3.2.42 \(\int \frac {c+d x^2+e x^4+f x^6}{x^{10} (a+b x^2)^3} \, dx\) [142]

Optimal. Leaf size=277 \[ -\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}-\frac {b^{3/2} \left (143 b^3 c-99 a b^2 d+63 a^2 b e-35 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{15/2}} \]

[Out]

-1/9*c/a^3/x^9+1/7*(-a*d+3*b*c)/a^4/x^7+1/5*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x^5+1/3*(-a^3*f+3*a^2*b*e-6*a*b^2*d+1
0*b^3*c)/a^6/x^3-b*(-3*a^3*f+6*a^2*b*e-10*a*b^2*d+15*b^3*c)/a^7/x-1/4*b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^6
/(b*x^2+a)^2-1/8*b^2*(-11*a^3*f+15*a^2*b*e-19*a*b^2*d+23*b^3*c)*x/a^7/(b*x^2+a)-1/8*b^(3/2)*(-35*a^3*f+63*a^2*
b*e-99*a*b^2*d+143*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(15/2)

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Rubi [A]
time = 0.41, antiderivative size = 277, 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}{7 a^4 x^7}-\frac {c}{9 a^3 x^9}-\frac {a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-35 a^3 f+63 a^2 b e-99 a b^2 d+143 b^3 c\right )}{8 a^{15/2}}-\frac {b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}-\frac {b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \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^10*(a + b*x^2)^3),x]

[Out]

-1/9*c/(a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(3*a^6*x^3) - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c -
a*b^2*d + a^2*b*e - a^3*f)*x)/(4*a^6*(a + b*x^2)^2) - (b^2*(23*b^3*c - 19*a*b^2*d + 15*a^2*b*e - 11*a^3*f)*x)/
(8*a^7*(a + b*x^2)) - (b^(3/2)*(143*b^3*c - 99*a*b^2*d + 63*a^2*b*e - 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
8*a^(15/2))

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^{10} \left (a+b x^2\right )^3} \, dx &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \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 {4 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}+\frac {3 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{10}}{a^5}}{x^{10} \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \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 {8 b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) x^8}{a^4}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x^{10}}{a^5}}{x^{10} \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}+\frac {\int \left (\frac {8 c}{a x^{10}}+\frac {8 (-3 b c+a d)}{a^2 x^8}+\frac {8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^6}+\frac {8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^4}-\frac {8 b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^5 x^2}+\frac {b^2 \left (-143 b^3 c+99 a b^2 d-63 a^2 b e+35 a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}-\frac {\left (b^2 \left (143 b^3 c-99 a b^2 d+63 a^2 b e-35 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^7}\\ &=-\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}-\frac {b^2 \left (23 b^3 c-19 a b^2 d+15 a^2 b e-11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}-\frac {b^{3/2} \left (143 b^3 c-99 a b^2 d+63 a^2 b e-35 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{15/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 276, normalized size = 1.00 \begin {gather*} -\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}+\frac {b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^7 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{4 a^6 \left (a+b x^2\right )^2}+\frac {b^2 \left (-23 b^3 c+19 a b^2 d-15 a^2 b e+11 a^3 f\right ) x}{8 a^7 \left (a+b x^2\right )}+\frac {b^{3/2} \left (-143 b^3 c+99 a b^2 d-63 a^2 b e+35 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{15/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]

[Out]

-1/9*c/(a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c
) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^6*(a + b*x^2)^2) + (b^2*(-23*b^3*c + 19*a*b^2*d - 15*a^2*b*e + 11*a^3*f
)*x)/(8*a^7*(a + b*x^2)) + (b^(3/2)*(-143*b^3*c + 99*a*b^2*d - 63*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[
a]])/(8*a^(15/2))

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Maple [A]
time = 0.17, size = 248, normalized size = 0.90

method	result	size
default	\(\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{3} b f -\frac {15}{8} a^{2} e \,b^{2}+\frac {19}{8} a d \,b^{3}-\frac {23}{8} c \,b^{4}\right ) x^{3}+\frac {a \left (13 a^{3} f -17 a^{2} b e +21 a \,b^{2} d -25 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (35 a^{3} f -63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{7}}-\frac {c}{9 a^{3} x^{9}}-\frac {a d -3 b c}{7 a^{4} x^{7}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{5 a^{5} x^{5}}-\frac {a^{3} f -3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c}{3 a^{6} x^{3}}+\frac {b \left (3 a^{3} f -6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right )}{a^{7} x}\)	\(248\)
risch	\(\frac {\frac {b^{3} \left (35 a^{3} f -63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{12}}{8 a^{7}}+\frac {5 b^{2} \left (35 a^{3} f -63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{10}}{24 a^{6}}+\frac {b \left (35 a^{3} f -63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{8}}{15 a^{5}}-\frac {\left (35 a^{3} f -63 a^{2} b e +99 a \,b^{2} d -143 b^{3} c \right ) x^{6}}{105 a^{4}}-\frac {\left (63 a^{2} e -99 a b d +143 b^{2} c \right ) x^{4}}{315 a^{3}}-\frac {\left (9 a d -13 b c \right ) x^{2}}{63 a^{2}}-\frac {c}{9 a}}{x^{9} \left (b \,x^{2}+a \right )^{2}}+\frac {35 \sqrt {-a b}\, b \ln \left (-b x -\sqrt {-a b}\right ) f}{16 a^{5}}-\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right ) e}{16 a^{6}}+\frac {99 \sqrt {-a b}\, b^{3} \ln \left (-b x -\sqrt {-a b}\right ) d}{16 a^{7}}-\frac {143 \sqrt {-a b}\, b^{4} \ln \left (-b x -\sqrt {-a b}\right ) c}{16 a^{8}}-\frac {35 \sqrt {-a b}\, b \ln \left (-b x +\sqrt {-a b}\right ) f}{16 a^{5}}+\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right ) e}{16 a^{6}}-\frac {99 \sqrt {-a b}\, b^{3} \ln \left (-b x +\sqrt {-a b}\right ) d}{16 a^{7}}+\frac {143 \sqrt {-a b}\, b^{4} \ln \left (-b x +\sqrt {-a b}\right ) c}{16 a^{8}}\)	\(432\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

b^2/a^7*(((11/8*a^3*b*f-15/8*a^2*e*b^2+19/8*a*d*b^3-23/8*c*b^4)*x^3+1/8*a*(13*a^3*f-17*a^2*b*e+21*a*b^2*d-25*b
^3*c)*x)/(b*x^2+a)^2+1/8*(35*a^3*f-63*a^2*b*e+99*a*b^2*d-143*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/9*c
/a^3/x^9-1/7*(a*d-3*b*c)/a^4/x^7-1/5*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^5-1/3*(a^3*f-3*a^2*b*e+6*a*b^2*d-10*b^3*c)/
a^6/x^3+b*(3*a^3*f-6*a^2*b*e+10*a*b^2*d-15*b^3*c)/a^7/x

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Maxima [A]
time = 0.50, size = 297, normalized size = 1.07 \begin {gather*} -\frac {315 \, {\left (143 \, b^{6} c - 99 \, a b^{5} d - 35 \, a^{3} b^{3} f + 63 \, a^{2} b^{4} e\right )} x^{12} + 525 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d - 35 \, a^{4} b^{2} f + 63 \, a^{3} b^{3} e\right )} x^{10} + 168 \, {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d - 35 \, a^{5} b f + 63 \, a^{4} b^{2} e\right )} x^{8} + 280 \, a^{6} c - 24 \, {\left (143 \, a^{3} b^{3} c - 99 \, a^{4} b^{2} d - 35 \, a^{6} f + 63 \, a^{5} b e\right )} x^{6} + 8 \, {\left (143 \, a^{4} b^{2} c - 99 \, a^{5} b d + 63 \, a^{6} e\right )} x^{4} - 40 \, {\left (13 \, a^{5} b c - 9 \, a^{6} d\right )} x^{2}}{2520 \, {\left (a^{7} b^{2} x^{13} + 2 \, a^{8} b x^{11} + a^{9} x^{9}\right )}} - \frac {{\left (143 \, b^{5} c - 99 \, a b^{4} d - 35 \, a^{3} b^{2} f + 63 \, a^{2} b^{3} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{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^10/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/2520*(315*(143*b^6*c - 99*a*b^5*d - 35*a^3*b^3*f + 63*a^2*b^4*e)*x^12 + 525*(143*a*b^5*c - 99*a^2*b^4*d - 3
5*a^4*b^2*f + 63*a^3*b^3*e)*x^10 + 168*(143*a^2*b^4*c - 99*a^3*b^3*d - 35*a^5*b*f + 63*a^4*b^2*e)*x^8 + 280*a^
6*c - 24*(143*a^3*b^3*c - 99*a^4*b^2*d - 35*a^6*f + 63*a^5*b*e)*x^6 + 8*(143*a^4*b^2*c - 99*a^5*b*d + 63*a^6*e
)*x^4 - 40*(13*a^5*b*c - 9*a^6*d)*x^2)/(a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*x^9) - 1/8*(143*b^5*c - 99*a*b^4*d -
35*a^3*b^2*f + 63*a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7)

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Fricas [A]
time = 6.08, size = 820, normalized size = 2.96 \begin {gather*} \left [-\frac {630 \, {\left (143 \, b^{6} c - 99 \, a b^{5} d - 35 \, a^{3} b^{3} f\right )} x^{12} + 1050 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d - 35 \, a^{4} b^{2} f\right )} x^{10} + 336 \, {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d - 35 \, a^{5} b f\right )} x^{8} + 560 \, a^{6} c - 48 \, {\left (143 \, a^{3} b^{3} c - 99 \, a^{4} b^{2} d - 35 \, a^{6} f\right )} x^{6} + 176 \, {\left (13 \, a^{4} b^{2} c - 9 \, a^{5} b d\right )} x^{4} - 80 \, {\left (13 \, a^{5} b c - 9 \, a^{6} d\right )} x^{2} - 315 \, {\left ({\left (143 \, b^{6} c - 99 \, a b^{5} d - 35 \, a^{3} b^{3} f\right )} x^{13} + 2 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d - 35 \, a^{4} b^{2} f\right )} x^{11} + {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d - 35 \, a^{5} b f\right )} x^{9} + 63 \, {\left (a^{2} b^{4} x^{13} + 2 \, a^{3} b^{3} x^{11} + a^{4} b^{2} x^{9}\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 ) + 126 \, {\left (315 \, a^{2} b^{4} x^{12} + 525 \, a^{3} b^{3} x^{10} + 168 \, a^{4} b^{2} x^{8} - 24 \, a^{5} b x^{6} + 8 \, a^{6} x^{4}\right )} e}{5040 \, {\left (a^{7} b^{2} x^{13} + 2 \, a^{8} b x^{11} + a^{9} x^{9}\right )}}, -\frac {315 \, {\left (143 \, b^{6} c - 99 \, a b^{5} d - 35 \, a^{3} b^{3} f\right )} x^{12} + 525 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d - 35 \, a^{4} b^{2} f\right )} x^{10} + 168 \, {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d - 35 \, a^{5} b f\right )} x^{8} + 280 \, a^{6} c - 24 \, {\left (143 \, a^{3} b^{3} c - 99 \, a^{4} b^{2} d - 35 \, a^{6} f\right )} x^{6} + 88 \, {\left (13 \, a^{4} b^{2} c - 9 \, a^{5} b d\right )} x^{4} - 40 \, {\left (13 \, a^{5} b c - 9 \, a^{6} d\right )} x^{2} + 315 \, {\left ({\left (143 \, b^{6} c - 99 \, a b^{5} d - 35 \, a^{3} b^{3} f\right )} x^{13} + 2 \, {\left (143 \, a b^{5} c - 99 \, a^{2} b^{4} d - 35 \, a^{4} b^{2} f\right )} x^{11} + {\left (143 \, a^{2} b^{4} c - 99 \, a^{3} b^{3} d - 35 \, a^{5} b f\right )} x^{9} + 63 \, {\left (a^{2} b^{4} x^{13} + 2 \, a^{3} b^{3} x^{11} + a^{4} b^{2} x^{9}\right )} e\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 63 \, {\left (315 \, a^{2} b^{4} x^{12} + 525 \, a^{3} b^{3} x^{10} + 168 \, a^{4} b^{2} x^{8} - 24 \, a^{5} b x^{6} + 8 \, a^{6} x^{4}\right )} e}{2520 \, {\left (a^{7} b^{2} x^{13} + 2 \, a^{8} b x^{11} + a^{9} x^{9}\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^10/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/5040*(630*(143*b^6*c - 99*a*b^5*d - 35*a^3*b^3*f)*x^12 + 1050*(143*a*b^5*c - 99*a^2*b^4*d - 35*a^4*b^2*f)*
x^10 + 336*(143*a^2*b^4*c - 99*a^3*b^3*d - 35*a^5*b*f)*x^8 + 560*a^6*c - 48*(143*a^3*b^3*c - 99*a^4*b^2*d - 35
*a^6*f)*x^6 + 176*(13*a^4*b^2*c - 9*a^5*b*d)*x^4 - 80*(13*a^5*b*c - 9*a^6*d)*x^2 - 315*((143*b^6*c - 99*a*b^5*
d - 35*a^3*b^3*f)*x^13 + 2*(143*a*b^5*c - 99*a^2*b^4*d - 35*a^4*b^2*f)*x^11 + (143*a^2*b^4*c - 99*a^3*b^3*d -
35*a^5*b*f)*x^9 + 63*(a^2*b^4*x^13 + 2*a^3*b^3*x^11 + a^4*b^2*x^9)*e)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a)
- a)/(b*x^2 + a)) + 126*(315*a^2*b^4*x^12 + 525*a^3*b^3*x^10 + 168*a^4*b^2*x^8 - 24*a^5*b*x^6 + 8*a^6*x^4)*e)
/(a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*x^9), -1/2520*(315*(143*b^6*c - 99*a*b^5*d - 35*a^3*b^3*f)*x^12 + 525*(143
*a*b^5*c - 99*a^2*b^4*d - 35*a^4*b^2*f)*x^10 + 168*(143*a^2*b^4*c - 99*a^3*b^3*d - 35*a^5*b*f)*x^8 + 280*a^6*c
- 24*(143*a^3*b^3*c - 99*a^4*b^2*d - 35*a^6*f)*x^6 + 88*(13*a^4*b^2*c - 9*a^5*b*d)*x^4 - 40*(13*a^5*b*c - 9*a
^6*d)*x^2 + 315*((143*b^6*c - 99*a*b^5*d - 35*a^3*b^3*f)*x^13 + 2*(143*a*b^5*c - 99*a^2*b^4*d - 35*a^4*b^2*f)*
x^11 + (143*a^2*b^4*c - 99*a^3*b^3*d - 35*a^5*b*f)*x^9 + 63*(a^2*b^4*x^13 + 2*a^3*b^3*x^11 + a^4*b^2*x^9)*e)*s
qrt(b/a)*arctan(x*sqrt(b/a)) + 63*(315*a^2*b^4*x^12 + 525*a^3*b^3*x^10 + 168*a^4*b^2*x^8 - 24*a^5*b*x^6 + 8*a^
6*x^4)*e)/(a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*x^9)]

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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**10/(b*x**2+a)**3,x)

[Out]

Timed out

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Giac [A]
time = 2.04, size = 301, normalized size = 1.09 \begin {gather*} -\frac {{\left (143 \, b^{5} c - 99 \, a b^{4} d - 35 \, a^{3} b^{2} f + 63 \, a^{2} b^{3} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{7}} - \frac {23 \, b^{6} c x^{3} - 19 \, a b^{5} d x^{3} - 11 \, a^{3} b^{3} f x^{3} + 15 \, a^{2} b^{4} x^{3} e + 25 \, a b^{5} c x - 21 \, a^{2} b^{4} d x - 13 \, a^{4} b^{2} f x + 17 \, a^{3} b^{3} x e}{8 \, {\left (b x^{2} + a\right )}^{2} a^{7}} - \frac {4725 \, b^{4} c x^{8} - 3150 \, a b^{3} d x^{8} - 945 \, a^{3} b f x^{8} + 1890 \, a^{2} b^{2} x^{8} e - 1050 \, a b^{3} c x^{6} + 630 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 315 \, a^{3} b x^{6} e + 378 \, a^{2} b^{2} c x^{4} - 189 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 135 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{7} 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)^3,x, algorithm="giac")

[Out]

-1/8*(143*b^5*c - 99*a*b^4*d - 35*a^3*b^2*f + 63*a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7) - 1/8*(23*b^
6*c*x^3 - 19*a*b^5*d*x^3 - 11*a^3*b^3*f*x^3 + 15*a^2*b^4*x^3*e + 25*a*b^5*c*x - 21*a^2*b^4*d*x - 13*a^4*b^2*f*
x + 17*a^3*b^3*x*e)/((b*x^2 + a)^2*a^7) - 1/315*(4725*b^4*c*x^8 - 3150*a*b^3*d*x^8 - 945*a^3*b*f*x^8 + 1890*a^
2*b^2*x^8*e - 1050*a*b^3*c*x^6 + 630*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 315*a^3*b*x^6*e + 378*a^2*b^2*c*x^4 - 189
*a^3*b*d*x^4 + 63*a^4*x^4*e - 135*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^7*x^9)

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Mupad [B]
time = 1.07, size = 268, normalized size = 0.97 \begin {gather*} -\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{105\,a^4}+\frac {x^2\,\left (9\,a\,d-13\,b\,c\right )}{63\,a^2}+\frac {x^4\,\left (63\,e\,a^2-99\,d\,a\,b+143\,c\,b^2\right )}{315\,a^3}+\frac {b\,x^8\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{15\,a^5}+\frac {5\,b^2\,x^{10}\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{24\,a^6}+\frac {b^3\,x^{12}\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{8\,a^7}}{a^2\,x^9+2\,a\,b\,x^{11}+b^2\,x^{13}}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-35\,f\,a^3+63\,e\,a^2\,b-99\,d\,a\,b^2+143\,c\,b^3\right )}{8\,a^{15/2}} \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)^3),x)

[Out]

- (c/(9*a) - (x^6*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(105*a^4) + (x^2*(9*a*d - 13*b*c))/(63*a^2
) + (x^4*(143*b^2*c + 63*a^2*e - 99*a*b*d))/(315*a^3) + (b*x^8*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e
))/(15*a^5) + (5*b^2*x^10*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(24*a^6) + (b^3*x^12*(143*b^3*c -
35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(8*a^7))/(a^2*x^9 + b^2*x^13 + 2*a*b*x^11) - (b^(3/2)*atan((b^(1/2)*x)/a^
(1/2))*(143*b^3*c - 35*a^3*f - 99*a*b^2*d + 63*a^2*b*e))/(8*a^(15/2))

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