3.2.23 \(\int \frac {c+d x^2+e x^4+f x^6}{x^{12} (a+b x^2)} \, dx\) [123]

Optimal. Leaf size=211 \[ -\frac {c}{11 a x^{11}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {b^{5/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{13/2}} \]

[Out]

-1/11*c/a/x^11+1/9*(-a*d+b*c)/a^2/x^9+1/7*(-a^2*e+a*b*d-b^2*c)/a^3/x^7+1/5*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^4/
x^5-1/3*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^5/x^3+b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^6/x+b^(5/2)*(-a^3*f+a^2*
b*e-a*b^2*d+b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(13/2)

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Rubi [A]
time = 0.12, antiderivative size = 211, 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}{9 a^2 x^9}-\frac {a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac {b^{5/2} \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^{13/2}}+\frac {b^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^6 x}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac {c}{11 a x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^12*(a + b*x^2)),x]

[Out]

-1/11*c/(a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e
- a^3*f)/(5*a^4*x^5) - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e -
a^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/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]

Rubi steps

\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx &=\int \left (\frac {c}{a x^{12}}+\frac {-b c+a d}{a^2 x^{10}}+\frac {b^2 c-a b d+a^2 e}{a^3 x^8}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^6}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^4}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 x^2}-\frac {b^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {\left (b^3 \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^6}\\ &=-\frac {c}{11 a x^{11}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {b^{5/2} \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^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 211, normalized size = 1.00 \begin {gather*} -\frac {c}{11 a x^{11}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}+\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{3 a^5 x^3}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac {b^{5/2} \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^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^12*(a + b*x^2)),x]

[Out]

-1/11*c/(a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e
- a^3*f)/(5*a^4*x^5) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*
e - a^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(13/2)

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Maple [A]
time = 0.13, size = 201, normalized size = 0.95

method	result	size
default	\(-\frac {b^{3} \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{6} \sqrt {a b}}-\frac {c}{11 a \,x^{11}}-\frac {a d -b c}{9 a^{2} x^{9}}-\frac {a^{2} e -a b d +b^{2} c}{7 a^{3} x^{7}}-\frac {a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c}{5 a^{4} x^{5}}-\frac {b^{2} \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{6} x}+\frac {b \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right )}{3 a^{5} x^{3}}\)	\(201\)
risch	\(\frac {-\frac {b^{2} \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{10}}{a^{6}}+\frac {b \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{8}}{3 a^{5}}-\frac {\left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) x^{6}}{5 a^{4}}-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{7 a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{9 a^{2}}-\frac {c}{11 a}}{x^{11}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{13} \textit {\_Z}^{2}+a^{6} b^{5} f^{2}-2 a^{5} b^{6} e f +2 a^{4} b^{7} d f +a^{4} b^{7} e^{2}-2 a^{3} b^{8} c f -2 a^{3} b^{8} d e +2 a^{2} b^{9} c e +a^{2} b^{9} d^{2}-2 a \,b^{10} c d +b^{11} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{13}+2 a^{6} b^{5} f^{2}-4 a^{5} b^{6} e f +4 a^{4} b^{7} d f +2 a^{4} b^{7} e^{2}-4 a^{3} b^{8} c f -4 a^{3} b^{8} d e +4 a^{2} b^{9} c e +2 a^{2} b^{9} d^{2}-4 a \,b^{10} c d +2 b^{11} c^{2}\right ) x +\left (a^{10} b^{2} f -a^{9} b^{3} e +a^{8} b^{4} d -a^{7} b^{5} c \right ) \textit {\_R} \right )\right )}{2}\)	\(416\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^12/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 219, normalized size = 1.04 \begin {gather*} \frac {{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{6}} + \frac {3465 \, {\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} x^{10} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d - a^{4} b f + a^{3} b^{2} e\right )} x^{8} + 693 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )} x^{6} - 315 \, a^{5} c - 495 \, {\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} + 385 \, {\left (a^{4} b c - a^{5} d\right )} x^{2}}{3465 \, a^{6} x^{11}} \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^12/(b*x^2+a),x, algorithm="maxima")

[Out]

(b^6*c - a*b^5*d - a^3*b^3*f + a^2*b^4*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) + 1/3465*(3465*(b^5*c - a*b^4*
d - a^3*b^2*f + a^2*b^3*e)*x^10 - 1155*(a*b^4*c - a^2*b^3*d - a^4*b*f + a^3*b^2*e)*x^8 + 693*(a^2*b^3*c - a^3*
b^2*d - a^5*f + a^4*b*e)*x^6 - 315*a^5*c - 495*(a^3*b^2*c - a^4*b*d + a^5*e)*x^4 + 385*(a^4*b*c - a^5*d)*x^2)/
(a^6*x^11)

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Fricas [A]
time = 4.18, size = 504, normalized size = 2.39 \begin {gather*} \left [\frac {6930 \, {\left (b^{5} c - a b^{4} d - a^{3} b^{2} f\right )} x^{10} - 2310 \, {\left (a b^{4} c - a^{2} b^{3} d - a^{4} b f\right )} x^{8} + 1386 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f\right )} x^{6} - 630 \, a^{5} c - 990 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x^{4} + 770 \, {\left (a^{4} b c - a^{5} d\right )} x^{2} + 3465 \, {\left (a^{2} b^{3} x^{11} e + {\left (b^{5} c - a b^{4} d - a^{3} b^{2} f\right )} x^{11}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 66 \, {\left (105 \, a^{2} b^{3} x^{10} - 35 \, a^{3} b^{2} x^{8} + 21 \, a^{4} b x^{6} - 15 \, a^{5} x^{4}\right )} e}{6930 \, a^{6} x^{11}}, \frac {3465 \, {\left (b^{5} c - a b^{4} d - a^{3} b^{2} f\right )} x^{10} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d - a^{4} b f\right )} x^{8} + 693 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f\right )} x^{6} - 315 \, a^{5} c - 495 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x^{4} + 385 \, {\left (a^{4} b c - a^{5} d\right )} x^{2} + 3465 \, {\left (a^{2} b^{3} x^{11} e + {\left (b^{5} c - a b^{4} d - a^{3} b^{2} f\right )} x^{11}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 33 \, {\left (105 \, a^{2} b^{3} x^{10} - 35 \, a^{3} b^{2} x^{8} + 21 \, a^{4} b x^{6} - 15 \, a^{5} x^{4}\right )} e}{3465 \, a^{6} x^{11}}\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^12/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/6930*(6930*(b^5*c - a*b^4*d - a^3*b^2*f)*x^10 - 2310*(a*b^4*c - a^2*b^3*d - a^4*b*f)*x^8 + 1386*(a^2*b^3*c
- a^3*b^2*d - a^5*f)*x^6 - 630*a^5*c - 990*(a^3*b^2*c - a^4*b*d)*x^4 + 770*(a^4*b*c - a^5*d)*x^2 + 3465*(a^2*b
^3*x^11*e + (b^5*c - a*b^4*d - a^3*b^2*f)*x^11)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 6
6*(105*a^2*b^3*x^10 - 35*a^3*b^2*x^8 + 21*a^4*b*x^6 - 15*a^5*x^4)*e)/(a^6*x^11), 1/3465*(3465*(b^5*c - a*b^4*d
- a^3*b^2*f)*x^10 - 1155*(a*b^4*c - a^2*b^3*d - a^4*b*f)*x^8 + 693*(a^2*b^3*c - a^3*b^2*d - a^5*f)*x^6 - 315*
a^5*c - 495*(a^3*b^2*c - a^4*b*d)*x^4 + 385*(a^4*b*c - a^5*d)*x^2 + 3465*(a^2*b^3*x^11*e + (b^5*c - a*b^4*d -
a^3*b^2*f)*x^11)*sqrt(b/a)*arctan(x*sqrt(b/a)) + 33*(105*a^2*b^3*x^10 - 35*a^3*b^2*x^8 + 21*a^4*b*x^6 - 15*a^5
*x^4)*e)/(a^6*x^11)]

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Sympy [A]
time = 40.34, size = 398, normalized size = 1.89 \begin {gather*} \frac {\sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} + \frac {- 315 a^{5} c + x^{10} \left (- 3465 a^{3} b^{2} f + 3465 a^{2} b^{3} e - 3465 a b^{4} d + 3465 b^{5} c\right ) + x^{8} \cdot \left (1155 a^{4} b f - 1155 a^{3} b^{2} e + 1155 a^{2} b^{3} d - 1155 a b^{4} c\right ) + x^{6} \left (- 693 a^{5} f + 693 a^{4} b e - 693 a^{3} b^{2} d + 693 a^{2} b^{3} c\right ) + x^{4} \left (- 495 a^{5} e + 495 a^{4} b d - 495 a^{3} b^{2} c\right ) + x^{2} \left (- 385 a^{5} d + 385 a^{4} b c\right )}{3465 a^{6} x^{11}} \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**12/(b*x**2+a),x)

[Out]

sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**7*sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b
**2*d - b**3*c)/(a**3*b**3*f - a**2*b**4*e + a*b**5*d - b**6*c) + x)/2 - sqrt(-b**5/a**13)*(a**3*f - a**2*b*e
+ a*b**2*d - b**3*c)*log(a**7*sqrt(-b**5/a**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**3*b**3*f - a**2*b*
*4*e + a*b**5*d - b**6*c) + x)/2 + (-315*a**5*c + x**10*(-3465*a**3*b**2*f + 3465*a**2*b**3*e - 3465*a*b**4*d
+ 3465*b**5*c) + x**8*(1155*a**4*b*f - 1155*a**3*b**2*e + 1155*a**2*b**3*d - 1155*a*b**4*c) + x**6*(-693*a**5*
f + 693*a**4*b*e - 693*a**3*b**2*d + 693*a**2*b**3*c) + x**4*(-495*a**5*e + 495*a**4*b*d - 495*a**3*b**2*c) +
x**2*(-385*a**5*d + 385*a**4*b*c))/(3465*a**6*x**11)

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Giac [A]
time = 3.08, size = 249, normalized size = 1.18 \begin {gather*} \frac {{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{6}} + \frac {3465 \, b^{5} c x^{10} - 3465 \, a b^{4} d x^{10} - 3465 \, a^{3} b^{2} f x^{10} + 3465 \, a^{2} b^{3} x^{10} e - 1155 \, a b^{4} c x^{8} + 1155 \, a^{2} b^{3} d x^{8} + 1155 \, a^{4} b f x^{8} - 1155 \, a^{3} b^{2} x^{8} e + 693 \, a^{2} b^{3} c x^{6} - 693 \, a^{3} b^{2} d x^{6} - 693 \, a^{5} f x^{6} + 693 \, a^{4} b x^{6} e - 495 \, a^{3} b^{2} c x^{4} + 495 \, a^{4} b d x^{4} - 495 \, a^{5} x^{4} e + 385 \, a^{4} b c x^{2} - 385 \, a^{5} d x^{2} - 315 \, a^{5} c}{3465 \, a^{6} x^{11}} \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^12/(b*x^2+a),x, algorithm="giac")

[Out]

(b^6*c - a*b^5*d - a^3*b^3*f + a^2*b^4*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) + 1/3465*(3465*b^5*c*x^10 - 34
65*a*b^4*d*x^10 - 3465*a^3*b^2*f*x^10 + 3465*a^2*b^3*x^10*e - 1155*a*b^4*c*x^8 + 1155*a^2*b^3*d*x^8 + 1155*a^4
*b*f*x^8 - 1155*a^3*b^2*x^8*e + 693*a^2*b^3*c*x^6 - 693*a^3*b^2*d*x^6 - 693*a^5*f*x^6 + 693*a^4*b*x^6*e - 495*
a^3*b^2*c*x^4 + 495*a^4*b*d*x^4 - 495*a^5*x^4*e + 385*a^4*b*c*x^2 - 385*a^5*d*x^2 - 315*a^5*c)/(a^6*x^11)

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Mupad [B]
time = 0.99, size = 197, normalized size = 0.93 \begin {gather*} \frac {b^{5/2}\,\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^{13/2}}-\frac {\frac {c}{11\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{5\,a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{9\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{7\,a^3}+\frac {b\,x^8\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^5}-\frac {b^2\,x^{10}\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^6}}{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^12*(a + b*x^2)),x)

[Out]

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

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