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

Optimal. Leaf size=147 \[ \frac{x \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac{x \left (9 a^3 f-5 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac{f x}{b^3} \]

[Out]

(f*x)/b^3 + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x)/(4*a*(a + b*x^2)^2) + ((3*
b^3*c + a*b^2*d - 5*a^2*b*e + 9*a^3*f)*x)/(8*a^2*b^3*(a + b*x^2)) + ((3*b^3*c +
a*b^2*d + 3*a^2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))

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Rubi [A]  time = 0.354676, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac{x \left (9 a^3 f-5 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac{f x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(f*x)/b^3 + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x)/(4*a*(a + b*x^2)^2) + ((3*
b^3*c + a*b^2*d - 5*a^2*b*e + 9*a^3*f)*x)/(8*a^2*b^3*(a + b*x^2)) + ((3*b^3*c +
a*b^2*d + 3*a^2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))

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Rubi in Sympy [A]  time = 84.4624, size = 146, normalized size = 0.99 \[ \frac{f x}{b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{4 a b^{3} \left (a + b x^{2}\right )^{2}} + \frac{x \left (9 a^{3} f - 5 a^{2} b e + a b^{2} d + 3 b^{3} c\right )}{8 a^{2} b^{3} \left (a + b x^{2}\right )} - \frac{\left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x/b**3 - x*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(4*a*b**3*(a + b*x**2)**2)
+ x*(9*a**3*f - 5*a**2*b*e + a*b**2*d + 3*b**3*c)/(8*a**2*b**3*(a + b*x**2)) - (
15*a**3*f - 3*a**2*b*e - a*b**2*d - 3*b**3*c)*atan(sqrt(b)*x/sqrt(a))/(8*a**(5/2
)*b**(7/2))

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Mathematica [A]  time = 0.223779, size = 141, normalized size = 0.96 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac{x \left (15 a^4 f+a^3 b \left (25 f x^2-3 e\right )-a^2 b^2 \left (d+5 e x^2-8 f x^4\right )+a b^3 \left (5 c+d x^2\right )+3 b^4 c x^2\right )}{8 a^2 b^3 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(15*a^4*f + 3*b^4*c*x^2 + a*b^3*(5*c + d*x^2) + a^3*b*(-3*e + 25*f*x^2) - a^2
*b^2*(d + 5*e*x^2 - 8*f*x^4)))/(8*a^2*b^3*(a + b*x^2)^2) + ((3*b^3*c + a*b^2*d +
3*a^2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))

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Maple [A]  time = 0.018, size = 234, normalized size = 1.6 \[{\frac{fx}{{b}^{3}}}+{\frac{9\,a{x}^{3}f}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{x}^{3}e}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{x}^{3}d}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,b{x}^{3}c}{8\, \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{7\,{a}^{2}fx}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,aex}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{dx}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,cx}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{15\,af}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,e}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239641, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, a^{2} b^{3} c + a^{3} b^{2} d + 3 \, a^{4} b e - 15 \, a^{5} f +{\left (3 \, b^{5} c + a b^{4} d + 3 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c + a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{2}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (8 \, a^{2} b^{2} f x^{5} +{\left (3 \, b^{4} c + a b^{3} d - 5 \, a^{2} b^{2} e + 25 \, a^{3} b f\right )} x^{3} +{\left (5 \, a b^{3} c - a^{2} b^{2} d - 3 \, a^{3} b e + 15 \, a^{4} f\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )} \sqrt{-a b}}, \frac{{\left (3 \, a^{2} b^{3} c + a^{3} b^{2} d + 3 \, a^{4} b e - 15 \, a^{5} f +{\left (3 \, b^{5} c + a b^{4} d + 3 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c + a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (8 \, a^{2} b^{2} f x^{5} +{\left (3 \, b^{4} c + a b^{3} d - 5 \, a^{2} b^{2} e + 25 \, a^{3} b f\right )} x^{3} +{\left (5 \, a b^{3} c - a^{2} b^{2} d - 3 \, a^{3} b e + 15 \, a^{4} f\right )} x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*((3*a^2*b^3*c + a^3*b^2*d + 3*a^4*b*e - 15*a^5*f + (3*b^5*c + a*b^4*d + 3
*a^2*b^3*e - 15*a^3*b^2*f)*x^4 + 2*(3*a*b^4*c + a^2*b^3*d + 3*a^3*b^2*e - 15*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^2*b^2*f
*x^5 + (3*b^4*c + a*b^3*d - 5*a^2*b^2*e + 25*a^3*b*f)*x^3 + (5*a*b^3*c - a^2*b^2
*d - 3*a^3*b*e + 15*a^4*f)*x)*sqrt(-a*b))/((a^2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^
3)*sqrt(-a*b)), 1/8*((3*a^2*b^3*c + a^3*b^2*d + 3*a^4*b*e - 15*a^5*f + (3*b^5*c
+ a*b^4*d + 3*a^2*b^3*e - 15*a^3*b^2*f)*x^4 + 2*(3*a*b^4*c + a^2*b^3*d + 3*a^3*b
^2*e - 15*a^4*b*f)*x^2)*arctan(sqrt(a*b)*x/a) + (8*a^2*b^2*f*x^5 + (3*b^4*c + a*
b^3*d - 5*a^2*b^2*e + 25*a^3*b*f)*x^3 + (5*a*b^3*c - a^2*b^2*d - 3*a^3*b*e + 15*
a^4*f)*x)*sqrt(a*b))/((a^2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4*b^3)*sqrt(a*b))]

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Sympy [A]  time = 15.3698, size = 243, normalized size = 1.65 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{7}}} \left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \log{\left (- a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a^{5} b^{7}}} \left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \log{\left (a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{16} + \frac{x^{3} \left (9 a^{3} b f - 5 a^{2} b^{2} e + a b^{3} d + 3 b^{4} c\right ) + x \left (7 a^{4} f - 3 a^{3} b e - a^{2} b^{2} d + 5 a b^{3} c\right )}{8 a^{4} b^{3} + 16 a^{3} b^{4} x^{2} + 8 a^{2} b^{5} x^{4}} + \frac{f x}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218685, size = 201, normalized size = 1.37 \[ \frac{f x}{b^{3}} + \frac{{\left (3 \, b^{3} c + a b^{2} d - 15 \, a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{3}} + \frac{3 \, b^{4} c x^{3} + a b^{3} d x^{3} + 9 \, a^{3} b f x^{3} - 5 \, a^{2} b^{2} x^{3} e + 5 \, a b^{3} c x - a^{2} b^{2} d x + 7 \, a^{4} f x - 3 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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