3.2.17 \(\int \frac {c+d x^2+e x^4+f x^6}{x^8 (a+b x^2)} \, dx\)

Optimal. Leaf size=137 \[ \frac {b c-a d}{5 a^2 x^5}-\frac {a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{9/2}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac {c}{7 a x^7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1802, 205} \begin {gather*} \frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{9/2}}-\frac {a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac {b c-a d}{5 a^2 x^5}-\frac {c}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx &=\int \left (\frac {c}{a x^8}+\frac {-b c+a d}{a^2 x^6}+\frac {b^2 c-a b d+a^2 e}{a^3 x^4}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{7 a x^7}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac {\left (b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{a^4}\\ &=-\frac {c}{7 a x^7}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac {\sqrt {b} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 139, normalized size = 1.01 \begin {gather*} \frac {b c-a d}{5 a^2 x^5}+\frac {a^2 (-e)+a b d-b^2 c}{3 a^3 x^3}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{9/2}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac {c}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]

[Out]

-1/7*c/(a*x^7) + (b*c - a*d)/(5*a^2*x^5) + (-(b^2*c) + a*b*d - a^2*e)/(3*a^3*x^3) + (b^3*c - a*b^2*d + a^2*b*e
- a^3*f)/(a^4*x) - (Sqrt[b]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]

[Out]

IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)), x]

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 292, normalized size = 2.13 \begin {gather*} \left [-\frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 210 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} + 70 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} + 30 \, a^{3} c - 42 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{210 \, a^{4} x^{7}}, \frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/210*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)
) - 210*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6 + 70*(a*b^2*c - a^2*b*d + a^3*e)*x^4 + 30*a^3*c - 42*(a^2*b*c
- a^3*d)*x^2)/(a^4*x^7), 1/105*(105*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7*sqrt(b/a)*arctan(x*sqrt(b/a)) + 10
5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6 - 35*(a*b^2*c - a^2*b*d + a^3*e)*x^4 - 15*a^3*c + 21*(a^2*b*c - a^3*
d)*x^2)/(a^4*x^7)]

________________________________________________________________________________________

giac [A]  time = 0.46, size = 151, normalized size = 1.10 \begin {gather*} \frac {{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} c x^{6} - 105 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 105 \, a^{2} b x^{6} e - 35 \, a b^{2} c x^{4} + 35 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 21 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x, algorithm="giac")

[Out]

(b^4*c - a*b^3*d - a^3*b*f + a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(105*b^3*c*x^6 - 105*a*b
^2*d*x^6 - 105*a^3*f*x^6 + 105*a^2*b*x^6*e - 35*a*b^2*c*x^4 + 35*a^2*b*d*x^4 - 35*a^3*x^4*e + 21*a^2*b*c*x^2 -
21*a^3*d*x^2 - 15*a^3*c)/(a^4*x^7)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 190, normalized size = 1.39 \begin {gather*} -\frac {b f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {b^{2} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {b^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {b^{4} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}-\frac {f}{a x}+\frac {b e}{a^{2} x}-\frac {b^{2} d}{a^{3} x}+\frac {b^{3} c}{a^{4} x}-\frac {e}{3 a \,x^{3}}+\frac {b d}{3 a^{2} x^{3}}-\frac {b^{2} c}{3 a^{3} x^{3}}-\frac {d}{5 a \,x^{5}}+\frac {b c}{5 a^{2} x^{5}}-\frac {c}{7 a \,x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.03, size = 134, normalized size = 0.98 \begin {gather*} \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a),x, algorithm="maxima")

[Out]

(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/105*(105*(b^3*c - a*b^2*d +
a^2*b*e - a^3*f)*x^6 - 35*(a*b^2*c - a^2*b*d + a^3*e)*x^4 - 15*a^3*c + 21*(a^2*b*c - a^3*d)*x^2)/(a^4*x^7)

________________________________________________________________________________________

mupad [B]  time = 0.98, size = 127, normalized size = 0.93 \begin {gather*} \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{9/2}}-\frac {\frac {c}{7\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{5\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,a^3}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x)

[Out]

(b^(1/2)*atan((b^(1/2)*x)/a^(1/2))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/a^(9/2) - (c/(7*a) - (x^6*(b^3*c - a^3
*f - a*b^2*d + a^2*b*e))/a^4 + (x^2*(a*d - b*c))/(5*a^2) + (x^4*(b^2*c + a^2*e - a*b*d))/(3*a^3))/x^7

________________________________________________________________________________________

sympy [B]  time = 21.65, size = 301, normalized size = 2.20 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} + \frac {- 15 a^{3} c + x^{6} \left (- 105 a^{3} f + 105 a^{2} b e - 105 a b^{2} d + 105 b^{3} c\right ) + x^{4} \left (- 35 a^{3} e + 35 a^{2} b d - 35 a b^{2} c\right ) + x^{2} \left (- 21 a^{3} d + 21 a^{2} b c\right )}{105 a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a),x)

[Out]

sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**5*sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d -
b**3*c)/(a**3*b*f - a**2*b**2*e + a*b**3*d - b**4*c) + x)/2 - sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b*
*3*c)*log(a**5*sqrt(-b/a**9)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b*f - a**2*b**2*e + a*b**3*d - b**4
*c) + x)/2 + (-15*a**3*c + x**6*(-105*a**3*f + 105*a**2*b*e - 105*a*b**2*d + 105*b**3*c) + x**4*(-35*a**3*e +
35*a**2*b*d - 35*a*b**2*c) + x**2*(-21*a**3*d + 21*a**2*b*c))/(105*a**4*x**7)

________________________________________________________________________________________