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

Optimal. Leaf size=196 \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{5 a^3 x^5}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^3 f+15 a^2 b e-35 a b^2 d+63 b^3 c\right )}{8 a^{11/2} \sqrt{b}}-\frac{x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{8 a^5 \left (a+b x^2\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2} \]

[Out]

-c/(5*a^3*x^5) + (3*b*c - a*d)/(3*a^4*x^3) - (6*b^2*c - 3*a*b*d + a^2*e)/(a^5*x)
- ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*a^4*(a + b*x^2)^2) - ((15*b^3*c -
11*a*b^2*d + 7*a^2*b*e - 3*a^3*f)*x)/(8*a^5*(a + b*x^2)) - ((63*b^3*c - 35*a*b^2
*d + 15*a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2)*Sqrt[b])

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Rubi [A]  time = 0.667145, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{5 a^3 x^5}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^3 f+15 a^2 b e-35 a b^2 d+63 b^3 c\right )}{8 a^{11/2} \sqrt{b}}-\frac{x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{8 a^5 \left (a+b x^2\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-c/(5*a^3*x^5) + (3*b*c - a*d)/(3*a^4*x^3) - (6*b^2*c - 3*a*b*d + a^2*e)/(a^5*x)
- ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*a^4*(a + b*x^2)^2) - ((15*b^3*c -
11*a*b^2*d + 7*a^2*b*e - 3*a^3*f)*x)/(8*a^5*(a + b*x^2)) - ((63*b^3*c - 35*a*b^2
*d + 15*a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2)*Sqrt[b])

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

[Out]

Timed out

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Mathematica [A]  time = 0.228875, size = 196, normalized size = 1. \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{5 a^3 x^5}+\frac{a^2 (-e)+3 a b d-6 b^2 c}{a^5 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-15 a^2 b e+35 a b^2 d-63 b^3 c\right )}{8 a^{11/2} \sqrt{b}}+\frac{x \left (3 a^3 f-7 a^2 b e+11 a b^2 d-15 b^3 c\right )}{8 a^5 \left (a+b x^2\right )}+\frac{x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-c/(5*a^3*x^5) + (3*b*c - a*d)/(3*a^4*x^3) + (-6*b^2*c + 3*a*b*d - a^2*e)/(a^5*x
) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^4*(a + b*x^2)^2) + ((-15*b^3
*c + 11*a*b^2*d - 7*a^2*b*e + 3*a^3*f)*x)/(8*a^5*(a + b*x^2)) + ((-63*b^3*c + 35
*a*b^2*d - 15*a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2)*Sqrt[b
])

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Maple [A]  time = 0.024, size = 300, normalized size = 1.5 \[ -{\frac{c}{5\,{a}^{3}{x}^{5}}}-{\frac{d}{3\,{a}^{3}{x}^{3}}}+{\frac{bc}{{a}^{4}{x}^{3}}}-{\frac{e}{{a}^{3}x}}+3\,{\frac{bd}{{a}^{4}x}}-6\,{\frac{{b}^{2}c}{{a}^{5}x}}+{\frac{3\,{x}^{3}bf}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{x}^{3}{b}^{2}e}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{x}^{3}{b}^{3}d}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{x}^{3}{b}^{4}c}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,fx}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bex}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,d{b}^{2}x}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,{b}^{3}cx}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,f}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,be}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,d{b}^{2}}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{b}^{3}c}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236501, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (63 \, b^{5} c - 35 \, a b^{4} d + 15 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \,{\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d + 15 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{7} +{\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 15 \, a^{4} b e - 3 \, a^{5} f\right )} x^{5}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (15 \,{\left (63 \, b^{4} c - 35 \, a b^{3} d + 15 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 25 \,{\left (63 \, a b^{3} c - 35 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} + 24 \, a^{4} c + 8 \,{\left (63 \, a^{2} b^{2} c - 35 \, a^{3} b d + 15 \, a^{4} e\right )} x^{4} - 8 \,{\left (9 \, a^{3} b c - 5 \, a^{4} d\right )} x^{2}\right )} \sqrt{-a b}}{240 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )} \sqrt{-a b}}, -\frac{15 \,{\left ({\left (63 \, b^{5} c - 35 \, a b^{4} d + 15 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \,{\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d + 15 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{7} +{\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 15 \, a^{4} b e - 3 \, a^{5} f\right )} x^{5}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (15 \,{\left (63 \, b^{4} c - 35 \, a b^{3} d + 15 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 25 \,{\left (63 \, a b^{3} c - 35 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} + 24 \, a^{4} c + 8 \,{\left (63 \, a^{2} b^{2} c - 35 \, a^{3} b d + 15 \, a^{4} e\right )} x^{4} - 8 \,{\left (9 \, a^{3} b c - 5 \, a^{4} d\right )} x^{2}\right )} \sqrt{a b}}{120 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216964, size = 267, normalized size = 1.36 \[ -\frac{{\left (63 \, b^{3} c - 35 \, a b^{2} d - 3 \, a^{3} f + 15 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} - \frac{15 \, b^{4} c x^{3} - 11 \, a b^{3} d x^{3} - 3 \, a^{3} b f x^{3} + 7 \, a^{2} b^{2} x^{3} e + 17 \, a b^{3} c x - 13 \, a^{2} b^{2} d x - 5 \, a^{4} f x + 9 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{5}} - \frac{90 \, b^{2} c x^{4} - 45 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 15 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{5} x^{5}} \]

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^6),x, algorithm="giac")

[Out]

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