3.2.30 \(\int \frac {x^6 (c+d x^2+e x^4+f x^6)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=247 \[ \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{8 b^{13/2}}+\frac {a x \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac {x \left (-21 a^3 f+13 a^2 b e-7 a b^2 d+3 b^3 c\right )}{2 b^6}-\frac {x^3 \left (-27 a^3 f+15 a^2 b e-7 a b^2 d+3 b^3 c\right )}{12 a b^5}+\frac {x^5 (b e-3 a f)}{5 b^4}+\frac {f x^7}{7 b^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.41, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1804, 1585, 1257, 1810, 205} \begin {gather*} \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {x^3 \left (15 a^2 b e-27 a^3 f-7 a b^2 d+3 b^3 c\right )}{12 a b^5}+\frac {a x \left (11 a^2 b e-15 a^3 f-7 a b^2 d+3 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac {x \left (13 a^2 b e-21 a^3 f-7 a b^2 d+3 b^3 c\right )}{2 b^6}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (63 a^2 b e-99 a^3 f-35 a b^2 d+15 b^3 c\right )}{8 b^{13/2}}+\frac {x^5 (b e-3 a f)}{5 b^4}+\frac {f x^7}{7 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*b^3*c - 7*a*b^2*d + 13*a^2*b*e - 21*a^3*f)*x)/(2*b^6) - ((3*b^3*c - 7*a*b^2*d + 15*a^2*b*e - 27*a^3*f)*x^3
)/(12*a*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^7)/(7*b^3) + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^7)/(4*
a*(a + b*x^2)^2) + (a*(3*b^3*c - 7*a*b^2*d + 11*a^2*b*e - 15*a^3*f)*x)/(8*b^6*(a + b*x^2)) - (Sqrt[a]*(15*b^3*
c - 35*a*b^2*d + 63*a^2*b*e - 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*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 1257

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^
(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[1/(2*e^(2*p +
m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4
)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]

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]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, 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 )}{\left (a+b x^2\right )^3} \, dx &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^5 \left (\left (3 b c-7 a d+\frac {7 a^2 e}{b}-\frac {7 a^3 f}{b^2}\right ) x-4 a \left (e-\frac {a f}{b}\right ) x^3-4 a f x^5\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^6 \left (3 b c-7 a d+\frac {7 a^2 e}{b}-\frac {7 a^3 f}{b^2}-4 a \left (e-\frac {a f}{b}\right ) x^2-4 a f x^4\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac {a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}+\frac {\int \frac {-a^2 \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right )+2 a b \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x^2-2 b^2 \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x^4+8 a b^3 (b e-2 a f) x^6+8 a b^4 f x^8}{a+b x^2} \, dx}{8 a b^6}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac {a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}+\frac {\int \left (4 a \left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right )-2 b \left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^2+8 a b^2 (b e-3 a f) x^4+8 a b^3 f x^6+\frac {-15 a^2 b^3 c+35 a^3 b^2 d-63 a^4 b e+99 a^5 f}{a+b x^2}\right ) \, dx}{8 a b^6}\\ &=\frac {\left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right ) x}{2 b^6}-\frac {\left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^3}{12 a b^5}+\frac {(b e-3 a f) x^5}{5 b^4}+\frac {f x^7}{7 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac {a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}-\frac {\left (a \left (15 b^3 c-35 a b^2 d+63 a^2 b e-99 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^6}\\ &=\frac {\left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right ) x}{2 b^6}-\frac {\left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^3}{12 a b^5}+\frac {(b e-3 a f) x^5}{5 b^4}+\frac {f x^7}{7 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac {a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}-\frac {\sqrt {a} \left (15 b^3 c-35 a b^2 d+63 a^2 b e-99 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 232, normalized size = 0.94 \begin {gather*} \frac {x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (99 a^3 f-63 a^2 b e+35 a b^2 d-15 b^3 c\right )}{8 b^{13/2}}+\frac {a x \left (-21 a^3 f+17 a^2 b e-13 a b^2 d+9 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac {a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 b^6 \left (a+b x^2\right )^2}+\frac {x \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{b^6}+\frac {x^5 (b e-3 a f)}{5 b^4}+\frac {f x^7}{7 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^3)/(3*b^5) + ((b*e - 3*a*f
)*x^5)/(5*b^4) + (f*x^7)/(7*b^3) + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*b^6*(a + b*x^2)^2) + (a*(
9*b^3*c - 13*a*b^2*d + 17*a^2*b*e - 21*a^3*f)*x)/(8*b^6*(a + b*x^2)) + (Sqrt[a]*(-15*b^3*c + 35*a*b^2*d - 63*a
^2*b*e + 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

________________________________________________________________________________________

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 )}{\left (a+b x^2\right )^3} \, 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)^3,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.14, size = 668, normalized size = 2.70 \begin {gather*} \left [\frac {240 \, b^{5} f x^{11} + 48 \, {\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 16 \, {\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 112 \, {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 350 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f + {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} 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 (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{1680 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {120 \, b^{5} f x^{11} + 24 \, {\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 8 \, {\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 56 \, {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 175 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f + {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{840 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\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)^3,x, algorithm="fricas")

[Out]

[1/1680*(240*b^5*f*x^11 + 48*(7*b^5*e - 11*a*b^4*f)*x^9 + 16*(35*b^5*d - 63*a*b^4*e + 99*a^2*b^3*f)*x^7 + 112*
(15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^5 + 350*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99
*a^4*b*f)*x^3 - 105*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f + (15*b^5*c - 35*a*b^4*d + 63*a^2*b^3
*e - 99*a^3*b^2*f)*x^4 + 2*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a^4*b*f)*x^2)*sqrt(-a/b)*log((b*x^2
+ 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f)*x)/(b^8*x^4 +
2*a*b^7*x^2 + a^2*b^6), 1/840*(120*b^5*f*x^11 + 24*(7*b^5*e - 11*a*b^4*f)*x^9 + 8*(35*b^5*d - 63*a*b^4*e + 99
*a^2*b^3*f)*x^7 + 56*(15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^5 + 175*(15*a*b^4*c - 35*a^2*b^3*
d + 63*a^3*b^2*e - 99*a^4*b*f)*x^3 - 105*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f + (15*b^5*c - 35
*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^4 + 2*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a^4*b*f)*x^2)*s
qrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f)*x)/(b^8*x^4 + 2*a
*b^7*x^2 + a^2*b^6)]

________________________________________________________________________________________

giac [A]  time = 0.47, size = 250, normalized size = 1.01 \begin {gather*} -\frac {{\left (15 \, a b^{3} c - 35 \, a^{2} b^{2} d - 99 \, a^{4} f + 63 \, a^{3} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {9 \, a b^{4} c x^{3} - 13 \, a^{2} b^{3} d x^{3} - 21 \, a^{4} b f x^{3} + 17 \, a^{3} b^{2} x^{3} e + 7 \, a^{2} b^{3} c x - 11 \, a^{3} b^{2} d x - 19 \, a^{5} f x + 15 \, a^{4} b x e}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {15 \, b^{18} f x^{7} - 63 \, a b^{17} f x^{5} + 21 \, b^{18} x^{5} e + 35 \, b^{18} d x^{3} + 210 \, a^{2} b^{16} f x^{3} - 105 \, a b^{17} x^{3} e + 105 \, b^{18} c x - 315 \, a b^{17} d x - 1050 \, a^{3} b^{15} f x + 630 \, a^{2} b^{16} x e}{105 \, b^{21}} \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)^3,x, algorithm="giac")

[Out]

-1/8*(15*a*b^3*c - 35*a^2*b^2*d - 99*a^4*f + 63*a^3*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/8*(9*a*b^4*
c*x^3 - 13*a^2*b^3*d*x^3 - 21*a^4*b*f*x^3 + 17*a^3*b^2*x^3*e + 7*a^2*b^3*c*x - 11*a^3*b^2*d*x - 19*a^5*f*x + 1
5*a^4*b*x*e)/((b*x^2 + a)^2*b^6) + 1/105*(15*b^18*f*x^7 - 63*a*b^17*f*x^5 + 21*b^18*x^5*e + 35*b^18*d*x^3 + 21
0*a^2*b^16*f*x^3 - 105*a*b^17*x^3*e + 105*b^18*c*x - 315*a*b^17*d*x - 1050*a^3*b^15*f*x + 630*a^2*b^16*x*e)/b^
21

________________________________________________________________________________________

maple [A]  time = 0.02, size = 343, normalized size = 1.39 \begin {gather*} \frac {f \,x^{7}}{7 b^{3}}-\frac {21 a^{4} f \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}+\frac {17 a^{3} e \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}-\frac {13 a^{2} d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {9 a c \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}-\frac {3 a f \,x^{5}}{5 b^{4}}+\frac {e \,x^{5}}{5 b^{3}}-\frac {19 a^{5} f x}{8 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {15 a^{4} e x}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {11 a^{3} d x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {7 a^{2} c x}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {2 a^{2} f \,x^{3}}{b^{5}}-\frac {a e \,x^{3}}{b^{4}}+\frac {d \,x^{3}}{3 b^{3}}+\frac {99 a^{4} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{6}}-\frac {63 a^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{5}}+\frac {35 a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}-\frac {15 a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{3}}-\frac {10 a^{3} f x}{b^{6}}+\frac {6 a^{2} e x}{b^{5}}-\frac {3 a d x}{b^{4}}+\frac {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)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.99, size = 237, normalized size = 0.96 \begin {gather*} \frac {{\left (9 \, a b^{4} c - 13 \, a^{2} b^{3} d + 17 \, a^{3} b^{2} e - 21 \, a^{4} b f\right )} x^{3} + {\left (7 \, a^{2} b^{3} c - 11 \, a^{3} b^{2} d + 15 \, a^{4} b e - 19 \, a^{5} f\right )} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} - \frac {{\left (15 \, a b^{3} c - 35 \, a^{2} b^{2} d + 63 \, a^{3} b e - 99 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - 3 \, a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x}{105 \, 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)^3,x, algorithm="maxima")

[Out]

1/8*((9*a*b^4*c - 13*a^2*b^3*d + 17*a^3*b^2*e - 21*a^4*b*f)*x^3 + (7*a^2*b^3*c - 11*a^3*b^2*d + 15*a^4*b*e - 1
9*a^5*f)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) - 1/8*(15*a*b^3*c - 35*a^2*b^2*d + 63*a^3*b*e - 99*a^4*f)*arctan
(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/105*(15*b^3*f*x^7 + 21*(b^3*e - 3*a*b^2*f)*x^5 + 35*(b^3*d - 3*a*b^2*e + 6
*a^2*b*f)*x^3 + 105*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 348, normalized size = 1.41 \begin {gather*} x^5\,\left (\frac {e}{5\,b^3}-\frac {3\,a\,f}{5\,b^4}\right )+x\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a^2\,f}{b^5}-\frac {d}{3\,b^3}+\frac {a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )-\frac {\left (\frac {21\,f\,a^4\,b}{8}-\frac {17\,e\,a^3\,b^2}{8}+\frac {13\,d\,a^2\,b^3}{8}-\frac {9\,c\,a\,b^4}{8}\right )\,x^3+\left (\frac {19\,f\,a^5}{8}-\frac {15\,e\,a^4\,b}{8}+\frac {11\,d\,a^3\,b^2}{8}-\frac {7\,c\,a^2\,b^3}{8}\right )\,x}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}+\frac {f\,x^7}{7\,b^3}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-99\,f\,a^3+63\,e\,a^2\,b-35\,d\,a\,b^2+15\,c\,b^3\right )}{99\,f\,a^4-63\,e\,a^3\,b+35\,d\,a^2\,b^2-15\,c\,a\,b^3}\right )\,\left (-99\,f\,a^3+63\,e\,a^2\,b-35\,d\,a\,b^2+15\,c\,b^3\right )}{8\,b^{13/2}} \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)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 18.22, size = 316, normalized size = 1.28 \begin {gather*} x^{5} \left (- \frac {3 a f}{5 b^{4}} + \frac {e}{5 b^{3}}\right ) + x^{3} \left (\frac {2 a^{2} f}{b^{5}} - \frac {a e}{b^{4}} + \frac {d}{3 b^{3}}\right ) + x \left (- \frac {10 a^{3} f}{b^{6}} + \frac {6 a^{2} e}{b^{5}} - \frac {3 a d}{b^{4}} + \frac {c}{b^{3}}\right ) - \frac {\sqrt {- \frac {a}{b^{13}}} \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log {\left (- b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{16} + \frac {\sqrt {- \frac {a}{b^{13}}} \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log {\left (b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{16} + \frac {x^{3} \left (- 21 a^{4} b f + 17 a^{3} b^{2} e - 13 a^{2} b^{3} d + 9 a b^{4} c\right ) + x \left (- 19 a^{5} f + 15 a^{4} b e - 11 a^{3} b^{2} d + 7 a^{2} b^{3} c\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac {f x^{7}}{7 b^{3}} \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)**3,x)

[Out]

x**5*(-3*a*f/(5*b**4) + e/(5*b**3)) + x**3*(2*a**2*f/b**5 - a*e/b**4 + d/(3*b**3)) + x*(-10*a**3*f/b**6 + 6*a*
*2*e/b**5 - 3*a*d/b**4 + c/b**3) - sqrt(-a/b**13)*(99*a**3*f - 63*a**2*b*e + 35*a*b**2*d - 15*b**3*c)*log(-b**
6*sqrt(-a/b**13) + x)/16 + sqrt(-a/b**13)*(99*a**3*f - 63*a**2*b*e + 35*a*b**2*d - 15*b**3*c)*log(b**6*sqrt(-a
/b**13) + x)/16 + (x**3*(-21*a**4*b*f + 17*a**3*b**2*e - 13*a**2*b**3*d + 9*a*b**4*c) + x*(-19*a**5*f + 15*a**
4*b*e - 11*a**3*b**2*d + 7*a**2*b**3*c))/(8*a**2*b**6 + 16*a*b**7*x**2 + 8*b**8*x**4) + f*x**7/(7*b**3)

________________________________________________________________________________________