3.2.12 \(\int \frac {x^7 (A+B x^2)}{(a+b x^2+c x^4)^2} \, dx\) [112]

Optimal. Leaf size=212 \[ \frac {\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]

[Out]

1/2*(-A*b*c-6*B*a*c+2*B*b^2)*x^2/c^2/(-4*a*c+b^2)-1/2*x^4*(a*(-2*A*c+B*b)+(-A*b*c-2*B*a*c+B*b^2)*x^2)/c/(-4*a*
c+b^2)/(c*x^4+b*x^2+a)-1/2*(6*A*a*b*c^2-A*b^3*c+12*B*a^2*c^2-12*B*a*b^2*c+2*B*b^4)*arctanh((2*c*x^2+b)/(-4*a*c
+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(3/2)-1/4*(-A*c+2*B*b)*ln(c*x^4+b*x^2+a)/c^3

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Rubi [A]
time = 0.25, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1265, 832, 787, 648, 632, 212, 642} \begin {gather*} -\frac {\left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {x^2 \left (-6 a B c-A b c+2 b^2 B\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac {x^4 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*b^2*B - A*b*c - 6*a*B*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (x^4*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^
2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)
*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*B - A*c)*Log[a + b*x^2 + c*x^4]
)/(4*c^3)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 787

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c
), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 1265

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

Rubi steps

\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {x \left (2 a (b B-2 A c)+\left (2 b^2 B-A b c-6 a B c\right ) x\right )}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {-a \left (2 b^2 B-A b c-6 a B c\right )+\left (2 a c (b B-2 A c)-b \left (2 b^2 B-A b c-6 a B c\right )\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^2 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 b B-A c) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}+\frac {\left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac {\left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 B-A b c-6 a B c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {x^4 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 208, normalized size = 0.98 \begin {gather*} \frac {2 B c x^2-\frac {2 \left (b^3 (b B-A c) x^2+a^2 c \left (-3 b B+2 c \left (A+B x^2\right )\right )+a b \left (b^2 B+3 A c^2 x^2-b c \left (A+4 B x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \left (2 b^4 B-A b^3 c-12 a b^2 B c+6 a A b c^2+12 a^2 B c^2\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+(-2 b B+A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*B*c*x^2 - (2*(b^3*(b*B - A*c)*x^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b^2*B + 3*A*c^2*x^2 - b*c*(A +
4*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*
B*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (-2*b*B + A*c)*Log[a + b*x^2 + c*x^4])
/(4*c^3)

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Maple [A]
time = 0.10, size = 282, normalized size = 1.33

method	result	size
default	\(\frac {B \,x^{2}}{2 c^{2}}+\frac {\frac {\frac {\left (3 A a b \,c^{2}-A \,b^{3} c +2 a^{2} B \,c^{2}-4 a \,b^{2} B c +b^{4} B \right ) x^{2}}{c \left (4 a c -b^{2}\right )}+\frac {a \left (2 c^{2} a A -A \,b^{2} c -3 a b B c +b^{3} B \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (4 c^{2} a A -A \,b^{2} c -8 a b B c +2 b^{3} B \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-A a b c -6 a^{2} c B +2 B a \,b^{2}-\frac {\left (4 c^{2} a A -A \,b^{2} c -8 a b B c +2 b^{3} B \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 c^{2}}\)	\(282\)
risch	\(\text {Expression too large to display}\)	\(3097\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*B*x^2/c^2+1/2/c^2*(((3*A*a*b*c^2-A*b^3*c+2*B*a^2*c^2-4*B*a*b^2*c+B*b^4)/c/(4*a*c-b^2)*x^2+a*(2*A*a*c^2-A*b
^2*c-3*B*a*b*c+B*b^3)/c/(4*a*c-b^2))/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)*(1/2*(4*A*a*c^2-A*b^2*c-8*B*a*b*c+2*B*b^3)/
c*ln(c*x^4+b*x^2+a)+2*(-A*a*b*c-6*a^2*c*B+2*B*a*b^2-1/2*(4*A*a*c^2-A*b^2*c-8*B*a*b*c+2*B*b^3)*b/c)/(4*a*c-b^2)
^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (200) = 400\).
time = 0.44, size = 1323, normalized size = 6.24 \begin {gather*} \left [-\frac {2 \, B a b^{5} - 16 \, A a^{3} c^{3} - 2 \, {\left (B b^{4} c^{2} - 8 \, B a b^{2} c^{3} + 16 \, B a^{2} c^{4}\right )} x^{6} - 2 \, {\left (B b^{5} c - 8 \, B a b^{3} c^{2} + 16 \, B a^{2} b c^{3}\right )} x^{4} + 12 \, {\left (2 \, B a^{3} b + A a^{2} b^{2}\right )} c^{2} + 2 \, {\left (B b^{6} - 12 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{3} + {\left (26 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} c^{2} - {\left (9 \, B a b^{4} + A b^{5}\right )} c\right )} x^{2} + {\left (2 \, B a b^{4} + {\left (2 \, B b^{4} c + 6 \, {\left (2 \, B a^{2} + A a b\right )} c^{3} - {\left (12 \, B a b^{2} + A b^{3}\right )} c^{2}\right )} x^{4} + 6 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{2} + {\left (2 \, B b^{5} + 6 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} c^{2} - {\left (12 \, B a b^{3} + A b^{4}\right )} c\right )} x^{2} - {\left (12 \, B a^{2} b^{2} + A a b^{3}\right )} c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 2 \, {\left (7 \, B a^{2} b^{3} + A a b^{4}\right )} c + {\left (2 \, B a b^{5} - 16 \, A a^{3} c^{3} + {\left (2 \, B b^{5} c - 16 \, A a^{2} c^{4} + 8 \, {\left (4 \, B a^{2} b + A a b^{2}\right )} c^{3} - {\left (16 \, B a b^{3} + A b^{4}\right )} c^{2}\right )} x^{4} + 8 \, {\left (4 \, B a^{3} b + A a^{2} b^{2}\right )} c^{2} + {\left (2 \, B b^{6} - 16 \, A a^{2} b c^{3} + 8 \, {\left (4 \, B a^{2} b^{2} + A a b^{3}\right )} c^{2} - {\left (16 \, B a b^{4} + A b^{5}\right )} c\right )} x^{2} - {\left (16 \, B a^{2} b^{3} + A a b^{4}\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{2}\right )}}, -\frac {2 \, B a b^{5} - 16 \, A a^{3} c^{3} - 2 \, {\left (B b^{4} c^{2} - 8 \, B a b^{2} c^{3} + 16 \, B a^{2} c^{4}\right )} x^{6} - 2 \, {\left (B b^{5} c - 8 \, B a b^{3} c^{2} + 16 \, B a^{2} b c^{3}\right )} x^{4} + 12 \, {\left (2 \, B a^{3} b + A a^{2} b^{2}\right )} c^{2} + 2 \, {\left (B b^{6} - 12 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{3} + {\left (26 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} c^{2} - {\left (9 \, B a b^{4} + A b^{5}\right )} c\right )} x^{2} + 2 \, {\left (2 \, B a b^{4} + {\left (2 \, B b^{4} c + 6 \, {\left (2 \, B a^{2} + A a b\right )} c^{3} - {\left (12 \, B a b^{2} + A b^{3}\right )} c^{2}\right )} x^{4} + 6 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{2} + {\left (2 \, B b^{5} + 6 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} c^{2} - {\left (12 \, B a b^{3} + A b^{4}\right )} c\right )} x^{2} - {\left (12 \, B a^{2} b^{2} + A a b^{3}\right )} c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (7 \, B a^{2} b^{3} + A a b^{4}\right )} c + {\left (2 \, B a b^{5} - 16 \, A a^{3} c^{3} + {\left (2 \, B b^{5} c - 16 \, A a^{2} c^{4} + 8 \, {\left (4 \, B a^{2} b + A a b^{2}\right )} c^{3} - {\left (16 \, B a b^{3} + A b^{4}\right )} c^{2}\right )} x^{4} + 8 \, {\left (4 \, B a^{3} b + A a^{2} b^{2}\right )} c^{2} + {\left (2 \, B b^{6} - 16 \, A a^{2} b c^{3} + 8 \, {\left (4 \, B a^{2} b^{2} + A a b^{3}\right )} c^{2} - {\left (16 \, B a b^{4} + A b^{5}\right )} c\right )} x^{2} - {\left (16 \, B a^{2} b^{3} + A a b^{4}\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

[-1/4*(2*B*a*b^5 - 16*A*a^3*c^3 - 2*(B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*x^6 - 2*(B*b^5*c - 8*B*a*b^3*c^
2 + 16*B*a^2*b*c^3)*x^4 + 12*(2*B*a^3*b + A*a^2*b^2)*c^2 + 2*(B*b^6 - 12*(2*B*a^3 + A*a^2*b)*c^3 + (26*B*a^2*b
^2 + 7*A*a*b^3)*c^2 - (9*B*a*b^4 + A*b^5)*c)*x^2 + (2*B*a*b^4 + (2*B*b^4*c + 6*(2*B*a^2 + A*a*b)*c^3 - (12*B*a
*b^2 + A*b^3)*c^2)*x^4 + 6*(2*B*a^3 + A*a^2*b)*c^2 + (2*B*b^5 + 6*(2*B*a^2*b + A*a*b^2)*c^2 - (12*B*a*b^3 + A*
b^4)*c)*x^2 - (12*B*a^2*b^2 + A*a*b^3)*c)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^
2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - 2*(7*B*a^2*b^3 + A*a*b^4)*c + (2*B*a*b^5 - 16*A*a^3*c^3 + (2*
B*b^5*c - 16*A*a^2*c^4 + 8*(4*B*a^2*b + A*a*b^2)*c^3 - (16*B*a*b^3 + A*b^4)*c^2)*x^4 + 8*(4*B*a^3*b + A*a^2*b^
2)*c^2 + (2*B*b^6 - 16*A*a^2*b*c^3 + 8*(4*B*a^2*b^2 + A*a*b^3)*c^2 - (16*B*a*b^4 + A*b^5)*c)*x^2 - (16*B*a^2*b
^3 + A*a*b^4)*c)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16
*a^2*c^6)*x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^2), -1/4*(2*B*a*b^5 - 16*A*a^3*c^3 - 2*(B*b^4*c^2 - 8
*B*a*b^2*c^3 + 16*B*a^2*c^4)*x^6 - 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*x^4 + 12*(2*B*a^3*b + A*a^2*b^
2)*c^2 + 2*(B*b^6 - 12*(2*B*a^3 + A*a^2*b)*c^3 + (26*B*a^2*b^2 + 7*A*a*b^3)*c^2 - (9*B*a*b^4 + A*b^5)*c)*x^2 +
2*(2*B*a*b^4 + (2*B*b^4*c + 6*(2*B*a^2 + A*a*b)*c^3 - (12*B*a*b^2 + A*b^3)*c^2)*x^4 + 6*(2*B*a^3 + A*a^2*b)*c
^2 + (2*B*b^5 + 6*(2*B*a^2*b + A*a*b^2)*c^2 - (12*B*a*b^3 + A*b^4)*c)*x^2 - (12*B*a^2*b^2 + A*a*b^3)*c)*sqrt(-
b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 2*(7*B*a^2*b^3 + A*a*b^4)*c + (2*B*a*b^
5 - 16*A*a^3*c^3 + (2*B*b^5*c - 16*A*a^2*c^4 + 8*(4*B*a^2*b + A*a*b^2)*c^3 - (16*B*a*b^3 + A*b^4)*c^2)*x^4 + 8
*(4*B*a^3*b + A*a^2*b^2)*c^2 + (2*B*b^6 - 16*A*a^2*b*c^3 + 8*(4*B*a^2*b^2 + A*a*b^3)*c^2 - (16*B*a*b^4 + A*b^5
)*c)*x^2 - (16*B*a^2*b^3 + A*a*b^4)*c)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*
c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^2)]

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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(x**7*(B*x**2+A)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 4.88, size = 239, normalized size = 1.13 \begin {gather*} \frac {B x^{2}}{2 \, c^{2}} + \frac {{\left (2 \, B b^{4} - 12 \, B a b^{2} c - A b^{3} c + 12 \, B a^{2} c^{2} + 6 \, A a b c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, B b^{3} x^{4} - 8 \, B a b c x^{4} - A b^{2} c x^{4} + 4 \, A a c^{2} x^{4} + A b^{3} x^{2} - 4 \, B a^{2} c x^{2} - 2 \, A a b c x^{2} - 2 \, B a^{2} b + A a b^{2}}{4 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} - \frac {{\left (2 \, B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*B*x^2/c^2 + 1/2*(2*B*b^4 - 12*B*a*b^2*c - A*b^3*c + 12*B*a^2*c^2 + 6*A*a*b*c^2)*arctan((2*c*x^2 + b)/sqrt(
-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + 1/4*(2*B*b^3*x^4 - 8*B*a*b*c*x^4 - A*b^2*c*x^4 + 4*A
*a*c^2*x^4 + A*b^3*x^2 - 4*B*a^2*c*x^2 - 2*A*a*b*c*x^2 - 2*B*a^2*b + A*a*b^2)/((c*x^4 + b*x^2 + a)*(b^2*c^2 -
4*a*c^3)) - 1/4*(2*B*b - A*c)*log(c*x^4 + b*x^2 + a)/c^3

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Mupad [B]
time = 0.83, size = 2282, normalized size = 10.76 \begin {gather*} \frac {\frac {a\,\left (B\,b^3-A\,b^2\,c-3\,B\,a\,b\,c+2\,A\,a\,c^2\right )}{2\,c\,\left (4\,a\,c-b^2\right )}+\frac {x^2\,\left (2\,B\,a^2\,c^2-4\,B\,a\,b^2\,c+3\,A\,a\,b\,c^2+B\,b^4-A\,b^3\,c\right )}{2\,c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^4+b\,c^2\,x^2+a\,c^2}+\frac {B\,x^2}{2\,c^2}+\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{2\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}-\frac {\mathrm {atan}\left (\frac {\left (8\,a\,c^5\,{\left (4\,a\,c-b^2\right )}^3-2\,b^2\,c^4\,{\left (4\,a\,c-b^2\right )}^3\right )\,\left (x^2\,\left (\frac {\frac {\left (\frac {24\,B\,a^2\,c^5-56\,B\,a\,b^2\,c^4+28\,A\,a\,b\,c^5+12\,B\,b^4\,c^3-6\,A\,b^3\,c^4}{4\,a\,c^5-b^2\,c^4}+\frac {\left (8\,b^3\,c^6-32\,a\,b\,c^7\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{2\,\left (4\,a\,c^5-b^2\,c^4\right )\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}\right )\,\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )}{8\,c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}}+\frac {\left (8\,b^3\,c^6-32\,a\,b\,c^7\right )\,\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{16\,c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}\,\left (4\,a\,c^5-b^2\,c^4\right )\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}}{a\,\left (4\,a\,c-b^2\right )}+\frac {b\,\left (\frac {-5\,A^2\,a\,b\,c^3+A^2\,b^3\,c^2-6\,A\,B\,a^2\,c^3+20\,A\,B\,a\,b^2\,c^2-4\,A\,B\,b^4\,c+12\,B^2\,a^2\,b\,c^2-20\,B^2\,a\,b^3\,c+4\,B^2\,b^5}{4\,a\,c^5-b^2\,c^4}+\frac {\left (\frac {24\,B\,a^2\,c^5-56\,B\,a\,b^2\,c^4+28\,A\,a\,b\,c^5+12\,B\,b^4\,c^3-6\,A\,b^3\,c^4}{4\,a\,c^5-b^2\,c^4}+\frac {\left (8\,b^3\,c^6-32\,a\,b\,c^7\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{2\,\left (4\,a\,c^5-b^2\,c^4\right )\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{2\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}-\frac {\left (\frac {b^3\,c^6}{2}-2\,a\,b\,c^7\right )\,{\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )}^2}{c^6\,{\left (4\,a\,c-b^2\right )}^3\,\left (4\,a\,c^5-b^2\,c^4\right )}\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )+\frac {\frac {\left (\frac {8\,A\,a\,c^4-16\,B\,a\,b\,c^3}{c^4}-\frac {8\,a\,c^2\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3}\right )\,\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )}{8\,c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {a\,\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{c\,{\left (4\,a\,c-b^2\right )}^{3/2}\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}}{a\,\left (4\,a\,c-b^2\right )}+\frac {b\,\left (\frac {\left (\frac {8\,A\,a\,c^4-16\,B\,a\,b\,c^3}{c^4}-\frac {8\,a\,c^2\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3}\right )\,\left (-256\,B\,a^3\,b\,c^3+128\,A\,a^3\,c^4+192\,B\,a^2\,b^3\,c^2-96\,A\,a^2\,b^2\,c^3-48\,B\,a\,b^5\,c+24\,A\,a\,b^4\,c^2+4\,B\,b^7-2\,A\,b^6\,c\right )}{2\,\left (256\,a^3\,c^6-192\,a^2\,b^2\,c^5+48\,a\,b^4\,c^4-4\,b^6\,c^3\right )}-\frac {a\,A^2\,c^2-4\,a\,A\,B\,b\,c+4\,a\,B^2\,b^2}{c^4}+\frac {a\,{\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )}^2}{c^4\,{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{36\,A^2\,a^2\,b^2\,c^4-12\,A^2\,a\,b^4\,c^3+A^2\,b^6\,c^2+144\,A\,B\,a^3\,b\,c^4-168\,A\,B\,a^2\,b^3\,c^3+48\,A\,B\,a\,b^5\,c^2-4\,A\,B\,b^7\,c+144\,B^2\,a^4\,c^4-288\,B^2\,a^3\,b^2\,c^3+192\,B^2\,a^2\,b^4\,c^2-48\,B^2\,a\,b^6\,c+4\,B^2\,b^8}\right )\,\left (12\,B\,a^2\,c^2-12\,B\,a\,b^2\,c+6\,A\,a\,b\,c^2+2\,B\,b^4-A\,b^3\,c\right )}{2\,c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x)

[Out]

((a*(B*b^3 + 2*A*a*c^2 - A*b^2*c - 3*B*a*b*c))/(2*c*(4*a*c - b^2)) + (x^2*(B*b^4 + 2*B*a^2*c^2 - A*b^3*c + 3*A
*a*b*c^2 - 4*B*a*b^2*c))/(2*c*(4*a*c - b^2)))/(a*c^2 + c^3*x^4 + b*c^2*x^2) + (B*x^2)/(2*c^2) + (log(a + b*x^2
+ c*x^4)*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^
2*c^3 + 192*B*a^2*b^3*c^2))/(2*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)) - (atan(((8*a*c^5*(
4*a*c - b^2)^3 - 2*b^2*c^4*(4*a*c - b^2)^3)*(x^2*(((((24*B*a^2*c^5 - 6*A*b^3*c^4 + 12*B*b^4*c^3 + 28*A*a*b*c^5
- 56*B*a*b^2*c^4)/(4*a*c^5 - b^2*c^4) + ((8*b^3*c^6 - 32*a*b*c^7)*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B
*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^2*c^3 + 192*B*a^2*b^3*c^2))/(2*(4*a*c^5 - b^2*c^4)*(2
56*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)))*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*a*b*c^2 - 1
2*B*a*b^2*c))/(8*c^3*(4*a*c - b^2)^(3/2)) + ((8*b^3*c^6 - 32*a*b*c^7)*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*
a*b*c^2 - 12*B*a*b^2*c)*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3
- 96*A*a^2*b^2*c^3 + 192*B*a^2*b^3*c^2))/(16*c^3*(4*a*c - b^2)^(3/2)*(4*a*c^5 - b^2*c^4)*(256*a^3*c^6 - 4*b^6
*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)))/(a*(4*a*c - b^2)) + (b*((4*B^2*b^5 + A^2*b^3*c^2 - 4*A*B*b^4*c - 6*A*
B*a^2*c^3 - 5*A^2*a*b*c^3 - 20*B^2*a*b^3*c + 12*B^2*a^2*b*c^2 + 20*A*B*a*b^2*c^2)/(4*a*c^5 - b^2*c^4) + (((24*
B*a^2*c^5 - 6*A*b^3*c^4 + 12*B*b^4*c^3 + 28*A*a*b*c^5 - 56*B*a*b^2*c^4)/(4*a*c^5 - b^2*c^4) + ((8*b^3*c^6 - 32
*a*b*c^7)*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^
2*c^3 + 192*B*a^2*b^3*c^2))/(2*(4*a*c^5 - b^2*c^4)*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5))
)*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^2*c^3 +
192*B*a^2*b^3*c^2))/(2*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)) - (((b^3*c^6)/2 - 2*a*b*c^7
)*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*a*b*c^2 - 12*B*a*b^2*c)^2)/(c^6*(4*a*c - b^2)^3*(4*a*c^5 - b^2*c^4))
))/(2*a*(4*a*c - b^2)^(3/2))) + ((((8*A*a*c^4 - 16*B*a*b*c^3)/c^4 - (8*a*c^2*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^
6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^2*c^3 + 192*B*a^2*b^3*c^2))/(256*a^3*c^6 -
4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5))*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*a*b*c^2 - 12*B*a*b^2*c))/
(8*c^3*(4*a*c - b^2)^(3/2)) - (a*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*a*b*c^2 - 12*B*a*b^2*c)*(4*B*b^7 + 12
8*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^2*c^3 + 192*B*a^2*b^3*c
^2))/(c*(4*a*c - b^2)^(3/2)*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)))/(a*(4*a*c - b^2)) + (
b*((((8*A*a*c^4 - 16*B*a*b*c^3)/c^4 - (8*a*c^2*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^
4*c^2 - 256*B*a^3*b*c^3 - 96*A*a^2*b^2*c^3 + 192*B*a^2*b^3*c^2))/(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192
*a^2*b^2*c^5))*(4*B*b^7 + 128*A*a^3*c^4 - 2*A*b^6*c - 48*B*a*b^5*c + 24*A*a*b^4*c^2 - 256*B*a^3*b*c^3 - 96*A*a
^2*b^2*c^3 + 192*B*a^2*b^3*c^2))/(2*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)) - (A^2*a*c^2 +
4*B^2*a*b^2 - 4*A*B*a*b*c)/c^4 + (a*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*a*b*c^2 - 12*B*a*b^2*c)^2)/(c^4*(
4*a*c - b^2)^3)))/(2*a*(4*a*c - b^2)^(3/2))))/(4*B^2*b^8 + A^2*b^6*c^2 + 144*B^2*a^4*c^4 - 4*A*B*b^7*c + 36*A^
2*a^2*b^2*c^4 + 192*B^2*a^2*b^4*c^2 - 288*B^2*a^3*b^2*c^3 - 48*B^2*a*b^6*c - 12*A^2*a*b^4*c^3 - 168*A*B*a^2*b^
3*c^3 + 48*A*B*a*b^5*c^2 + 144*A*B*a^3*b*c^4))*(2*B*b^4 + 12*B*a^2*c^2 - A*b^3*c + 6*A*a*b*c^2 - 12*B*a*b^2*c)
)/(2*c^3*(4*a*c - b^2)^(3/2))

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