3.2.27 \(\int \frac {x^5 (A+B x^2)}{(a+b x^2+c x^4)^3} \, dx\) [127]

Optimal. Leaf size=185 \[ -\frac {x^4 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 a b B-A \left (b^2+2 a c\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

-1/4*x^4*(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+1/4*(-a*(-6*A*b*c+8*B*a*c+B*b^2)-(4*A*a*c
^2-4*A*b^2*c+2*B*a*b*c+B*b^3)*x^2)/c/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+(3*a*b*B-A*(2*a*c+b^2))*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 = 0.17, antiderivative size = 185, 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, 834, 791, 632, 212} \begin {gather*} \frac {\left (3 a b B-A \left (2 a c+b^2\right )\right ) \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 {x^4 \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}-\frac {a \left (8 a B c-6 A b c+b^2 B\right )+x^2 \left (4 a A c^2+2 a b B c-4 A b^2 c+b^3 B\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

-1/4*(x^4*(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) - (a*(b^2*B - 6*A*b*c + 8*a
*B*c) + (b^3*B - 4*A*b^2*c + 2*a*b*B*c + 4*a*A*c^2)*x^2)/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*a*b*B
- A*(b^2 + 2*a*c))*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 791

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

Rule 834

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)*((f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*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, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

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^5 \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^2 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^4 \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 {\text {Subst}\left (\int \frac {x (-2 (A b-2 a B)-(b B-2 A c) x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac {x^4 \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 {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 a b B-A \left (b^2+2 a c\right )\right ) \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^4 \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 {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 a b B-A \left (b^2+2 a c\right )\right ) \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^4 \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 {a \left (b^2 B-6 A b c+8 a B c\right )+\left (b^3 B-4 A b^2 c+2 a b B c+4 a A c^2\right ) x^2}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 a b B-A \left (b^2+2 a c\right )\right ) \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 = 0.17, size = 233, normalized size = 1.26 \begin {gather*} \frac {1}{4} \left (\frac {-b^4 B+A b^3 c+2 a b c^2 \left (A-3 B x^2\right )+4 a c^2 \left (-4 a B+A c x^2\right )+b^2 c \left (5 a B+2 A c x^2\right )}{c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {2 a^2 B c+b^2 (-b B+A c) x^2+a \left (-b^2 B-2 A c^2 x^2+b c \left (A+3 B x^2\right )\right )}{c^2 \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {4 \left (-3 a b B+A \left (b^2+2 a c\right )\right ) \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^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((-(b^4*B) + A*b^3*c + 2*a*b*c^2*(A - 3*B*x^2) + 4*a*c^2*(-4*a*B + A*c*x^2) + b^2*c*(5*a*B + 2*A*c*x^2))/(c^2*
(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^2*B*c + b^2*(-(b*B) + A*c)*x^2 + a*(-(b^2*B) - 2*A*c^2*x^2 + b*c*(
A + 3*B*x^2)))/(c^2*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (4*(-3*a*b*B + A*(b^2 + 2*a*c))*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 [A]
time = 0.07, size = 303, normalized size = 1.64

method	result	size
default	\(\frac {\frac {c \left (2 a c A +A \,b^{2}-3 a b B \right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {\left (6 A a b \,c^{2}+3 A \,b^{3} c -16 a^{2} B \,c^{2}-a \,b^{2} B c -b^{4} B \right ) x^{4}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (2 c^{2} a A -5 A \,b^{2} c +5 a b B c +b^{3} B \right ) x^{2}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (6 b c A -8 a c B -b^{2} B \right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (2 a c A +A \,b^{2}-3 a b B \right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\)	\(303\)
risch	\(\frac {\frac {c \left (2 a c A +A \,b^{2}-3 a b B \right ) x^{6}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {\left (6 A a b \,c^{2}+3 A \,b^{3} c -16 a^{2} B \,c^{2}-a \,b^{2} B c -b^{4} B \right ) x^{4}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (2 c^{2} a A -5 A \,b^{2} c +5 a b B c +b^{3} B \right ) x^{2}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (6 b c A -8 a c B -b^{2} B \right )}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) a c A}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) A \,b^{2}}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right ) a b B}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) a c A}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) A \,b^{2}}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right ) a b B}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\)	\(682\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(c*(2*A*a*c+A*b^2-3*B*a*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*(6*A*a*b*c^2+3*A*b^3*c-16*B*a^2*c^2-B*a*b^2*
c-B*b^4)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-a/c*(2*A*a*c^2-5*A*b^2*c+5*B*a*b*c+B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)
*x^2+1/2*a^2*(6*A*b*c-8*B*a*c-B*b^2)/c/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+(2*A*a*c+A*b^2-3*B*a*b)/(
16*a^2*c^2-8*a*b^2*c+b^4)/(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^5*(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. 672 vs. \(2 (177) = 354\).
time = 0.40, size = 1369, normalized size = 7.40 \begin {gather*} \left [-\frac {B a^{2} b^{4} + 2 \, {\left (8 \, A a^{2} c^{4} - 2 \, {\left (6 \, B a^{2} b - A a b^{2}\right )} c^{3} + {\left (3 \, B a b^{3} - A b^{4}\right )} c^{2}\right )} x^{6} + {\left (B b^{6} - 8 \, {\left (8 \, B a^{3} - 3 \, A a^{2} b\right )} c^{3} + 6 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} c^{2} - 3 \, {\left (B a b^{4} + A b^{5}\right )} c\right )} x^{4} - 8 \, {\left (4 \, B a^{4} - 3 \, A a^{3} b\right )} c^{2} + 2 \, {\left (B a b^{5} - 8 \, A a^{3} c^{3} - 2 \, {\left (10 \, B a^{3} b - 11 \, A a^{2} b^{2}\right )} c^{2} + {\left (B a^{2} b^{3} - 5 \, A a b^{4}\right )} c\right )} x^{2} - 2 \, {\left ({\left (2 \, A a c^{4} - {\left (3 \, B a b - A b^{2}\right )} c^{3}\right )} x^{8} + 2 \, {\left (2 \, A a b c^{3} - {\left (3 \, B a b^{2} - A b^{3}\right )} c^{2}\right )} x^{6} + 2 \, A a^{3} c^{2} + {\left (4 \, A a^{2} c^{3} - 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} c^{2} - {\left (3 \, B a b^{3} - A b^{4}\right )} c\right )} x^{4} + 2 \, {\left (2 \, A a^{2} b c^{2} - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} x^{2} - {\left (3 \, B a^{3} b - A a^{2} b^{2}\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 (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} c}{4 \, {\left (a^{2} b^{6} c - 12 \, a^{3} b^{4} c^{2} + 48 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{8} + 2 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{6} + {\left (b^{8} c - 10 \, a b^{6} c^{2} + 24 \, a^{2} b^{4} c^{3} + 32 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5}\right )} x^{4} + 2 \, {\left (a b^{7} c - 12 \, a^{2} b^{5} c^{2} + 48 \, a^{3} b^{3} c^{3} - 64 \, a^{4} b c^{4}\right )} x^{2}\right )}}, -\frac {B a^{2} b^{4} + 2 \, {\left (8 \, A a^{2} c^{4} - 2 \, {\left (6 \, B a^{2} b - A a b^{2}\right )} c^{3} + {\left (3 \, B a b^{3} - A b^{4}\right )} c^{2}\right )} x^{6} + {\left (B b^{6} - 8 \, {\left (8 \, B a^{3} - 3 \, A a^{2} b\right )} c^{3} + 6 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} c^{2} - 3 \, {\left (B a b^{4} + A b^{5}\right )} c\right )} x^{4} - 8 \, {\left (4 \, B a^{4} - 3 \, A a^{3} b\right )} c^{2} + 2 \, {\left (B a b^{5} - 8 \, A a^{3} c^{3} - 2 \, {\left (10 \, B a^{3} b - 11 \, A a^{2} b^{2}\right )} c^{2} + {\left (B a^{2} b^{3} - 5 \, A a b^{4}\right )} c\right )} x^{2} + 4 \, {\left ({\left (2 \, A a c^{4} - {\left (3 \, B a b - A b^{2}\right )} c^{3}\right )} x^{8} + 2 \, {\left (2 \, A a b c^{3} - {\left (3 \, B a b^{2} - A b^{3}\right )} c^{2}\right )} x^{6} + 2 \, A a^{3} c^{2} + {\left (4 \, A a^{2} c^{3} - 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} c^{2} - {\left (3 \, B a b^{3} - A b^{4}\right )} c\right )} x^{4} + 2 \, {\left (2 \, A a^{2} b c^{2} - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} x^{2} - {\left (3 \, B a^{3} b - A a^{2} b^{2}\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 (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} c}{4 \, {\left (a^{2} b^{6} c - 12 \, a^{3} b^{4} c^{2} + 48 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{8} + 2 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{6} + {\left (b^{8} c - 10 \, a b^{6} c^{2} + 24 \, a^{2} b^{4} c^{3} + 32 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5}\right )} x^{4} + 2 \, {\left (a b^{7} c - 12 \, a^{2} b^{5} c^{2} + 48 \, a^{3} b^{3} c^{3} - 64 \, a^{4} b c^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/4*(B*a^2*b^4 + 2*(8*A*a^2*c^4 - 2*(6*B*a^2*b - A*a*b^2)*c^3 + (3*B*a*b^3 - A*b^4)*c^2)*x^6 + (B*b^6 - 8*(8
*B*a^3 - 3*A*a^2*b)*c^3 + 6*(2*B*a^2*b^2 + A*a*b^3)*c^2 - 3*(B*a*b^4 + A*b^5)*c)*x^4 - 8*(4*B*a^4 - 3*A*a^3*b)
*c^2 + 2*(B*a*b^5 - 8*A*a^3*c^3 - 2*(10*B*a^3*b - 11*A*a^2*b^2)*c^2 + (B*a^2*b^3 - 5*A*a*b^4)*c)*x^2 - 2*((2*A
*a*c^4 - (3*B*a*b - A*b^2)*c^3)*x^8 + 2*(2*A*a*b*c^3 - (3*B*a*b^2 - A*b^3)*c^2)*x^6 + 2*A*a^3*c^2 + (4*A*a^2*c
^3 - 2*(3*B*a^2*b - 2*A*a*b^2)*c^2 - (3*B*a*b^3 - A*b^4)*c)*x^4 + 2*(2*A*a^2*b*c^2 - (3*B*a^2*b^2 - A*a*b^3)*c
)*x^2 - (3*B*a^3*b - A*a^2*b^2)*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*(2*B*a^3*b^2 - 3*A*a^2*b^3)*c)/(a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^
4*b^2*c^3 - 64*a^5*c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^8 + 2*(b^7*c^2 - 12*a*b^5*c^
3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + (b^8*c - 10*a*b^6*c^2 + 24*a^2*b^4*c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5
)*x^4 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*x^2), -1/4*(B*a^2*b^4 + 2*(8*A*a^2*c^4 -
2*(6*B*a^2*b - A*a*b^2)*c^3 + (3*B*a*b^3 - A*b^4)*c^2)*x^6 + (B*b^6 - 8*(8*B*a^3 - 3*A*a^2*b)*c^3 + 6*(2*B*a^2
*b^2 + A*a*b^3)*c^2 - 3*(B*a*b^4 + A*b^5)*c)*x^4 - 8*(4*B*a^4 - 3*A*a^3*b)*c^2 + 2*(B*a*b^5 - 8*A*a^3*c^3 - 2*
(10*B*a^3*b - 11*A*a^2*b^2)*c^2 + (B*a^2*b^3 - 5*A*a*b^4)*c)*x^2 + 4*((2*A*a*c^4 - (3*B*a*b - A*b^2)*c^3)*x^8
+ 2*(2*A*a*b*c^3 - (3*B*a*b^2 - A*b^3)*c^2)*x^6 + 2*A*a^3*c^2 + (4*A*a^2*c^3 - 2*(3*B*a^2*b - 2*A*a*b^2)*c^2 -
(3*B*a*b^3 - A*b^4)*c)*x^4 + 2*(2*A*a^2*b*c^2 - (3*B*a^2*b^2 - A*a*b^3)*c)*x^2 - (3*B*a^3*b - A*a^2*b^2)*c)*s
qrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + 2*(2*B*a^3*b^2 - 3*A*a^2*b^3)*c)/(
a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5*c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c
^6)*x^8 + 2*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + (b^8*c - 10*a*b^6*c^2 + 24*a^2*b^4*
c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5)*x^4 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*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**5*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Giac [A]
time = 4.39, size = 268, normalized size = 1.45 \begin {gather*} -\frac {{\left (3 \, B a b - A b^{2} - 2 \, A a c\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 {6 \, B a b c^{2} x^{6} - 2 \, A b^{2} c^{2} x^{6} - 4 \, A a c^{3} x^{6} + B b^{4} x^{4} + B a b^{2} c x^{4} - 3 \, A b^{3} c x^{4} + 16 \, B a^{2} c^{2} x^{4} - 6 \, A a b c^{2} x^{4} + 2 \, B a b^{3} x^{2} + 10 \, B a^{2} b c x^{2} - 10 \, A a b^{2} c x^{2} + 4 \, A a^{2} c^{2} x^{2} + B a^{2} b^{2} + 8 \, B a^{3} c - 6 \, A a^{2} b c}{4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\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^5*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

-(3*B*a*b - A*b^2 - 2*A*a*c)*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*(6*B*a*b*c^2*x^6 - 2*A*b^2*c^2*x^6 - 4*A*a*c^3*x^6 + B*b^4*x^4 + B*a*b^2*c*x^4 - 3*A*b^3*c*x
^4 + 16*B*a^2*c^2*x^4 - 6*A*a*b*c^2*x^4 + 2*B*a*b^3*x^2 + 10*B*a^2*b*c*x^2 - 10*A*a*b^2*c*x^2 + 4*A*a^2*c^2*x^
2 + B*a^2*b^2 + 8*B*a^3*c - 6*A*a^2*b*c)/((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*(c*x^4 + b*x^2 + a)^2)

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Mupad [B]
time = 0.68, size = 625, normalized size = 3.38 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {\left (A\,b^2\,c^2-3\,B\,a\,b\,c^2+2\,A\,a\,c^3\right )\,\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,{\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}^2\,\left (32\,a^2\,b\,c^4-16\,a\,b^3\,c^3+2\,b^5\,c^2\right )}{2\,a\,{\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 {2\,b\,c^2\,{\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\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 )}{8\,A^2\,a^2\,c^4+8\,A^2\,a\,b^2\,c^3+2\,A^2\,b^4\,c^2-24\,A\,B\,a^2\,b\,c^3-12\,A\,B\,a\,b^3\,c^2+18\,B^2\,a^2\,b^2\,c^2}\right )\,\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {x^4\,\left (16\,B\,a^2\,c^2+B\,a\,b^2\,c-6\,A\,a\,b\,c^2+B\,b^4-3\,A\,b^3\,c\right )}{4\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {c\,x^6\,\left (A\,b^2-3\,B\,a\,b+2\,A\,a\,c\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,\left (8\,B\,c\,a^2+B\,a\,b^2-6\,A\,c\,a\,b\right )}{4\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (5\,B\,a^2\,b\,c+2\,A\,a^2\,c^2+B\,a\,b^3-5\,A\,a\,b^2\,c\right )}{2\,c\,\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^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x)

[Out]

(atan(((x^2*(((A*b^2*c^2 + 2*A*a*c^3 - 3*B*a*b*c^2)*(A*b^2 + 2*A*a*c - 3*B*a*b))/(a*(4*a*c - b^2)^(9/2)*(b^4 +
16*a^2*c^2 - 8*a*b^2*c)) + (b*(A*b^2 + 2*A*a*c - 3*B*a*b)^2*(2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4))/(2*a*(
4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))) + (2*b*c^2*(A*b^2 + 2*A*a*c - 3*B*a*b)^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))/(8*A^2*a^2*c^4 + 2*A^2*
b^4*c^2 + 18*B^2*a^2*b^2*c^2 + 8*A^2*a*b^2*c^3 - 12*A*B*a*b^3*c^2 - 24*A*B*a^2*b*c^3))*(A*b^2 + 2*A*a*c - 3*B*
a*b))/(4*a*c - b^2)^(5/2) - ((x^4*(B*b^4 + 16*B*a^2*c^2 - 3*A*b^3*c - 6*A*a*b*c^2 + B*a*b^2*c))/(4*c*(b^4 + 16
*a^2*c^2 - 8*a*b^2*c)) - (c*x^6*(A*b^2 + 2*A*a*c - 3*B*a*b))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*(B*a*b^2
+ 8*B*a^2*c - 6*A*a*b*c))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(2*A*a^2*c^2 + B*a*b^3 - 5*A*a*b^2*c + 5
*B*a^2*b*c))/(2*c*(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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