3.2.3 \(\int \frac {x^3 (A+B x^2)}{a+b x^2+c x^4} \, dx\) [103]

Optimal. Leaf size=97 \[ \frac {B x^2}{2 c}-\frac {\left (b^2 B-A b c-2 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1265, 787, 648, 632, 212, 642} \begin {gather*} -\frac {\left (-2 a B c-A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac {B x^2}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.05, size = 93, normalized size = 0.96 \begin {gather*} \frac {2 B c x^2+\frac {2 \left (b^2 B-A b c-2 a B c\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+(-b B+A c) \log \left (a+b x^2+c x^4\right )}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 98, normalized size = 1.01

method	result	size
default	\(\frac {B \,x^{2}}{2 c}+\frac {\frac {\left (A c -b B \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-a B -\frac {\left (A c -b B \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c}\)	\(98\)
risch	\(\frac {B \,x^{2}}{2 c}+\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B -\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) a A}{4 a c -b^{2}}-\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B -\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) A \,b^{2}}{4 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B -\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) a b B}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B -\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) b^{3} B}{4 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B -\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B +\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) a A}{4 a c -b^{2}}-\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B +\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) A \,b^{2}}{4 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B +\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) a b B}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B +\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) b^{3} B}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (4 A a b \,c^{2}-A \,b^{3} c +8 a^{2} B \,c^{2}-6 a \,b^{2} B c +b^{4} B +\sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (b c A +2 a c B -b^{2} B \right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}\)	\(1398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*x^2/c+1/2/c*(1/2*(A*c-B*b)/c*ln(c*x^4+b*x^2+a)+2*(-a*B-1/2*(A*c-B*b)*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^3*(B*x^2+A)/(c*x^4+b*x^2+a),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 [A]
time = 0.37, size = 312, normalized size = 3.22 \begin {gather*} \left [\frac {2 \, {\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} - {\left (B b^{2} - {\left (2 \, B a + A b\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 ) - {\left (B b^{3} + 4 \, A a c^{2} - {\left (4 \, B a b + A b^{2}\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} - 2 \, {\left (B b^{2} - {\left (2 \, B a + A b\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 ) - {\left (B b^{3} + 4 \, A a c^{2} - {\left (4 \, B a b + A b^{2}\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(B*b^2*c - 4*B*a*c^2)*x^2 - (B*b^2 - (2*B*a + A*b)*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)) - (B*b^3 + 4*A*a*c^2 - (4*B*a*b + A*b^2)*c)
*log(c*x^4 + b*x^2 + a))/(b^2*c^2 - 4*a*c^3), 1/4*(2*(B*b^2*c - 4*B*a*c^2)*x^2 - 2*(B*b^2 - (2*B*a + A*b)*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)) - (B*b^3 + 4*A*a*c^2 - (4*B*a*b + A*
b^2)*c)*log(c*x^4 + b*x^2 + a))/(b^2*c^2 - 4*a*c^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (90) = 180\).
time = 71.09, size = 434, normalized size = 4.47 \begin {gather*} \frac {B x^{2}}{2 c} + \left (- \frac {- A c + B b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {2 A a c - B a b - 8 a c^{2} \left (- \frac {- A c + B b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac {- A c + B b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{A b c + 2 B a c - B b^{2}} \right )} + \left (- \frac {- A c + B b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {2 A a c - B a b - 8 a c^{2} \left (- \frac {- A c + B b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac {- A c + B b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{A b c + 2 B a c - B b^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x**2+A)/(c*x**4+b*x**2+a),x)

[Out]

B*x**2/(2*c) + (-(-A*c + B*b)/(4*c**2) - sqrt(-4*a*c + b**2)*(A*b*c + 2*B*a*c - B*b**2)/(4*c**2*(4*a*c - b**2)
))*log(x**2 + (2*A*a*c - B*a*b - 8*a*c**2*(-(-A*c + B*b)/(4*c**2) - sqrt(-4*a*c + b**2)*(A*b*c + 2*B*a*c - B*b
**2)/(4*c**2*(4*a*c - b**2))) + 2*b**2*c*(-(-A*c + B*b)/(4*c**2) - sqrt(-4*a*c + b**2)*(A*b*c + 2*B*a*c - B*b*
*2)/(4*c**2*(4*a*c - b**2))))/(A*b*c + 2*B*a*c - B*b**2)) + (-(-A*c + B*b)/(4*c**2) + sqrt(-4*a*c + b**2)*(A*b
*c + 2*B*a*c - B*b**2)/(4*c**2*(4*a*c - b**2)))*log(x**2 + (2*A*a*c - B*a*b - 8*a*c**2*(-(-A*c + B*b)/(4*c**2)
+ sqrt(-4*a*c + b**2)*(A*b*c + 2*B*a*c - B*b**2)/(4*c**2*(4*a*c - b**2))) + 2*b**2*c*(-(-A*c + B*b)/(4*c**2)
+ sqrt(-4*a*c + b**2)*(A*b*c + 2*B*a*c - B*b**2)/(4*c**2*(4*a*c - b**2))))/(A*b*c + 2*B*a*c - B*b**2))

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Giac [A]
time = 3.40, size = 91, normalized size = 0.94 \begin {gather*} \frac {B x^{2}}{2 \, c} - \frac {{\left (B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac {{\left (B b^{2} - 2 \, B a c - A b c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.65, size = 979, normalized size = 10.09 \begin {gather*} \frac {B\,x^2}{2\,c}+\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {\mathrm {atan}\left (\frac {2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {\left (\frac {8\,A\,a\,c^3-8\,B\,a\,b\,c^2}{c^2}-\frac {8\,a\,c^2\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}-\frac {a\,\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+x^2\,\left (\frac {\frac {\left (\frac {-6\,B\,b^2\,c^2+6\,A\,b\,c^3+4\,B\,a\,c^3}{c^2}-\frac {4\,b\,c^2\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}-\frac {b\,\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+\frac {b\,\left (\frac {\left (\frac {-6\,B\,b^2\,c^2+6\,A\,b\,c^3+4\,B\,a\,c^3}{c^2}-\frac {4\,b\,c^2\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {A^2\,b\,c^2-2\,A\,B\,b^2\,c+a\,A\,B\,c^2+B^2\,b^3-a\,B^2\,b\,c}{c^2}+\frac {b\,{\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )}^2}{2\,c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {\left (\frac {8\,A\,a\,c^3-8\,B\,a\,b\,c^2}{c^2}-\frac {8\,a\,c^2\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (2\,B\,b^3-2\,A\,b^2\,c-8\,B\,a\,b\,c+8\,A\,a\,c^2\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {a\,A^2\,c^2-2\,a\,A\,B\,b\,c+a\,B^2\,b^2}{c^2}+\frac {a\,{\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )}^2}{c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{A^2\,b^2\,c^2+4\,A\,B\,a\,b\,c^2-2\,A\,B\,b^3\,c+4\,B^2\,a^2\,c^2-4\,B^2\,a\,b^2\,c+B^2\,b^4}\right )\,\left (-B\,b^2+A\,c\,b+2\,B\,a\,c\right )}{2\,c^2\,\sqrt {4\,a\,c-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*x^2)/(2*c) + (log(a + b*x^2 + c*x^4)*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/(2*(16*a*c^3 - 4*b^2*c^
2)) - (atan((2*c^2*(4*a*c - b^2)*(((((8*A*a*c^3 - 8*B*a*b*c^2)/c^2 - (8*a*c^2*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c
- 8*B*a*b*c))/(16*a*c^3 - 4*b^2*c^2))*(A*b*c - B*b^2 + 2*B*a*c))/(8*c^2*(4*a*c - b^2)^(1/2)) - (a*(A*b*c - B*
b^2 + 2*B*a*c)*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/((4*a*c - b^2)^(1/2)*(16*a*c^3 - 4*b^2*c^2)))/a
+ x^2*(((((6*A*b*c^3 - 6*B*b^2*c^2 + 4*B*a*c^3)/c^2 - (4*b*c^2*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/
(16*a*c^3 - 4*b^2*c^2))*(A*b*c - B*b^2 + 2*B*a*c))/(8*c^2*(4*a*c - b^2)^(1/2)) - (b*(A*b*c - B*b^2 + 2*B*a*c)*
(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/(2*(4*a*c - b^2)^(1/2)*(16*a*c^3 - 4*b^2*c^2)))/a + (b*((((6*A*
b*c^3 - 6*B*b^2*c^2 + 4*B*a*c^3)/c^2 - (4*b*c^2*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/(16*a*c^3 - 4*b
^2*c^2))*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/(2*(16*a*c^3 - 4*b^2*c^2)) - (B^2*b^3 + A^2*b*c^2 + A*
B*a*c^2 - 2*A*B*b^2*c - B^2*a*b*c)/c^2 + (b*(A*b*c - B*b^2 + 2*B*a*c)^2)/(2*c^2*(4*a*c - b^2))))/(2*a*(4*a*c -
b^2)^(1/2))) + (b*((((8*A*a*c^3 - 8*B*a*b*c^2)/c^2 - (8*a*c^2*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/
(16*a*c^3 - 4*b^2*c^2))*(2*B*b^3 + 8*A*a*c^2 - 2*A*b^2*c - 8*B*a*b*c))/(2*(16*a*c^3 - 4*b^2*c^2)) - (A^2*a*c^2
+ B^2*a*b^2 - 2*A*B*a*b*c)/c^2 + (a*(A*b*c - B*b^2 + 2*B*a*c)^2)/(c^2*(4*a*c - b^2))))/(2*a*(4*a*c - b^2)^(1/
2))))/(B^2*b^4 + A^2*b^2*c^2 + 4*B^2*a^2*c^2 - 2*A*B*b^3*c - 4*B^2*a*b^2*c + 4*A*B*a*b*c^2))*(A*b*c - B*b^2 +
2*B*a*c))/(2*c^2*(4*a*c - b^2)^(1/2))

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