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

Optimal. Leaf size=210 \[ \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac {a x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac {x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}-\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{13/2}}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b} \]

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Rubi [A]  time = 0.16, antiderivative size = 210, 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 {x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}-\frac {a x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac {a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac {a^{5/2} \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 )}{b^{13/2}}+\frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.16, size = 210, normalized size = 1.00 \begin {gather*} \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}-\frac {a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^6}+\frac {a x^3 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^5}+\frac {x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}+\frac {a^{5/2} \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 )}{b^{13/2}}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, size = 452, normalized size = 2.15 \begin {gather*} \left [\frac {630 \, b^{5} f x^{11} + 770 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 990 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 1386 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 2310 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6930 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{6930 \, b^{6}}, \frac {315 \, b^{5} f x^{11} + 385 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6930*(630*b^5*f*x^11 + 770*(b^5*e - a*b^4*f)*x^9 + 990*(b^5*d - a*b^4*e + a^2*b^3*f)*x^7 + 1386*(b^5*c - a*
b^4*d + a^2*b^3*e - a^3*b^2*f)*x^5 - 2310*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 - 3465*(a^2*b^3*c -
a^3*b^2*d + a^4*b*e - a^5*f)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 6930*(a^2*b^3*c - a^
3*b^2*d + a^4*b*e - a^5*f)*x)/b^6, 1/3465*(315*b^5*f*x^11 + 385*(b^5*e - a*b^4*f)*x^9 + 495*(b^5*d - a*b^4*e +
a^2*b^3*f)*x^7 + 693*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^5 - 1155*(a*b^4*c - a^2*b^3*d + a^3*b^2*e -
a^4*b*f)*x^3 - 3465*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 3465*(a^2*b^
3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x)/b^6]

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giac [A]  time = 0.46, size = 250, normalized size = 1.19 \begin {gather*} -\frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d - a^{6} f + a^{5} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, b^{10} f x^{11} - 385 \, a b^{9} f x^{9} + 385 \, b^{10} x^{9} e + 495 \, b^{10} d x^{7} + 495 \, a^{2} b^{8} f x^{7} - 495 \, a b^{9} x^{7} e + 693 \, b^{10} c x^{5} - 693 \, a b^{9} d x^{5} - 693 \, a^{3} b^{7} f x^{5} + 693 \, a^{2} b^{8} x^{5} e - 1155 \, a b^{9} c x^{3} + 1155 \, a^{2} b^{8} d x^{3} + 1155 \, a^{4} b^{6} f x^{3} - 1155 \, a^{3} b^{7} x^{3} e + 3465 \, a^{2} b^{8} c x - 3465 \, a^{3} b^{7} d x - 3465 \, a^{5} b^{5} f x + 3465 \, a^{4} b^{6} x e}{3465 \, b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*b^3*c - a^4*b^2*d - a^6*f + a^5*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/3465*(315*b^10*f*x^11 - 3
85*a*b^9*f*x^9 + 385*b^10*x^9*e + 495*b^10*d*x^7 + 495*a^2*b^8*f*x^7 - 495*a*b^9*x^7*e + 693*b^10*c*x^5 - 693*
a*b^9*d*x^5 - 693*a^3*b^7*f*x^5 + 693*a^2*b^8*x^5*e - 1155*a*b^9*c*x^3 + 1155*a^2*b^8*d*x^3 + 1155*a^4*b^6*f*x
^3 - 1155*a^3*b^7*x^3*e + 3465*a^2*b^8*c*x - 3465*a^3*b^7*d*x - 3465*a^5*b^5*f*x + 3465*a^4*b^6*x*e)/b^11

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maple [A]  time = 0.00, size = 278, normalized size = 1.32 \begin {gather*} \frac {f \,x^{11}}{11 b}-\frac {a f \,x^{9}}{9 b^{2}}+\frac {e \,x^{9}}{9 b}+\frac {a^{2} f \,x^{7}}{7 b^{3}}-\frac {a e \,x^{7}}{7 b^{2}}+\frac {d \,x^{7}}{7 b}-\frac {a^{3} f \,x^{5}}{5 b^{4}}+\frac {a^{2} e \,x^{5}}{5 b^{3}}-\frac {a d \,x^{5}}{5 b^{2}}+\frac {c \,x^{5}}{5 b}+\frac {a^{4} f \,x^{3}}{3 b^{5}}-\frac {a^{3} e \,x^{3}}{3 b^{4}}+\frac {a^{2} d \,x^{3}}{3 b^{3}}-\frac {a c \,x^{3}}{3 b^{2}}+\frac {a^{6} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{6}}-\frac {a^{5} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {a^{4} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {a^{3} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}-\frac {a^{5} f x}{b^{6}}+\frac {a^{4} e x}{b^{5}}-\frac {a^{3} d x}{b^{4}}+\frac {a^{2} c x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.90, size = 213, normalized size = 1.01 \begin {gather*} -\frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, b^{5} f x^{11} + 385 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} + 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/3465*(315*b^5*f*x^11 + 38
5*(b^5*e - a*b^4*f)*x^9 + 495*(b^5*d - a*b^4*e + a^2*b^3*f)*x^7 + 693*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f
)*x^5 - 1155*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 + 3465*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*
x)/b^6

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mupad [B]  time = 0.93, size = 289, normalized size = 1.38 \begin {gather*} x^9\,\left (\frac {e}{9\,b}-\frac {a\,f}{9\,b^2}\right )+x^7\,\left (\frac {d}{7\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{7\,b}\right )+x^5\,\left (\frac {c}{5\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{5\,b}\right )+\frac {f\,x^{11}}{11\,b}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^6-e\,a^5\,b+d\,a^4\,b^2-c\,a^3\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{13/2}}-\frac {a\,x^3\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{3\,b}+\frac {a^2\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.65, size = 384, normalized size = 1.83 \begin {gather*} x^{9} \left (- \frac {a f}{9 b^{2}} + \frac {e}{9 b}\right ) + x^{7} \left (\frac {a^{2} f}{7 b^{3}} - \frac {a e}{7 b^{2}} + \frac {d}{7 b}\right ) + x^{5} \left (- \frac {a^{3} f}{5 b^{4}} + \frac {a^{2} e}{5 b^{3}} - \frac {a d}{5 b^{2}} + \frac {c}{5 b}\right ) + x^{3} \left (\frac {a^{4} f}{3 b^{5}} - \frac {a^{3} e}{3 b^{4}} + \frac {a^{2} d}{3 b^{3}} - \frac {a c}{3 b^{2}}\right ) + x \left (- \frac {a^{5} f}{b^{6}} + \frac {a^{4} e}{b^{5}} - \frac {a^{3} d}{b^{4}} + \frac {a^{2} c}{b^{3}}\right ) - \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac {f x^{11}}{11 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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