3.4.67 \(\int \frac {x^3 (d+e x^2)^{3/2}}{a+b x^2+c x^4} \, dx\) [367]

Optimal. Leaf size=460 \[ \frac {(c d-b e) \sqrt {d+e x^2}}{c^2}+\frac {\left (d+e x^2\right )^{3/2}}{3 c}+\frac {\left (b^3 e^2-b^2 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )\right )+b c \left (c d^2+e \left (2 \sqrt {b^2-4 a c} d-3 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b^3 e^2-b^2 e \left (2 c d-\sqrt {b^2-4 a c} e\right )+b c \left (c d^2-e \left (2 \sqrt {b^2-4 a c} d+3 a e\right )\right )-c \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

1/3*(e*x^2+d)^(3/2)/c+(-b*e+c*d)*(e*x^2+d)^(1/2)/c^2+1/2*arctanh(2^(1/2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b-(
-4*a*c+b^2)^(1/2)))^(1/2))*(b^3*e^2-b^2*e*(2*c*d+e*(-4*a*c+b^2)^(1/2))+c*(a*e^2*(-4*a*c+b^2)^(1/2)-c*d*(-4*a*e
+d*(-4*a*c+b^2)^(1/2)))+b*c*(c*d^2+e*(-3*a*e+2*d*(-4*a*c+b^2)^(1/2))))/c^(5/2)*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c
*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*arctanh(2^(1/2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))
)^(1/2))*(b^3*e^2-b^2*e*(2*c*d-e*(-4*a*c+b^2)^(1/2))-c*(a*e^2*(-4*a*c+b^2)^(1/2)-c*d*(4*a*e+d*(-4*a*c+b^2)^(1/
2)))+b*c*(c*d^2-e*(3*a*e+2*d*(-4*a*c+b^2)^(1/2))))/c^(5/2)*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)
^(1/2)))^(1/2)

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Rubi [A]
time = 3.58, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1265, 838, 840, 1180, 214} \begin {gather*} \frac {\left (b c \left (e \left (2 d \sqrt {b^2-4 a c}-3 a e\right )+c d^2\right )+c \left (a e^2 \sqrt {b^2-4 a c}-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )\right )-b^2 e \left (e \sqrt {b^2-4 a c}+2 c d\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (b c \left (c d^2-e \left (2 d \sqrt {b^2-4 a c}+3 a e\right )\right )-c \left (a e^2 \sqrt {b^2-4 a c}-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )\right )-b^2 e \left (2 c d-e \sqrt {b^2-4 a c}\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d+e x^2} (c d-b e)}{c^2}+\frac {\left (d+e x^2\right )^{3/2}}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d - b*e)*Sqrt[d + e*x^2])/c^2 + (d + e*x^2)^(3/2)/(3*c) + ((b^3*e^2 - b^2*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e)
+ c*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e)) + b*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 3*
a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*Sq
rt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b^3*e^2 - b^2*e*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b
*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 3*a*e)) - c*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e
)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*Sqrt[
b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 214

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

Rule 838

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

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 1180

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.60, size = 501, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {c} \sqrt {d+e x^2} \left (4 c d-3 b e+c e x^2\right )+\frac {3 \left (i b^3 e^2+b^2 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b c \left (c d^2+e \left (2 i \sqrt {-b^2+4 a c} d-3 a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {3 \left (-i b^3 e^2+b^2 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+b c \left (-i c d^2+e \left (-2 \sqrt {-b^2+4 a c} d+3 i a e\right )\right )+c \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-4 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{6 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[d + e*x^2]*(4*c*d - 3*b*e + c*e*x^2) + (3*(I*b^3*e^2 + b^2*e*((-2*I)*c*d + Sqrt[-b^2 + 4*a*c]*
e) + I*b*c*(c*d^2 + e*((2*I)*Sqrt[-b^2 + 4*a*c]*d - 3*a*e)) + c*(-(a*Sqrt[-b^2 + 4*a*c]*e^2) + c*d*(Sqrt[-b^2
+ 4*a*c]*d + (4*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]
])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (3*((-I)*b^3*e^2 + b^2*e*((2*I)*c*d
+ Sqrt[-b^2 + 4*a*c]*e) + b*c*((-I)*c*d^2 + e*(-2*Sqrt[-b^2 + 4*a*c]*d + (3*I)*a*e)) + c*(-(a*Sqrt[-b^2 + 4*a*
c]*e^2) + c*d*(Sqrt[-b^2 + 4*a*c]*d - (4*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e
+ I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(6*c^(5/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 444, normalized size = 0.97

method	result	size
risch	\(-\frac {\left (-c e \,x^{2}+3 e b -4 c d \right ) \sqrt {e \,x^{2}+d}}{3 c^{2}}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (\left (a c \,e^{2}-e^{2} b^{2}+2 b c d e -c^{2} d^{2}\right ) \textit {\_R}^{6}+\left (-4 e^{3} a b +5 a c d \,e^{2}+3 b^{2} d \,e^{2}-6 d^{2} e b c +3 c^{2} d^{3}\right ) \textit {\_R}^{4}+d \left (4 e^{3} a b -5 a c d \,e^{2}-3 b^{2} d \,e^{2}+6 d^{2} e b c -3 c^{2} d^{3}\right ) \textit {\_R}^{2}-a c \,d^{3} e^{2}+b^{2} d^{3} e^{2}-2 b c e \,d^{4}+c^{2} d^{5}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 c^{2}}\)	\(348\)
default	\(-\frac {-\frac {\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{3} c}{3}+4 e b \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )-5 c d \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{8 c^{2}}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (\left (a c \,e^{2}-e^{2} b^{2}+2 b c d e -c^{2} d^{2}\right ) \textit {\_R}^{6}+\left (-4 e^{3} a b +5 a c d \,e^{2}+3 b^{2} d \,e^{2}-6 d^{2} e b c +3 c^{2} d^{3}\right ) \textit {\_R}^{4}+d \left (4 e^{3} a b -5 a c d \,e^{2}-3 b^{2} d \,e^{2}+6 d^{2} e b c -3 c^{2} d^{3}\right ) \textit {\_R}^{2}-a c \,d^{3} e^{2}+b^{2} d^{3} e^{2}-2 b c e \,d^{4}+c^{2} d^{5}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 c^{2}}+\frac {d^{3}}{24 c \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{3}}-\frac {d \left (4 e b -5 c d \right )}{8 c^{2} \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}\)	\(444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/c^2*(-1/3*((e*x^2+d)^(1/2)-e^(1/2)*x)^3*c+4*e*b*((e*x^2+d)^(1/2)-e^(1/2)*x)-5*c*d*((e*x^2+d)^(1/2)-e^(1/2
)*x))-1/4/c^2*sum(((a*c*e^2-b^2*e^2+2*b*c*d*e-c^2*d^2)*_R^6+(-4*a*b*e^3+5*a*c*d*e^2+3*b^2*d*e^2-6*b*c*d^2*e+3*
c^2*d^3)*_R^4+d*(4*a*b*e^3-5*a*c*d*e^2-3*b^2*d*e^2+6*b*c*d^2*e-3*c^2*d^3)*_R^2-a*c*d^3*e^2+b^2*d^3*e^2-2*b*c*e
*d^4+c^2*d^5)/(_R^7*c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_R^3*c*d^2+_R*b*d^2*e-_R*c*d^3)*ln((e*
x^2+d)^(1/2)-e^(1/2)*x-_R),_R=RootOf(c*_Z^8+(4*b*e-4*c*d)*_Z^6+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^4+(4*b*d^2*e-4*c*
d^3)*_Z^2+d^4*c))+1/24/c*d^3/((e*x^2+d)^(1/2)-e^(1/2)*x)^3-1/8/c^2*d*(4*b*e-5*c*d)/((e*x^2+d)^(1/2)-e^(1/2)*x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate((x^2*e + d)^(3/2)*x^3/(c*x^4 + b*x^2 + a), x)

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Fricas [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^3*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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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**3*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (414) = 828\).
time = 4.96, size = 857, normalized size = 1.86 \begin {gather*} -\frac {{\left (2 \, b c^{5} d^{3} - {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d^{2} e + {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} c^{2} + 2 \, {\left (2 \, b^{3} c^{3} - 5 \, a b c^{4}\right )} d e^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{3} - 2 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{3} + {\left (b^{2} c^{2} + a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{2}\right )} {\left | c \right |} - {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{3}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c^{4} d - b c^{3} e + \sqrt {-4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{4} + {\left (2 \, c^{4} d - b c^{3} e\right )}^{2}}}{c^{4}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d - {\left (b^{2} c^{2} - 4 \, a c^{3} + \sqrt {b^{2} - 4 \, a c} b c^{2}\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2}} + \frac {{\left (2 \, b c^{5} d^{3} - {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d^{2} e + {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} c^{2} + 2 \, {\left (2 \, b^{3} c^{3} - 5 \, a b c^{4}\right )} d e^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{3} - 2 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{3} + {\left (b^{2} c^{2} + a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{2}\right )} {\left | c \right |} - {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{3}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c^{4} d - b c^{3} e - \sqrt {-4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{4} + {\left (2 \, c^{4} d - b c^{3} e\right )}^{2}}}{c^{4}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d + {\left (b^{2} c^{2} - 4 \, a c^{3} - \sqrt {b^{2} - 4 \, a c} b c^{2}\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2}} + \frac {{\left (x^{2} e + d\right )}^{\frac {3}{2}} c^{2} + 3 \, \sqrt {x^{2} e + d} c^{2} d - 3 \, \sqrt {x^{2} e + d} b c e}{3 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*c^5*d^3 - (5*b^2*c^4 - 8*a*c^5)*d^2*e + ((b^2*c^2 - 4*a*c^3)*d^2*e - 2*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4
- 5*a*b^2*c + 4*a^2*c^2)*e^3)*c^2 + 2*(2*b^3*c^3 - 5*a*b*c^4)*d*e^2 - 2*(sqrt(b^2 - 4*a*c)*c^4*d^3 - 2*sqrt(b^
2 - 4*a*c)*b*c^3*d^2*e - sqrt(b^2 - 4*a*c)*a*b*c^2*e^3 + (b^2*c^2 + a*c^3)*sqrt(b^2 - 4*a*c)*d*e^2)*abs(c) - (
b^4*c^2 - 3*a*b^2*c^3)*e^3)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c^4*d - b*c^3*e + sqrt(-4*(c^4*d^2 - b
*c^3*d*e + a*c^3*e^2)*c^4 + (2*c^4*d - b*c^3*e)^2))/c^4))/((2*sqrt(b^2 - 4*a*c)*c^3*d - (b^2*c^2 - 4*a*c^3 + s
qrt(b^2 - 4*a*c)*b*c^2)*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2) + (2*b*c^5*d^3 - (5*b^2*c^4 -
8*a*c^5)*d^2*e + ((b^2*c^2 - 4*a*c^3)*d^2*e - 2*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^3
)*c^2 + 2*(2*b^3*c^3 - 5*a*b*c^4)*d*e^2 + 2*(sqrt(b^2 - 4*a*c)*c^4*d^3 - 2*sqrt(b^2 - 4*a*c)*b*c^3*d^2*e - sqr
t(b^2 - 4*a*c)*a*b*c^2*e^3 + (b^2*c^2 + a*c^3)*sqrt(b^2 - 4*a*c)*d*e^2)*abs(c) - (b^4*c^2 - 3*a*b^2*c^3)*e^3)*
arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c^4*d - b*c^3*e - sqrt(-4*(c^4*d^2 - b*c^3*d*e + a*c^3*e^2)*c^4 +
(2*c^4*d - b*c^3*e)^2))/c^4))/((2*sqrt(b^2 - 4*a*c)*c^3*d + (b^2*c^2 - 4*a*c^3 - sqrt(b^2 - 4*a*c)*b*c^2)*e)*s
qrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2) + 1/3*((x^2*e + d)^(3/2)*c^2 + 3*sqrt(x^2*e + d)*c^2*d -
3*sqrt(x^2*e + d)*b*c*e)/c^3

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Mupad [B]
time = 3.41, size = 2500, normalized size = 5.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x)

[Out]

(d + e*x^2)^(3/2)/(3*c) - atan(((((4*a*b^3*c^3*e^5 - 16*a^2*b*c^4*e^5 + 16*a*c^6*d^3*e^2 + 16*a^2*c^5*d*e^4 -
4*b^4*c^3*d*e^4 - 4*b^2*c^5*d^3*e^2 + 8*b^3*c^4*d^2*e^3 - 32*a*b*c^5*d^2*e^3 + 12*a*b^2*c^4*d*e^4)/c^3 - (2*(d
+ e*x^2)^(1/2)*(-(((4*b^7*e^3 - 32*a^2*c^5*d^3 - 4*b^4*c^3*d^3 + 24*a*b^2*c^4*d^3 - 80*a^3*b*c^3*e^3 + 96*a^3
*c^4*d*e^2 + 12*b^5*c^2*d^2*e + 100*a^2*b^3*c^2*e^3 - 36*a*b^5*c*e^3 - 12*b^6*c*d*e^2 - 84*a*b^3*c^3*d^2*e + 9
6*a*b^4*c^2*d*e^2 + 144*a^2*b*c^4*d^2*e - 216*a^2*b^2*c^3*d*e^2)^2/4 - (256*a^2*c^7 + 16*b^4*c^5 - 128*a*b^2*c
^6)*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 - a^2*b^3*d^3*e^3 + 3*a^3*b^2*d^2*e^4 + 3*a^3*c^2*d^4*e^2 - 3*a^4
*b*d*e^5 - 3*a^2*b*c^2*d^5*e - 6*a^3*b*c*d^3*e^3 + 3*a^2*b^2*c*d^4*e^2))^(1/2) + 2*b^7*e^3 - 16*a^2*c^5*d^3 -
2*b^4*c^3*d^3 + 12*a*b^2*c^4*d^3 - 40*a^3*b*c^3*e^3 + 48*a^3*c^4*d*e^2 + 6*b^5*c^2*d^2*e + 50*a^2*b^3*c^2*e^3
- 18*a*b^5*c*e^3 - 6*b^6*c*d*e^2 - 42*a*b^3*c^3*d^2*e + 48*a*b^4*c^2*d*e^2 + 72*a^2*b*c^4*d^2*e - 108*a^2*b^2*
c^3*d*e^2)/(16*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*(4*b^3*c^5*e^3 - 8*b^2*c^6*d*e^2 - 16*a*b*c^6*e^3
+ 32*a*c^7*d*e^2))/c^3)*(-(((4*b^7*e^3 - 32*a^2*c^5*d^3 - 4*b^4*c^3*d^3 + 24*a*b^2*c^4*d^3 - 80*a^3*b*c^3*e^3
+ 96*a^3*c^4*d*e^2 + 12*b^5*c^2*d^2*e + 100*a^2*b^3*c^2*e^3 - 36*a*b^5*c*e^3 - 12*b^6*c*d*e^2 - 84*a*b^3*c^3*d
^2*e + 96*a*b^4*c^2*d*e^2 + 144*a^2*b*c^4*d^2*e - 216*a^2*b^2*c^3*d*e^2)^2/4 - (256*a^2*c^7 + 16*b^4*c^5 - 128
*a*b^2*c^6)*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 - a^2*b^3*d^3*e^3 + 3*a^3*b^2*d^2*e^4 + 3*a^3*c^2*d^4*e^2
- 3*a^4*b*d*e^5 - 3*a^2*b*c^2*d^5*e - 6*a^3*b*c*d^3*e^3 + 3*a^2*b^2*c*d^4*e^2))^(1/2) + 2*b^7*e^3 - 16*a^2*c^
5*d^3 - 2*b^4*c^3*d^3 + 12*a*b^2*c^4*d^3 - 40*a^3*b*c^3*e^3 + 48*a^3*c^4*d*e^2 + 6*b^5*c^2*d^2*e + 50*a^2*b^3*
c^2*e^3 - 18*a*b^5*c*e^3 - 6*b^6*c*d*e^2 - 42*a*b^3*c^3*d^2*e + 48*a*b^4*c^2*d*e^2 + 72*a^2*b*c^4*d^2*e - 108*
a^2*b^2*c^3*d*e^2)/(16*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2) - (2*(d + e*x^2)^(1/2)*(b^6*e^6 - 2*a^3*c^
3*e^6 - 2*a*c^5*d^4*e^2 + 9*a^2*b^2*c^2*e^6 + 12*a^2*c^4*d^2*e^4 + b^2*c^4*d^4*e^2 - 4*b^3*c^3*d^3*e^3 + 6*b^4
*c^2*d^2*e^4 - 6*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 12*a*b*c^4*d^3*e^3 + 20*a*b^3*c^2*d*e^5 - 20*a^2*b*c^3*d*e^5 -
24*a*b^2*c^3*d^2*e^4))/c^3)*(-(((4*b^7*e^3 - 32*a^2*c^5*d^3 - 4*b^4*c^3*d^3 + 24*a*b^2*c^4*d^3 - 80*a^3*b*c^3*
e^3 + 96*a^3*c^4*d*e^2 + 12*b^5*c^2*d^2*e + 100*a^2*b^3*c^2*e^3 - 36*a*b^5*c*e^3 - 12*b^6*c*d*e^2 - 84*a*b^3*c
^3*d^2*e + 96*a*b^4*c^2*d*e^2 + 144*a^2*b*c^4*d^2*e - 216*a^2*b^2*c^3*d*e^2)^2/4 - (256*a^2*c^7 + 16*b^4*c^5 -
128*a*b^2*c^6)*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 - a^2*b^3*d^3*e^3 + 3*a^3*b^2*d^2*e^4 + 3*a^3*c^2*d^4
*e^2 - 3*a^4*b*d*e^5 - 3*a^2*b*c^2*d^5*e - 6*a^3*b*c*d^3*e^3 + 3*a^2*b^2*c*d^4*e^2))^(1/2) + 2*b^7*e^3 - 16*a^
2*c^5*d^3 - 2*b^4*c^3*d^3 + 12*a*b^2*c^4*d^3 - 40*a^3*b*c^3*e^3 + 48*a^3*c^4*d*e^2 + 6*b^5*c^2*d^2*e + 50*a^2*
b^3*c^2*e^3 - 18*a*b^5*c*e^3 - 6*b^6*c*d*e^2 - 42*a*b^3*c^3*d^2*e + 48*a*b^4*c^2*d*e^2 + 72*a^2*b*c^4*d^2*e -
108*a^2*b^2*c^3*d*e^2)/(16*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*1i - (((4*a*b^3*c^3*e^5 - 16*a^2*b*c^4
*e^5 + 16*a*c^6*d^3*e^2 + 16*a^2*c^5*d*e^4 - 4*b^4*c^3*d*e^4 - 4*b^2*c^5*d^3*e^2 + 8*b^3*c^4*d^2*e^3 - 32*a*b*
c^5*d^2*e^3 + 12*a*b^2*c^4*d*e^4)/c^3 + (2*(d + e*x^2)^(1/2)*(-(((4*b^7*e^3 - 32*a^2*c^5*d^3 - 4*b^4*c^3*d^3 +
24*a*b^2*c^4*d^3 - 80*a^3*b*c^3*e^3 + 96*a^3*c^4*d*e^2 + 12*b^5*c^2*d^2*e + 100*a^2*b^3*c^2*e^3 - 36*a*b^5*c*
e^3 - 12*b^6*c*d*e^2 - 84*a*b^3*c^3*d^2*e + 96*a*b^4*c^2*d*e^2 + 144*a^2*b*c^4*d^2*e - 216*a^2*b^2*c^3*d*e^2)^
2/4 - (256*a^2*c^7 + 16*b^4*c^5 - 128*a*b^2*c^6)*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 - a^2*b^3*d^3*e^3 +
3*a^3*b^2*d^2*e^4 + 3*a^3*c^2*d^4*e^2 - 3*a^4*b*d*e^5 - 3*a^2*b*c^2*d^5*e - 6*a^3*b*c*d^3*e^3 + 3*a^2*b^2*c*d^
4*e^2))^(1/2) + 2*b^7*e^3 - 16*a^2*c^5*d^3 - 2*b^4*c^3*d^3 + 12*a*b^2*c^4*d^3 - 40*a^3*b*c^3*e^3 + 48*a^3*c^4*
d*e^2 + 6*b^5*c^2*d^2*e + 50*a^2*b^3*c^2*e^3 - 18*a*b^5*c*e^3 - 6*b^6*c*d*e^2 - 42*a*b^3*c^3*d^2*e + 48*a*b^4*
c^2*d*e^2 + 72*a^2*b*c^4*d^2*e - 108*a^2*b^2*c^3*d*e^2)/(16*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*(4*b^
3*c^5*e^3 - 8*b^2*c^6*d*e^2 - 16*a*b*c^6*e^3 + 32*a*c^7*d*e^2))/c^3)*(-(((4*b^7*e^3 - 32*a^2*c^5*d^3 - 4*b^4*c
^3*d^3 + 24*a*b^2*c^4*d^3 - 80*a^3*b*c^3*e^3 + 96*a^3*c^4*d*e^2 + 12*b^5*c^2*d^2*e + 100*a^2*b^3*c^2*e^3 - 36*
a*b^5*c*e^3 - 12*b^6*c*d*e^2 - 84*a*b^3*c^3*d^2*e + 96*a*b^4*c^2*d*e^2 + 144*a^2*b*c^4*d^2*e - 216*a^2*b^2*c^3
*d*e^2)^2/4 - (256*a^2*c^7 + 16*b^4*c^5 - 128*a*b^2*c^6)*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 - a^2*b^3*d^
3*e^3 + 3*a^3*b^2*d^2*e^4 + 3*a^3*c^2*d^4*e^2 - 3*a^4*b*d*e^5 - 3*a^2*b*c^2*d^5*e - 6*a^3*b*c*d^3*e^3 + 3*a^2*
b^2*c*d^4*e^2))^(1/2) + 2*b^7*e^3 - 16*a^2*c^5*d^3 - 2*b^4*c^3*d^3 + 12*a*b^2*c^4*d^3 - 40*a^3*b*c^3*e^3 + 48*
a^3*c^4*d*e^2 + 6*b^5*c^2*d^2*e + 50*a^2*b^3*c^2*e^3 - 18*a*b^5*c*e^3 - 6*b^6*c*d*e^2 - 42*a*b^3*c^3*d^2*e + 4
8*a*b^4*c^2*d*e^2 + 72*a^2*b*c^4*d^2*e - 108*a^2*b^2*c^3*d*e^2)/(16*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/
2) + (2*(d + e*x^2)^(1/2)*(b^6*e^6 - 2*a^3*c^3*...

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