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

Optimal. Leaf size=202 \[ \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{2 b^{11/2}}+\frac {x \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{2 b^5}-\frac {x^3 \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{6 a b^4}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^7}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1804, 1585, 1261, 205} \[ \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {x^3 \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{6 a b^4}+\frac {x \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^5}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^{11/2}}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1804

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

Rubi steps

\begin {align*} \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^3 \left (\left (3 b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 a^3 f}{b^2}\right ) x-2 a \left (e-\frac {a f}{b}\right ) x^3-2 a f x^5\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^4 \left (3 b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 a^3 f}{b^2}-2 a \left (e-\frac {a f}{b}\right ) x^2-2 a f x^4\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\int \left (-\frac {a \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right )}{b^4}+\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^2}{b^3}-\frac {2 a (b e-2 a f) x^4}{b^2}-\frac {2 a f x^6}{b}+\frac {3 a^2 b^3 c-5 a^3 b^2 d+7 a^4 b e-9 a^5 f}{b^4 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^7}{7 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\left (a \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^5}\\ &=\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^7}{7 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\sqrt {a} \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 187, normalized size = 0.93 \[ \frac {x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (9 a^3 f-7 a^2 b e+5 a b^2 d-3 b^3 c\right )}{2 b^{11/2}}+\frac {x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{b^5}+\frac {x \left (a^4 (-f)+a^3 b e-a^2 b^2 d+a b^3 c\right )}{2 b^5 \left (a+b x^2\right )}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.71, size = 478, normalized size = 2.37 \[ \left [\frac {60 \, b^{4} f x^{9} + 12 \, {\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 28 \, {\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 140 \, {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f + {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{420 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {30 \, b^{4} f x^{9} + 6 \, {\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 14 \, {\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 70 \, {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f + {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{210 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/420*(60*b^4*f*x^9 + 12*(7*b^4*e - 9*a*b^3*f)*x^7 + 28*(5*b^4*d - 7*a*b^3*e + 9*a^2*b^2*f)*x^5 + 140*(3*b^4*
c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^3 - 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f + (3*b^4*c -
5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(3
*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f)*x)/(b^6*x^2 + a*b^5), 1/210*(30*b^4*f*x^9 + 6*(7*b^4*e - 9*a*b^3
*f)*x^7 + 14*(5*b^4*d - 7*a*b^3*e + 9*a^2*b^2*f)*x^5 + 70*(3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^3
- 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f + (3*b^4*c - 5*a*b^3*d + 7*a^2*b^2*e - 9*a^3*b*f)*x^2)*sq
rt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9*a^4*f)*x)/(b^6*x^2 + a*b^5)]

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giac [A]  time = 0.42, size = 201, normalized size = 1.00 \[ -\frac {{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d - 9 \, a^{4} f + 7 \, a^{3} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {a b^{3} c x - a^{2} b^{2} d x - a^{4} f x + a^{3} b x e}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {15 \, b^{12} f x^{7} - 42 \, a b^{11} f x^{5} + 21 \, b^{12} x^{5} e + 35 \, b^{12} d x^{3} + 105 \, a^{2} b^{10} f x^{3} - 70 \, a b^{11} x^{3} e + 105 \, b^{12} c x - 210 \, a b^{11} d x - 420 \, a^{3} b^{9} f x + 315 \, a^{2} b^{10} x e}{105 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*a*b^3*c - 5*a^2*b^2*d - 9*a^4*f + 7*a^3*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/2*(a*b^3*c*x -
a^2*b^2*d*x - a^4*f*x + a^3*b*x*e)/((b*x^2 + a)*b^5) + 1/105*(15*b^12*f*x^7 - 42*a*b^11*f*x^5 + 21*b^12*x^5*e
+ 35*b^12*d*x^3 + 105*a^2*b^10*f*x^3 - 70*a*b^11*x^3*e + 105*b^12*c*x - 210*a*b^11*d*x - 420*a^3*b^9*f*x + 315
*a^2*b^10*x*e)/b^14

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maple [A]  time = 0.01, size = 258, normalized size = 1.28 \[ \frac {f \,x^{7}}{7 b^{2}}-\frac {2 a f \,x^{5}}{5 b^{3}}+\frac {e \,x^{5}}{5 b^{2}}+\frac {a^{2} f \,x^{3}}{b^{4}}-\frac {2 a e \,x^{3}}{3 b^{3}}+\frac {d \,x^{3}}{3 b^{2}}-\frac {a^{4} f x}{2 \left (b \,x^{2}+a \right ) b^{5}}+\frac {9 a^{4} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{5}}+\frac {a^{3} e x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {a^{2} d x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {a c x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {4 a^{3} f x}{b^{5}}+\frac {3 a^{2} e x}{b^{4}}-\frac {2 a d x}{b^{3}}+\frac {c x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, size = 183, normalized size = 0.91 \[ \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x}{105 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x/(b^6*x^2 + a*b^5) - 1/2*(3*a*b^3*c - 5*a^2*b^2*d + 7*a^3*b*e - 9
*a^4*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/105*(15*b^3*f*x^7 + 21*(b^3*e - 2*a*b^2*f)*x^5 + 35*(b^3*d -
2*a*b^2*e + 3*a^2*b*f)*x^3 + 105*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5

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mupad [B]  time = 0.97, size = 288, normalized size = 1.43 \[ x^5\,\left (\frac {e}{5\,b^2}-\frac {2\,a\,f}{5\,b^3}\right )+x\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a^2\,f}{3\,b^4}-\frac {d}{3\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{3\,b}\right )-\frac {x\,\left (\frac {f\,a^4}{2}-\frac {e\,a^3\,b}{2}+\frac {d\,a^2\,b^2}{2}-\frac {c\,a\,b^3}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {f\,x^7}{7\,b^2}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-9\,f\,a^3+7\,e\,a^2\,b-5\,d\,a\,b^2+3\,c\,b^3\right )}{9\,f\,a^4-7\,e\,a^3\,b+5\,d\,a^2\,b^2-3\,c\,a\,b^3}\right )\,\left (-9\,f\,a^3+7\,e\,a^2\,b-5\,d\,a\,b^2+3\,c\,b^3\right )}{2\,b^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.76, size = 257, normalized size = 1.27 \[ x^{5} \left (- \frac {2 a f}{5 b^{3}} + \frac {e}{5 b^{2}}\right ) + x^{3} \left (\frac {a^{2} f}{b^{4}} - \frac {2 a e}{3 b^{3}} + \frac {d}{3 b^{2}}\right ) + x \left (- \frac {4 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{b^{4}} - \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {\sqrt {- \frac {a}{b^{11}}} \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log {\left (- b^{5} \sqrt {- \frac {a}{b^{11}}} + x \right )}}{4} + \frac {\sqrt {- \frac {a}{b^{11}}} \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log {\left (b^{5} \sqrt {- \frac {a}{b^{11}}} + x \right )}}{4} + \frac {f x^{7}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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