3.2.26 \(\int \frac {x^7 (A+B x^2)}{(a+b x^2+c x^4)^3} \, dx\) [126]

Optimal. Leaf size=146 \[ -\frac {x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 a (A b-2 a B) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

-1/4*x^6*(A*b-2*a*B-(-2*A*c+B*b)*x^2)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/4*(A*b-2*B*a)*x^2*(b*x^2+2*a)/(-4*a*c+b
^2)^2/(c*x^4+b*x^2+a)+3*a*(A*b-2*B*a)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)

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Rubi [A]
time = 0.10, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1265, 818, 736, 632, 212} \begin {gather*} \frac {3 a (A b-2 a B) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {x^6 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

-1/4*(x^6*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*(A*b - 2*a*B)*x^2*(2*a
+ b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*a*(A*b - 2*a*B)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c
]])/(b^2 - 4*a*c)^(5/2)

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 736

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

Rule 818

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

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 )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {(3 (A b-2 a B)) \text {Subst}\left (\int \frac {x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac {x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {(3 a (A b-2 a B)) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac {x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {(3 a (A b-2 a B)) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {x^6 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 (A b-2 a B) x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 a (A b-2 a B) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 261, normalized size = 1.79 \begin {gather*} \frac {1}{4} \left (\frac {b^5 B-8 a b^3 B c-b^4 c \left (A+2 B x^2\right )-4 a^2 c^3 \left (4 A+5 B x^2\right )+a b^2 c^2 \left (5 A+16 B x^2\right )+2 a b c^2 \left (11 a B-3 A c x^2\right )}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {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 )}{c^3 \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {12 a (A b-2 a B) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((b^5*B - 8*a*b^3*B*c - b^4*c*(A + 2*B*x^2) - 4*a^2*c^3*(4*A + 5*B*x^2) + a*b^2*c^2*(5*A + 16*B*x^2) + 2*a*b*c
^2*(11*a*B - 3*A*c*x^2))/(c^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (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)))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) - (
12*a*(A*b - 2*a*B)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(138)=276\).
time = 0.07, size = 343, normalized size = 2.35

method	result	size
default	\(\frac {-\frac {\left (3 A a b \,c^{2}+10 a^{2} B \,c^{2}-8 a \,b^{2} B c +b^{4} B \right ) x^{6}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (16 A \,a^{2} c^{3}+A a \,b^{2} c^{2}+A \,b^{4} c -2 B \,a^{2} b \,c^{2}-8 B a \,b^{3} c +B \,b^{5}\right ) x^{4}}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (5 A a b \,c^{2}+A \,b^{3} c +6 a^{2} B \,c^{2}-10 a \,b^{2} B c +b^{4} B \right ) x^{2}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}-\frac {a^{2} \left (8 c^{2} a A +A \,b^{2} c -10 a b B c +b^{3} B \right )}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 a \left (A b -2 a B \right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\)	\(343\)
risch	\(\frac {-\frac {\left (3 A a b \,c^{2}+10 a^{2} B \,c^{2}-8 a \,b^{2} B c +b^{4} B \right ) x^{6}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (16 A \,a^{2} c^{3}+A a \,b^{2} c^{2}+A \,b^{4} c -2 B \,a^{2} b \,c^{2}-8 B a \,b^{3} c +B \,b^{5}\right ) x^{4}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (5 A a b \,c^{2}+A \,b^{3} c +6 a^{2} B \,c^{2}-10 a \,b^{2} B c +b^{4} B \right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}-\frac {a^{2} \left (8 c^{2} a A +A \,b^{2} c -10 a b B c +b^{3} B \right )}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 a \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) A b}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 a^{2} \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) B}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 a \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) A b}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {3 a^{2} \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) B}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\)	\(586\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(-(3*A*a*b*c^2+10*B*a^2*c^2-8*B*a*b^2*c+B*b^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-1/2*(16*A*a^2*c^3+A*a*b^2*
c^2+A*b^4*c-2*B*a^2*b*c^2-8*B*a*b^3*c+B*b^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-a*(5*A*a*b*c^2+A*b^3*c+6*B*a^2
*c^2-10*B*a*b^2*c+B*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2-1/2*a^2/c^2*(8*A*a*c^2+A*b^2*c-10*B*a*b*c+B*b^3)/(
16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2-3*a*(A*b-2*B*a)/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arct
an((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)^3,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. 677 vs. \(2 (140) = 280\).
time = 0.40, size = 1378, normalized size = 9.44 \begin {gather*} \left [-\frac {B a^{2} b^{5} - 32 \, A a^{4} c^{3} + 2 \, {\left (B b^{6} c - 12 \, B a b^{4} c^{2} - 4 \, {\left (10 \, B a^{3} + 3 \, A a^{2} b\right )} c^{4} + 3 \, {\left (14 \, B a^{2} b^{2} + A a b^{3}\right )} c^{3}\right )} x^{6} + {\left (B b^{7} - 64 \, A a^{3} c^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 3 \, {\left (10 \, B a^{2} b^{3} - A a b^{4}\right )} c^{2} - {\left (12 \, B a b^{5} - A b^{6}\right )} c\right )} x^{4} + 4 \, {\left (10 \, B a^{4} b + A a^{3} b^{2}\right )} c^{2} + 2 \, {\left (B a b^{6} - 4 \, {\left (6 \, B a^{4} + 5 \, A a^{3} b\right )} c^{3} + {\left (46 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} c^{2} - {\left (14 \, B a^{2} b^{4} - A a b^{5}\right )} c\right )} x^{2} + 6 \, {\left ({\left (2 \, B a^{2} - A a b\right )} c^{4} x^{8} + 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} c^{3} x^{6} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c^{2} x^{2} + {\left (2 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{3} + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c^{2}\right )} x^{4} + {\left (2 \, B a^{4} - A a^{3} b\right )} c^{2}\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 ) - {\left (14 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} c}{4 \, {\left (a^{2} b^{6} c^{2} - 12 \, a^{3} b^{4} c^{3} + 48 \, a^{4} b^{2} c^{4} - 64 \, a^{5} c^{5} + {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} x^{8} + 2 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} x^{6} + {\left (b^{8} c^{2} - 10 \, a b^{6} c^{3} + 24 \, a^{2} b^{4} c^{4} + 32 \, a^{3} b^{2} c^{5} - 128 \, a^{4} c^{6}\right )} x^{4} + 2 \, {\left (a b^{7} c^{2} - 12 \, a^{2} b^{5} c^{3} + 48 \, a^{3} b^{3} c^{4} - 64 \, a^{4} b c^{5}\right )} x^{2}\right )}}, -\frac {B a^{2} b^{5} - 32 \, A a^{4} c^{3} + 2 \, {\left (B b^{6} c - 12 \, B a b^{4} c^{2} - 4 \, {\left (10 \, B a^{3} + 3 \, A a^{2} b\right )} c^{4} + 3 \, {\left (14 \, B a^{2} b^{2} + A a b^{3}\right )} c^{3}\right )} x^{6} + {\left (B b^{7} - 64 \, A a^{3} c^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 3 \, {\left (10 \, B a^{2} b^{3} - A a b^{4}\right )} c^{2} - {\left (12 \, B a b^{5} - A b^{6}\right )} c\right )} x^{4} + 4 \, {\left (10 \, B a^{4} b + A a^{3} b^{2}\right )} c^{2} + 2 \, {\left (B a b^{6} - 4 \, {\left (6 \, B a^{4} + 5 \, A a^{3} b\right )} c^{3} + {\left (46 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} c^{2} - {\left (14 \, B a^{2} b^{4} - A a b^{5}\right )} c\right )} x^{2} + 12 \, {\left ({\left (2 \, B a^{2} - A a b\right )} c^{4} x^{8} + 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} c^{3} x^{6} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c^{2} x^{2} + {\left (2 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{3} + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c^{2}\right )} x^{4} + {\left (2 \, B a^{4} - A a^{3} b\right )} c^{2}\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 ) - {\left (14 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} c}{4 \, {\left (a^{2} b^{6} c^{2} - 12 \, a^{3} b^{4} c^{3} + 48 \, a^{4} b^{2} c^{4} - 64 \, a^{5} c^{5} + {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} x^{8} + 2 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} x^{6} + {\left (b^{8} c^{2} - 10 \, a b^{6} c^{3} + 24 \, a^{2} b^{4} c^{4} + 32 \, a^{3} b^{2} c^{5} - 128 \, a^{4} c^{6}\right )} x^{4} + 2 \, {\left (a b^{7} c^{2} - 12 \, a^{2} b^{5} c^{3} + 48 \, a^{3} b^{3} c^{4} - 64 \, a^{4} 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)^3,x, algorithm="fricas")

[Out]

[-1/4*(B*a^2*b^5 - 32*A*a^4*c^3 + 2*(B*b^6*c - 12*B*a*b^4*c^2 - 4*(10*B*a^3 + 3*A*a^2*b)*c^4 + 3*(14*B*a^2*b^2
+ A*a*b^3)*c^3)*x^6 + (B*b^7 - 64*A*a^3*c^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 3*(10*B*a^2*b^3 - A*a*b^4)*c^
2 - (12*B*a*b^5 - A*b^6)*c)*x^4 + 4*(10*B*a^4*b + A*a^3*b^2)*c^2 + 2*(B*a*b^6 - 4*(6*B*a^4 + 5*A*a^3*b)*c^3 +
(46*B*a^3*b^2 + A*a^2*b^3)*c^2 - (14*B*a^2*b^4 - A*a*b^5)*c)*x^2 + 6*((2*B*a^2 - A*a*b)*c^4*x^8 + 2*(2*B*a^2*b
- A*a*b^2)*c^3*x^6 + 2*(2*B*a^3*b - A*a^2*b^2)*c^2*x^2 + (2*(2*B*a^3 - A*a^2*b)*c^3 + (2*B*a^2*b^2 - A*a*b^3)
*c^2)*x^4 + (2*B*a^4 - A*a^3*b)*c^2)*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)) - (14*B*a^3*b^3 - A*a^2*b^4)*c)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a
^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^8 + 2*(b^7*c^3 - 12*a*b^5*c
^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*
c^6)*x^4 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c^5)*x^2), -1/4*(B*a^2*b^5 - 32*A*a^4*c^3
+ 2*(B*b^6*c - 12*B*a*b^4*c^2 - 4*(10*B*a^3 + 3*A*a^2*b)*c^4 + 3*(14*B*a^2*b^2 + A*a*b^3)*c^3)*x^6 + (B*b^7 -
64*A*a^3*c^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 3*(10*B*a^2*b^3 - A*a*b^4)*c^2 - (12*B*a*b^5 - A*b^6)*c)*x^4
+ 4*(10*B*a^4*b + A*a^3*b^2)*c^2 + 2*(B*a*b^6 - 4*(6*B*a^4 + 5*A*a^3*b)*c^3 + (46*B*a^3*b^2 + A*a^2*b^3)*c^2
- (14*B*a^2*b^4 - A*a*b^5)*c)*x^2 + 12*((2*B*a^2 - A*a*b)*c^4*x^8 + 2*(2*B*a^2*b - A*a*b^2)*c^3*x^6 + 2*(2*B*a
^3*b - A*a^2*b^2)*c^2*x^2 + (2*(2*B*a^3 - A*a^2*b)*c^3 + (2*B*a^2*b^2 - A*a*b^3)*c^2)*x^4 + (2*B*a^4 - A*a^3*b
)*c^2)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (14*B*a^3*b^3 - A*a^2*b^4)
*c)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 6
4*a^3*c^7)*x^8 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + (b^8*c^2 - 10*a*b^6*c^3 + 24
*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^4 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*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)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (140) = 280\).
time = 7.42, size = 318, normalized size = 2.18 \begin {gather*} \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, B b^{4} c x^{6} - 16 \, B a b^{2} c^{2} x^{6} + 20 \, B a^{2} c^{3} x^{6} + 6 \, A a b c^{3} x^{6} + B b^{5} x^{4} - 8 \, B a b^{3} c x^{4} + A b^{4} c x^{4} - 2 \, B a^{2} b c^{2} x^{4} + A a b^{2} c^{2} x^{4} + 16 \, A a^{2} c^{3} x^{4} + 2 \, B a b^{4} x^{2} - 20 \, B a^{2} b^{2} c x^{2} + 2 \, A a b^{3} c x^{2} + 12 \, B a^{3} c^{2} x^{2} + 10 \, A a^{2} b c^{2} x^{2} + B a^{2} b^{3} - 10 \, B a^{3} b c + A a^{2} b^{2} c + 8 \, A a^{3} c^{2}}{4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

3*(2*B*a^2 - A*a*b)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c
)) - 1/4*(2*B*b^4*c*x^6 - 16*B*a*b^2*c^2*x^6 + 20*B*a^2*c^3*x^6 + 6*A*a*b*c^3*x^6 + B*b^5*x^4 - 8*B*a*b^3*c*x^
4 + A*b^4*c*x^4 - 2*B*a^2*b*c^2*x^4 + A*a*b^2*c^2*x^4 + 16*A*a^2*c^3*x^4 + 2*B*a*b^4*x^2 - 20*B*a^2*b^2*c*x^2
+ 2*A*a*b^3*c*x^2 + 12*B*a^3*c^2*x^2 + 10*A*a^2*b*c^2*x^2 + B*a^2*b^3 - 10*B*a^3*b*c + A*a^2*b^2*c + 8*A*a^3*c
^2)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^4 + b*x^2 + a)^2)

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Mupad [B]
time = 0.69, size = 593, normalized size = 4.06 \begin {gather*} \frac {3\,a\,\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {3\,\left (A\,b-2\,B\,a\right )\,\left (6\,B\,a^2\,c^2-3\,A\,a\,b\,c^2\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {9\,a\,b\,{\left (A\,b-2\,B\,a\right )}^2\,\left (32\,a^2\,b\,c^4-16\,a\,b^3\,c^3+2\,b^5\,c^2\right )}{2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )-\frac {18\,a^2\,b\,c^2\,{\left (A\,b-2\,B\,a\right )}^2}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{18\,A^2\,a^2\,b^2\,c^2-72\,A\,B\,a^3\,b\,c^2+72\,B^2\,a^4\,c^2}\right )\,\left (A\,b-2\,B\,a\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {x^4\,\left (-2\,B\,a^2\,b\,c^2+16\,A\,a^2\,c^3-8\,B\,a\,b^3\,c+A\,a\,b^2\,c^2+B\,b^5+A\,b^4\,c\right )}{4\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (B\,b^3+A\,b^2\,c-10\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{4\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^6\,\left (10\,B\,a^2\,c^2-8\,B\,a\,b^2\,c+3\,A\,a\,b\,c^2+B\,b^4\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,x^2\,\left (6\,B\,a^2\,c^2-10\,B\,a\,b^2\,c+5\,A\,a\,b\,c^2+B\,b^4+A\,b^3\,c\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6} \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)^3,x)

[Out]

(3*a*atan(((x^2*((3*(A*b - 2*B*a)*(6*B*a^2*c^2 - 3*A*a*b*c^2))/((4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8*a*b^
2*c)) - (9*a*b*(A*b - 2*B*a)^2*(2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4))/(2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^
2*c^2 - 8*a*b^2*c))) - (18*a^2*b*c^2*(A*b - 2*B*a)^2)/(4*a*c - b^2)^(15/2))*(b^4*(4*a*c - b^2)^5 + 16*a^2*c^2*
(4*a*c - b^2)^5 - 8*a*b^2*c*(4*a*c - b^2)^5))/(72*B^2*a^4*c^2 + 18*A^2*a^2*b^2*c^2 - 72*A*B*a^3*b*c^2))*(A*b -
2*B*a))/(4*a*c - b^2)^(5/2) - ((x^4*(B*b^5 + 16*A*a^2*c^3 + A*b^4*c - 8*B*a*b^3*c + A*a*b^2*c^2 - 2*B*a^2*b*c
^2))/(4*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a^2*(B*b^3 + 8*A*a*c^2 + A*b^2*c - 10*B*a*b*c))/(4*c^2*(b^4 + 1
6*a^2*c^2 - 8*a*b^2*c)) + (x^6*(B*b^4 + 10*B*a^2*c^2 + 3*A*a*b*c^2 - 8*B*a*b^2*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*
a*b^2*c)) + (a*x^2*(B*b^4 + 6*B*a^2*c^2 + A*b^3*c + 5*A*a*b*c^2 - 10*B*a*b^2*c))/(2*c^2*(b^4 + 16*a^2*c^2 - 8*
a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6)

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