3.2.41 \(\int \frac {c+d x^2+e x^4+f x^6}{x^8 (a+b x^2)^3} \, dx\) [141]

Optimal. Leaf size=234 \[ -\frac {c}{7 a^3 x^7}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (99 b^3 c-63 a b^2 d+35 a^2 b e-15 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{13/2}} \]

[Out]

-1/7*c/a^3/x^7+1/5*(-a*d+3*b*c)/a^4/x^5+1/3*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x^3+(-a^3*f+3*a^2*b*e-6*a*b^2*d+10*b^
3*c)/a^6/x+1/4*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^5/(b*x^2+a)^2+1/8*b*(-7*a^3*f+11*a^2*b*e-15*a*b^2*d+19*b^3
*c)*x/a^6/(b*x^2+a)+1/8*(-15*a^3*f+35*a^2*b*e-63*a*b^2*d+99*b^3*c)*arctan(x*b^(1/2)/a^(1/2))*b^(1/2)/a^(13/2)

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Rubi [A]
time = 0.33, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1819, 1816, 211} \begin {gather*} \frac {3 b c-a d}{5 a^4 x^5}-\frac {c}{7 a^3 x^7}-\frac {a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-15 a^3 f+35 a^2 b e-63 a b^2 d+99 b^3 c\right )}{8 a^{13/2}}+\frac {b x \left (-7 a^3 f+11 a^2 b e-15 a b^2 d+19 b^3 c\right )}{8 a^6 \left (a+b x^2\right )}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^3),x]

[Out]

-1/7*c/(a^3*x^7) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*x^3) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*a^5*(a + b*x^2)^2) + (b*(19*b^3*c
- 15*a*b^2*d + 11*a^2*b*e - 7*a^3*f)*x)/(8*a^6*(a + b*x^2)) + (Sqrt[b]*(99*b^3*c - 63*a*b^2*d + 35*a^2*b*e -
15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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]

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )^3} \, dx &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}-\frac {\int \frac {-4 c+4 \left (\frac {b c}{a}-d\right ) x^2-\frac {4 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac {4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac {3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac {\int \frac {8 c-8 \left (\frac {2 b c}{a}-d\right ) x^2+8 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^4-\frac {8 \left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) x^6}{a^3}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac {\int \left (\frac {8 c}{a x^8}+\frac {8 (-3 b c+a d)}{a^2 x^6}+\frac {8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^4}+\frac {8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^2}-\frac {b \left (-99 b^3 c+63 a b^2 d-35 a^2 b e+15 a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac {c}{7 a^3 x^7}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac {\left (b \left (99 b^3 c-63 a b^2 d+35 a^2 b e-15 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^6}\\ &=-\frac {c}{7 a^3 x^7}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (99 b^3 c-63 a b^2 d+35 a^2 b e-15 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 234, normalized size = 1.00 \begin {gather*} -\frac {c}{7 a^3 x^7}+\frac {3 b c-a d}{5 a^4 x^5}-\frac {6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac {b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (99 b^3 c-63 a b^2 d+35 a^2 b e-15 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^3),x]

[Out]

-1/7*c/(a^3*x^7) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*x^3) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*a^5*(a + b*x^2)^2) + (b*(19*b^3*c
- 15*a*b^2*d + 11*a^2*b*e - 7*a^3*f)*x)/(8*a^6*(a + b*x^2)) + (Sqrt[b]*(99*b^3*c - 63*a*b^2*d + 35*a^2*b*e -
15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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Maple [A]
time = 0.14, size = 212, normalized size = 0.91

method	result	size
default	\(-\frac {b \left (\frac {\left (\frac {7}{8} a^{3} b f -\frac {11}{8} a^{2} e \,b^{2}+\frac {15}{8} a d \,b^{3}-\frac {19}{8} c \,b^{4}\right ) x^{3}+\frac {a \left (9 a^{3} f -13 a^{2} b e +17 a \,b^{2} d -21 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (15 a^{3} f -35 a^{2} b e +63 a \,b^{2} d -99 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{6}}-\frac {c}{7 a^{3} x^{7}}-\frac {a d -3 b c}{5 a^{4} x^{5}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{3 a^{5} x^{3}}-\frac {a^{3} f -3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c}{a^{6} x}\)	\(212\)
risch	\(\frac {-\frac {b^{2} \left (15 a^{3} f -35 a^{2} b e +63 a \,b^{2} d -99 b^{3} c \right ) x^{10}}{8 a^{6}}-\frac {5 b \left (15 a^{3} f -35 a^{2} b e +63 a \,b^{2} d -99 b^{3} c \right ) x^{8}}{24 a^{5}}-\frac {\left (15 a^{3} f -35 a^{2} b e +63 a \,b^{2} d -99 b^{3} c \right ) x^{6}}{15 a^{4}}-\frac {\left (35 a^{2} e -63 a b d +99 b^{2} c \right ) x^{4}}{105 a^{3}}-\frac {\left (7 a d -11 b c \right ) x^{2}}{35 a^{2}}-\frac {c}{7 a}}{x^{7} \left (b \,x^{2}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{13} \textit {\_Z}^{2}+225 a^{6} b \,f^{2}-1050 a^{5} b^{2} e f +1890 a^{4} b^{3} d f +1225 a^{4} b^{3} e^{2}-2970 a^{3} b^{4} c f -4410 a^{3} b^{4} d e +6930 a^{2} b^{5} c e +3969 a^{2} b^{5} d^{2}-12474 a \,b^{6} c d +9801 b^{7} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{13}+450 a^{6} b \,f^{2}-2100 a^{5} b^{2} e f +3780 a^{4} b^{3} d f +2450 a^{4} b^{3} e^{2}-5940 a^{3} b^{4} c f -8820 a^{3} b^{4} d e +13860 a^{2} b^{5} c e +7938 a^{2} b^{5} d^{2}-24948 a \,b^{6} c d +19602 b^{7} c^{2}\right ) x +\left (15 a^{10} f -35 a^{9} b e +63 a^{8} b^{2} d -99 a^{7} b^{3} c \right ) \textit {\_R} \right )\right )}{16}\)	\(431\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

-b/a^6*(((7/8*a^3*b*f-11/8*a^2*e*b^2+15/8*a*d*b^3-19/8*c*b^4)*x^3+1/8*a*(9*a^3*f-13*a^2*b*e+17*a*b^2*d-21*b^3*
c)*x)/(b*x^2+a)^2+1/8*(15*a^3*f-35*a^2*b*e+63*a*b^2*d-99*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/7*c/a^3
/x^7-1/5*(a*d-3*b*c)/a^4/x^5-1/3*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^3-(a^3*f-3*a^2*b*e+6*a*b^2*d-10*b^3*c)/a^6/x

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Maxima [A]
time = 0.53, size = 252, normalized size = 1.08 \begin {gather*} \frac {105 \, {\left (99 \, b^{5} c - 63 \, a b^{4} d - 15 \, a^{3} b^{2} f + 35 \, a^{2} b^{3} e\right )} x^{10} + 175 \, {\left (99 \, a b^{4} c - 63 \, a^{2} b^{3} d - 15 \, a^{4} b f + 35 \, a^{3} b^{2} e\right )} x^{8} + 56 \, {\left (99 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d - 15 \, a^{5} f + 35 \, a^{4} b e\right )} x^{6} - 120 \, a^{5} c - 8 \, {\left (99 \, a^{3} b^{2} c - 63 \, a^{4} b d + 35 \, a^{5} e\right )} x^{4} + 24 \, {\left (11 \, a^{4} b c - 7 \, a^{5} d\right )} x^{2}}{840 \, {\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}} + \frac {{\left (99 \, b^{4} c - 63 \, a b^{3} d - 15 \, a^{3} b f + 35 \, a^{2} b^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{6}} \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)^3,x, algorithm="maxima")

[Out]

1/840*(105*(99*b^5*c - 63*a*b^4*d - 15*a^3*b^2*f + 35*a^2*b^3*e)*x^10 + 175*(99*a*b^4*c - 63*a^2*b^3*d - 15*a^
4*b*f + 35*a^3*b^2*e)*x^8 + 56*(99*a^2*b^3*c - 63*a^3*b^2*d - 15*a^5*f + 35*a^4*b*e)*x^6 - 120*a^5*c - 8*(99*a
^3*b^2*c - 63*a^4*b*d + 35*a^5*e)*x^4 + 24*(11*a^4*b*c - 7*a^5*d)*x^2)/(a^6*b^2*x^11 + 2*a^7*b*x^9 + a^8*x^7)
+ 1/8*(99*b^4*c - 63*a*b^3*d - 15*a^3*b*f + 35*a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6)

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Fricas [A]
time = 4.15, size = 726, normalized size = 3.10 \begin {gather*} \left [\frac {630 \, {\left (33 \, b^{5} c - 21 \, a b^{4} d - 5 \, a^{3} b^{2} f\right )} x^{10} + 1050 \, {\left (33 \, a b^{4} c - 21 \, a^{2} b^{3} d - 5 \, a^{4} b f\right )} x^{8} + 336 \, {\left (33 \, a^{2} b^{3} c - 21 \, a^{3} b^{2} d - 5 \, a^{5} f\right )} x^{6} - 240 \, a^{5} c - 144 \, {\left (11 \, a^{3} b^{2} c - 7 \, a^{4} b d\right )} x^{4} + 48 \, {\left (11 \, a^{4} b c - 7 \, a^{5} d\right )} x^{2} + 105 \, {\left (3 \, {\left (33 \, b^{5} c - 21 \, a b^{4} d - 5 \, a^{3} b^{2} f\right )} x^{11} + 6 \, {\left (33 \, a b^{4} c - 21 \, a^{2} b^{3} d - 5 \, a^{4} b f\right )} x^{9} + 3 \, {\left (33 \, a^{2} b^{3} c - 21 \, a^{3} b^{2} d - 5 \, a^{5} f\right )} x^{7} + 35 \, {\left (a^{2} b^{3} x^{11} + 2 \, a^{3} b^{2} x^{9} + a^{4} b x^{7}\right )} e\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 70 \, {\left (105 \, a^{2} b^{3} x^{10} + 175 \, a^{3} b^{2} x^{8} + 56 \, a^{4} b x^{6} - 8 \, a^{5} x^{4}\right )} e}{1680 \, {\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}, \frac {315 \, {\left (33 \, b^{5} c - 21 \, a b^{4} d - 5 \, a^{3} b^{2} f\right )} x^{10} + 525 \, {\left (33 \, a b^{4} c - 21 \, a^{2} b^{3} d - 5 \, a^{4} b f\right )} x^{8} + 168 \, {\left (33 \, a^{2} b^{3} c - 21 \, a^{3} b^{2} d - 5 \, a^{5} f\right )} x^{6} - 120 \, a^{5} c - 72 \, {\left (11 \, a^{3} b^{2} c - 7 \, a^{4} b d\right )} x^{4} + 24 \, {\left (11 \, a^{4} b c - 7 \, a^{5} d\right )} x^{2} + 105 \, {\left (3 \, {\left (33 \, b^{5} c - 21 \, a b^{4} d - 5 \, a^{3} b^{2} f\right )} x^{11} + 6 \, {\left (33 \, a b^{4} c - 21 \, a^{2} b^{3} d - 5 \, a^{4} b f\right )} x^{9} + 3 \, {\left (33 \, a^{2} b^{3} c - 21 \, a^{3} b^{2} d - 5 \, a^{5} f\right )} x^{7} + 35 \, {\left (a^{2} b^{3} x^{11} + 2 \, a^{3} b^{2} x^{9} + a^{4} b x^{7}\right )} e\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 35 \, {\left (105 \, a^{2} b^{3} x^{10} + 175 \, a^{3} b^{2} x^{8} + 56 \, a^{4} b x^{6} - 8 \, a^{5} x^{4}\right )} e}{840 \, {\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}\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)^3,x, algorithm="fricas")

[Out]

[1/1680*(630*(33*b^5*c - 21*a*b^4*d - 5*a^3*b^2*f)*x^10 + 1050*(33*a*b^4*c - 21*a^2*b^3*d - 5*a^4*b*f)*x^8 + 3
36*(33*a^2*b^3*c - 21*a^3*b^2*d - 5*a^5*f)*x^6 - 240*a^5*c - 144*(11*a^3*b^2*c - 7*a^4*b*d)*x^4 + 48*(11*a^4*b
*c - 7*a^5*d)*x^2 + 105*(3*(33*b^5*c - 21*a*b^4*d - 5*a^3*b^2*f)*x^11 + 6*(33*a*b^4*c - 21*a^2*b^3*d - 5*a^4*b
*f)*x^9 + 3*(33*a^2*b^3*c - 21*a^3*b^2*d - 5*a^5*f)*x^7 + 35*(a^2*b^3*x^11 + 2*a^3*b^2*x^9 + a^4*b*x^7)*e)*sqr
t(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 70*(105*a^2*b^3*x^10 + 175*a^3*b^2*x^8 + 56*a^4*b*x^
6 - 8*a^5*x^4)*e)/(a^6*b^2*x^11 + 2*a^7*b*x^9 + a^8*x^7), 1/840*(315*(33*b^5*c - 21*a*b^4*d - 5*a^3*b^2*f)*x^1
0 + 525*(33*a*b^4*c - 21*a^2*b^3*d - 5*a^4*b*f)*x^8 + 168*(33*a^2*b^3*c - 21*a^3*b^2*d - 5*a^5*f)*x^6 - 120*a^
5*c - 72*(11*a^3*b^2*c - 7*a^4*b*d)*x^4 + 24*(11*a^4*b*c - 7*a^5*d)*x^2 + 105*(3*(33*b^5*c - 21*a*b^4*d - 5*a^
3*b^2*f)*x^11 + 6*(33*a*b^4*c - 21*a^2*b^3*d - 5*a^4*b*f)*x^9 + 3*(33*a^2*b^3*c - 21*a^3*b^2*d - 5*a^5*f)*x^7
+ 35*(a^2*b^3*x^11 + 2*a^3*b^2*x^9 + a^4*b*x^7)*e)*sqrt(b/a)*arctan(x*sqrt(b/a)) + 35*(105*a^2*b^3*x^10 + 175*
a^3*b^2*x^8 + 56*a^4*b*x^6 - 8*a^5*x^4)*e)/(a^6*b^2*x^11 + 2*a^7*b*x^9 + a^8*x^7)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.80, size = 250, normalized size = 1.07 \begin {gather*} \frac {{\left (99 \, b^{4} c - 63 \, a b^{3} d - 15 \, a^{3} b f + 35 \, a^{2} b^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{6}} + \frac {19 \, b^{5} c x^{3} - 15 \, a b^{4} d x^{3} - 7 \, a^{3} b^{2} f x^{3} + 11 \, a^{2} b^{3} x^{3} e + 21 \, a b^{4} c x - 17 \, a^{2} b^{3} d x - 9 \, a^{4} b f x + 13 \, a^{3} b^{2} x e}{8 \, {\left (b x^{2} + a\right )}^{2} a^{6}} + \frac {1050 \, b^{3} c x^{6} - 630 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 315 \, a^{2} b x^{6} e - 210 \, a b^{2} c x^{4} + 105 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 63 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{6} 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)^3,x, algorithm="giac")

[Out]

1/8*(99*b^4*c - 63*a*b^3*d - 15*a^3*b*f + 35*a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) + 1/8*(19*b^5*c*
x^3 - 15*a*b^4*d*x^3 - 7*a^3*b^2*f*x^3 + 11*a^2*b^3*x^3*e + 21*a*b^4*c*x - 17*a^2*b^3*d*x - 9*a^4*b*f*x + 13*a
^3*b^2*x*e)/((b*x^2 + a)^2*a^6) + 1/105*(1050*b^3*c*x^6 - 630*a*b^2*d*x^6 - 105*a^3*f*x^6 + 315*a^2*b*x^6*e -
210*a*b^2*c*x^4 + 105*a^2*b*d*x^4 - 35*a^3*x^4*e + 63*a^2*b*c*x^2 - 21*a^3*d*x^2 - 15*a^3*c)/(a^6*x^7)

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Mupad [B]
time = 1.05, size = 230, normalized size = 0.98 \begin {gather*} \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-15\,f\,a^3+35\,e\,a^2\,b-63\,d\,a\,b^2+99\,c\,b^3\right )}{8\,a^{13/2}}-\frac {\frac {c}{7\,a}-\frac {x^6\,\left (-15\,f\,a^3+35\,e\,a^2\,b-63\,d\,a\,b^2+99\,c\,b^3\right )}{15\,a^4}+\frac {x^2\,\left (7\,a\,d-11\,b\,c\right )}{35\,a^2}+\frac {x^4\,\left (35\,e\,a^2-63\,d\,a\,b+99\,c\,b^2\right )}{105\,a^3}-\frac {5\,b\,x^8\,\left (-15\,f\,a^3+35\,e\,a^2\,b-63\,d\,a\,b^2+99\,c\,b^3\right )}{24\,a^5}-\frac {b^2\,x^{10}\,\left (-15\,f\,a^3+35\,e\,a^2\,b-63\,d\,a\,b^2+99\,c\,b^3\right )}{8\,a^6}}{a^2\,x^7+2\,a\,b\,x^9+b^2\,x^{11}} \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)^3),x)

[Out]

(b^(1/2)*atan((b^(1/2)*x)/a^(1/2))*(99*b^3*c - 15*a^3*f - 63*a*b^2*d + 35*a^2*b*e))/(8*a^(13/2)) - (c/(7*a) -
(x^6*(99*b^3*c - 15*a^3*f - 63*a*b^2*d + 35*a^2*b*e))/(15*a^4) + (x^2*(7*a*d - 11*b*c))/(35*a^2) + (x^4*(99*b^
2*c + 35*a^2*e - 63*a*b*d))/(105*a^3) - (5*b*x^8*(99*b^3*c - 15*a^3*f - 63*a*b^2*d + 35*a^2*b*e))/(24*a^5) - (
b^2*x^10*(99*b^3*c - 15*a^3*f - 63*a*b^2*d + 35*a^2*b*e))/(8*a^6))/(a^2*x^7 + b^2*x^11 + 2*a*b*x^9)

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