} e - 175 \, a^{3} b f\right )} x^{3} - 3 \,{\left (a b^{3} c + 3 \, a^{2} b^{2} d - 15 \, a^{3} b e + 35 \, a^{4} f\right )} x\right )} \sqrt{a b}}{24 \,{\left (a b^{6} x^{4} + 2 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*(a^2*b^3*c + 3*a^3*b^2*d - 15*a^4*b*e + 35*a^5*f + (b^5*c + 3*a*b^4*d -
15*a^2*b^3*e + 35*a^3*b^2*f)*x^4 + 2*(a*b^4*c + 3*a^2*b^3*d - 15*a^3*b^2*e + 35
*a^4*b*f)*x^2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(8*a*b^3*
f*x^7 + 8*(3*a*b^3*e - 7*a^2*b^2*f)*x^5 + (3*b^4*c - 15*a*b^3*d + 75*a^2*b^2*e -
175*a^3*b*f)*x^3 - 3*(a*b^3*c + 3*a^2*b^2*d - 15*a^3*b*e + 35*a^4*f)*x)*sqrt(-a
*b))/((a*b^6*x^4 + 2*a^2*b^5*x^2 + a^3*b^4)*sqrt(-a*b)), 1/24*(3*(a^2*b^3*c + 3*
a^3*b^2*d - 15*a^4*b*e + 35*a^5*f + (b^5*c + 3*a*b^4*d - 15*a^2*b^3*e + 35*a^3*b
^2*f)*x^4 + 2*(a*b^4*c + 3*a^2*b^3*d - 15*a^3*b^2*e + 35*a^4*b*f)*x^2)*arctan(sq
rt(a*b)*x/a) + (8*a*b^3*f*x^7 + 8*(3*a*b^3*e - 7*a^2*b^2*f)*x^5 + (3*b^4*c - 15*
a*b^3*d + 75*a^2*b^2*e - 175*a^3*b*f)*x^3 - 3*(a*b^3*c + 3*a^2*b^2*d - 15*a^3*b*
e + 35*a^4*f)*x)*sqrt(a*b))/((a*b^6*x^4 + 2*a^2*b^5*x^2 + a^3*b^4)*sqrt(a*b))]

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Sympy [A]  time = 21.23, size = 258, normalized size = 1.34 \[ - \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (35 a^{3} f - 15 a^{2} b e + 3 a b^{2} d + b^{3} c\right ) \log{\left (- a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (35 a^{3} f - 15 a^{2} b e + 3 a b^{2} d + b^{3} c\right ) \log{\left (a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{16} - \frac{x^{3} \left (13 a^{3} b f - 9 a^{2} b^{2} e + 5 a b^{3} d - b^{4} c\right ) + x \left (11 a^{4} f - 7 a^{3} b e + 3 a^{2} b^{2} d + a b^{3} c\right )}{8 a^{3} b^{4} + 16 a^{2} b^{5} x^{2} + 8 a b^{6} x^{4}} + \frac{f x^{3}}{3 b^{3}} - \frac{x \left (3 a f - b e\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217039, size = 234, normalized size = 1.21 \[ \frac{{\left (b^{3} c + 3 \, a b^{2} d + 35 \, a^{3} f - 15 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{4}} + \frac{b^{4} c x^{3} - 5 \, a b^{3} d x^{3} - 13 \, a^{3} b f x^{3} + 9 \, a^{2} b^{2} x^{3} e - a b^{3} c x - 3 \, a^{2} b^{2} d x - 11 \, a^{4} f x + 7 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{4}} + \frac{b^{6} f x^{3} - 9 \, a b^{5} f x + 3 \, b^{6} x e}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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