3.2.22 \(\int \frac {c+d x^2+e x^4+f x^6}{x^{10} (a+b x^2)} \, dx\) [122]

Optimal. Leaf size=175 \[ -\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac {b^{3/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{11/2}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 175, 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 = {1816, 211} \begin {gather*} \frac {b c-a d}{7 a^2 x^7}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac {b^{3/2} \text {ArcTan}\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^{11/2}}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5 x}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {c}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)),x]

[Out]

-1/9*c/(a*x^9) + (b*c - a*d)/(7*a^2*x^7) - (b^2*c - a*b*d + a^2*e)/(5*a^3*x^5) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(3*a^4*x^3) - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a^5*x) - (b^(3/2)*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(11/2)

Rule 211

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

Rule 1816

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^{10} \left (a+b x^2\right )} \, dx &=\int \left (\frac {c}{a x^{10}}+\frac {-b c+a d}{a^2 x^8}+\frac {b^2 c-a b d+a^2 e}{a^3 x^6}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^4}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^2}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac {\left (b^2 \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^5}\\ &=-\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac {b^{3/2} \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^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 174, normalized size = 0.99 \begin {gather*} -\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}+\frac {-b^2 c+a b d-a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}+\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x}+\frac {b^{3/2} \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^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)),x]

[Out]

-1/9*c/(a*x^9) + (b*c - a*d)/(7*a^2*x^7) + (-(b^2*c) + a*b*d - a^2*e)/(5*a^3*x^5) + (b^3*c - a*b^2*d + a^2*b*e
- a^3*f)/(3*a^4*x^3) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a^5*x) + (b^(3/2)*(-(b^3*c) + a*b^2*d - a^
2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(11/2)

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Maple [A]
time = 0.13, size = 163, normalized size = 0.93

method	result	size
default	\(\frac {b^{2} \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{5} \sqrt {a b}}-\frac {c}{9 a \,x^{9}}-\frac {a d -b c}{7 a^{2} x^{7}}-\frac {a^{2} e -a b d +b^{2} c}{5 a^{3} x^{5}}-\frac {a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c}{3 a^{4} x^{3}}+\frac {b \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{5} x}\)	\(163\)
risch	\(\frac {\frac {b \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{8}}{a^{5}}-\frac {\left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{6}}{3 a^{4}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{5 a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{7 a^{2}}-\frac {c}{9 a}}{x^{9}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{11} \textit {\_Z}^{2}+a^{6} b^{3} f^{2}-2 a^{5} b^{4} e f +2 a^{4} b^{5} d f +a^{4} b^{5} e^{2}-2 a^{3} b^{6} c f -2 a^{3} b^{6} d e +2 a^{2} b^{7} c e +a^{2} b^{7} d^{2}-2 a \,b^{8} c d +b^{9} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{11}+2 a^{6} b^{3} f^{2}-4 a^{5} b^{4} e f +4 a^{4} b^{5} d f +2 a^{4} b^{5} e^{2}-4 a^{3} b^{6} c f -4 a^{3} b^{6} d e +4 a^{2} b^{7} c e +2 a^{2} b^{7} d^{2}-4 a \,b^{8} c d +2 b^{9} c^{2}\right ) x +\left (-a^{9} b f +a^{8} b^{2} e -a^{7} b^{3} d +a^{6} b^{4} c \right ) \textit {\_R} \right )\right )}{2}\)	\(377\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 179, normalized size = 1.02 \begin {gather*} -\frac {{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {315 \, {\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} x^{8} - 105 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} x^{6} + 35 \, a^{4} c + 63 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 45 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{315 \, a^{5} x^{9}} \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^10/(b*x^2+a),x, algorithm="maxima")

[Out]

-(b^5*c - a*b^4*d - a^3*b^2*f + a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/315*(315*(b^4*c - a*b^3*d
- a^3*b*f + a^2*b^2*e)*x^8 - 105*(a*b^3*c - a^2*b^2*d - a^4*f + a^3*b*e)*x^6 + 35*a^4*c + 63*(a^2*b^2*c - a^3
*b*d + a^4*e)*x^4 - 45*(a^3*b*c - a^4*d)*x^2)/(a^5*x^9)

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Fricas [A]
time = 6.66, size = 414, normalized size = 2.37 \begin {gather*} \left [-\frac {630 \, {\left (b^{4} c - a b^{3} d - a^{3} b f\right )} x^{8} - 210 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f\right )} x^{6} + 70 \, a^{4} c + 126 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{4} - 90 \, {\left (a^{3} b c - a^{4} d\right )} x^{2} - 315 \, {\left (a^{2} b^{2} x^{9} e + {\left (b^{4} c - a b^{3} d - a^{3} b f\right )} x^{9}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 42 \, {\left (15 \, a^{2} b^{2} x^{8} - 5 \, a^{3} b x^{6} + 3 \, a^{4} x^{4}\right )} e}{630 \, a^{5} x^{9}}, -\frac {315 \, {\left (b^{4} c - a b^{3} d - a^{3} b f\right )} x^{8} - 105 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f\right )} x^{6} + 35 \, a^{4} c + 63 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{4} - 45 \, {\left (a^{3} b c - a^{4} d\right )} x^{2} + 315 \, {\left (a^{2} b^{2} x^{9} e + {\left (b^{4} c - a b^{3} d - a^{3} b f\right )} x^{9}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 21 \, {\left (15 \, a^{2} b^{2} x^{8} - 5 \, a^{3} b x^{6} + 3 \, a^{4} x^{4}\right )} e}{315 \, a^{5} x^{9}}\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^10/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/630*(630*(b^4*c - a*b^3*d - a^3*b*f)*x^8 - 210*(a*b^3*c - a^2*b^2*d - a^4*f)*x^6 + 70*a^4*c + 126*(a^2*b^2
*c - a^3*b*d)*x^4 - 90*(a^3*b*c - a^4*d)*x^2 - 315*(a^2*b^2*x^9*e + (b^4*c - a*b^3*d - a^3*b*f)*x^9)*sqrt(-b/a
)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 42*(15*a^2*b^2*x^8 - 5*a^3*b*x^6 + 3*a^4*x^4)*e)/(a^5*x^9)
, -1/315*(315*(b^4*c - a*b^3*d - a^3*b*f)*x^8 - 105*(a*b^3*c - a^2*b^2*d - a^4*f)*x^6 + 35*a^4*c + 63*(a^2*b^2
*c - a^3*b*d)*x^4 - 45*(a^3*b*c - a^4*d)*x^2 + 315*(a^2*b^2*x^9*e + (b^4*c - a*b^3*d - a^3*b*f)*x^9)*sqrt(b/a)
*arctan(x*sqrt(b/a)) + 21*(15*a^2*b^2*x^8 - 5*a^3*b*x^6 + 3*a^4*x^4)*e)/(a^5*x^9)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (167) = 334\).
time = 25.30, size = 354, normalized size = 2.02 \begin {gather*} - \frac {\sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac {- 35 a^{4} c + x^{8} \cdot \left (315 a^{3} b f - 315 a^{2} b^{2} e + 315 a b^{3} d - 315 b^{4} c\right ) + x^{6} \left (- 105 a^{4} f + 105 a^{3} b e - 105 a^{2} b^{2} d + 105 a b^{3} c\right ) + x^{4} \left (- 63 a^{4} e + 63 a^{3} b d - 63 a^{2} b^{2} c\right ) + x^{2} \left (- 45 a^{4} d + 45 a^{3} b c\right )}{315 a^{5} x^{9}} \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**10/(b*x**2+a),x)

[Out]

-sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**6*sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*
b**2*d - b**3*c)/(a**3*b**2*f - a**2*b**3*e + a*b**4*d - b**5*c) + x)/2 + sqrt(-b**3/a**11)*(a**3*f - a**2*b*e
+ a*b**2*d - b**3*c)*log(a**6*sqrt(-b**3/a**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b**2*f - a**2*b
**3*e + a*b**4*d - b**5*c) + x)/2 + (-35*a**4*c + x**8*(315*a**3*b*f - 315*a**2*b**2*e + 315*a*b**3*d - 315*b*
*4*c) + x**6*(-105*a**4*f + 105*a**3*b*e - 105*a**2*b**2*d + 105*a*b**3*c) + x**4*(-63*a**4*e + 63*a**3*b*d -
63*a**2*b**2*c) + x**2*(-45*a**4*d + 45*a**3*b*c))/(315*a**5*x**9)

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Giac [A]
time = 1.55, size = 201, normalized size = 1.15 \begin {gather*} -\frac {{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {315 \, b^{4} c x^{8} - 315 \, a b^{3} d x^{8} - 315 \, a^{3} b f x^{8} + 315 \, a^{2} b^{2} x^{8} e - 105 \, a b^{3} c x^{6} + 105 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 105 \, a^{3} b x^{6} e + 63 \, a^{2} b^{2} c x^{4} - 63 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 45 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{5} x^{9}} \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^10/(b*x^2+a),x, algorithm="giac")

[Out]

-(b^5*c - a*b^4*d - a^3*b^2*f + a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/315*(315*b^4*c*x^8 - 315*
a*b^3*d*x^8 - 315*a^3*b*f*x^8 + 315*a^2*b^2*x^8*e - 105*a*b^3*c*x^6 + 105*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 105*
a^3*b*x^6*e + 63*a^2*b^2*c*x^4 - 63*a^3*b*d*x^4 + 63*a^4*x^4*e - 45*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^
5*x^9)

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Mupad [B]
time = 1.02, size = 161, normalized size = 0.92 \begin {gather*} -\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{7\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{5\,a^3}+\frac {b\,x^8\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^5}}{x^9}-\frac {b^{3/2}\,\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^{11/2}} \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^10*(a + b*x^2)),x)

[Out]

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

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