3.4.44 \(\int \frac {x^3}{(d+e x^2) (a+b x^2+c x^4)^{3/2}} \, dx\) [344]

Optimal. Leaf size=159 \[ \frac {a (2 c d-b e)+c (b d-2 a e) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {d e \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1265, 836, 12, 738, 212} \begin {gather*} \frac {c x^2 (b d-2 a e)+a (2 c d-b e)}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac {d e \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 738

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

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (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^3}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {a (2 c d-b e)+c (b d-2 a e) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\text {Subst}\left (\int \frac {\left (b^2-4 a c\right ) d e}{2 (d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac {a (2 c d-b e)+c (b d-2 a e) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {(d e) \text {Subst}\left (\int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {a (2 c d-b e)+c (b d-2 a e) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {(d e) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x^2}{\sqrt {a+b x^2+c x^4}}\right )}{c d^2-b d e+a e^2}\\ &=\frac {a (2 c d-b e)+c (b d-2 a e) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {d e \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 172, normalized size = 1.08 \begin {gather*} \frac {-b c d x^2+a \left (-2 c d+b e+2 c e x^2\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+b x^2+c x^4}}-\frac {d e \sqrt {-c d^2+e (b d-a e)} \tan ^{-1}\left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\left (c d^2+e (-b d+a e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b*c*d*x^2) + a*(-2*c*d + b*e + 2*c*e*x^2))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + b*x^2 + c*x^4
]) - (d*e*Sqrt[-(c*d^2) + e*(b*d - a*e)]*ArcTan[(Sqrt[c]*(d + e*x^2) - e*Sqrt[a + b*x^2 + c*x^4])/Sqrt[-(c*d^2
) + e*(b*d - a*e)]])/(c*d^2 + e*(-(b*d) + a*e))^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(147)=294\).
time = 0.14, size = 499, normalized size = 3.14

method	result	size
elliptic	\(-\frac {2 c d \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\)	\(478\)
default	\(\frac {2 c \,x^{2}+b}{e \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-\frac {d \left (\frac {2 c e \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {2 c \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{e}\)	\(499\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (152) = 304\).
time = 0.65, size = 1367, normalized size = 8.60 \begin {gather*} \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d x^{4} + {\left (b^{3} - 4 \, a b c\right )} d x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} d\right )} \sqrt {c d^{2} - b d e + a e^{2}} e \log \left (-\frac {8 \, c^{2} d^{2} x^{4} + 8 \, b c d^{2} x^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {c d^{2} - b d e + a e^{2}} + {\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{4} + {\left (3 \, b^{2} + 4 \, a c\right )} d x^{2} + 4 \, a b d\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) + 4 \, {\left (b c^{2} d^{3} x^{2} + 2 \, a c^{2} d^{3} - {\left (2 \, a^{2} c x^{2} + a^{2} b\right )} e^{3} + {\left (3 \, a b c d x^{2} + {\left (a b^{2} + 2 \, a^{2} c\right )} d\right )} e^{2} - {\left (3 \, a b c d^{2} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} x^{2}\right )} e\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{4} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{4} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x^{2} + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{4} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x^{2} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d x^{4} + {\left (b^{3} - 4 \, a b c\right )} d x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} d\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {-c d^{2} + b d e - a e^{2}}}{2 \, {\left (c^{2} d^{2} x^{4} + b c d^{2} x^{2} + a c d^{2} + {\left (a c x^{4} + a b x^{2} + a^{2}\right )} e^{2} - {\left (b c d x^{4} + b^{2} d x^{2} + a b d\right )} e\right )}}\right ) e - 2 \, {\left (b c^{2} d^{3} x^{2} + 2 \, a c^{2} d^{3} - {\left (2 \, a^{2} c x^{2} + a^{2} b\right )} e^{3} + {\left (3 \, a b c d x^{2} + {\left (a b^{2} + 2 \, a^{2} c\right )} d\right )} e^{2} - {\left (3 \, a b c d^{2} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} x^{2}\right )} e\right )} \sqrt {c x^{4} + b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{4} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{4} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x^{2} + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{4} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x^{2} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((b^2*c - 4*a*c^2)*d*x^4 + (b^3 - 4*a*b*c)*d*x^2 + (a*b^2 - 4*a^2*c)*d)*sqrt(c*d^2 - b*d*e + a*e^2)*e*lo
g(-(8*c^2*d^2*x^4 + 8*b*c*d^2*x^2 + (b^2 + 4*a*c)*d^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*d*x^2 + b*d - (b*x^2 +
2*a)*e)*sqrt(c*d^2 - b*d*e + a*e^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^4 + (3*b^2 +
4*a*c)*d*x^2 + 4*a*b*d)*e)/(x^4*e^2 + 2*d*x^2*e + d^2)) + 4*(b*c^2*d^3*x^2 + 2*a*c^2*d^3 - (2*a^2*c*x^2 + a^2*
b)*e^3 + (3*a*b*c*d*x^2 + (a*b^2 + 2*a^2*c)*d)*e^2 - (3*a*b*c*d^2 + (b^2*c + 2*a*c^2)*d^2*x^2)*e)*sqrt(c*x^4 +
b*x^2 + a))/((b^2*c^3 - 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3
*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x
^4 + (a*b^4 - 4*a^2*b^2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^4 +
(b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^
3)*d^3*x^4 + (b^4*c - 4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e), -1/2*(((b^2*c - 4*a*c^2)*d*x^4 +
(b^3 - 4*a*b*c)*d*x^2 + (a*b^2 - 4*a^2*c)*d)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)
*(2*c*d*x^2 + b*d - (b*x^2 + 2*a)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)/(c^2*d^2*x^4 + b*c*d^2*x^2 + a*c*d^2 + (a*c*
x^4 + a*b*x^2 + a^2)*e^2 - (b*c*d*x^4 + b^2*d*x^2 + a*b*d)*e))*e - 2*(b*c^2*d^3*x^2 + 2*a*c^2*d^3 - (2*a^2*c*x
^2 + a^2*b)*e^3 + (3*a*b*c*d*x^2 + (a*b^2 + 2*a^2*c)*d)*e^2 - (3*a*b*c*d^2 + (b^2*c + 2*a*c^2)*d^2*x^2)*e)*sqr
t(c*x^4 + b*x^2 + a))/((b^2*c^3 - 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d
^4 + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b
*c^2)*d*x^4 + (a*b^4 - 4*a^2*b^2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*
d^2*x^4 + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 -
4*a*b*c^3)*d^3*x^4 + (b^4*c - 4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (152) = 304\).
time = 2.94, size = 441, normalized size = 2.77 \begin {gather*} -\frac {d \arctan \left (-\frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right ) e}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} + \frac {\frac {{\left (b c^{2} d^{3} - b^{2} c d^{2} e - 2 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - 2 \, a^{2} c e^{3}\right )} x^{2}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac {2 \, a c^{2} d^{3} - 3 \, a b c d^{2} e + a b^{2} d e^{2} + 2 \, a^{2} c d e^{2} - a^{2} b e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}}{\sqrt {c x^{4} + b x^{2} + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*arctan(-((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))*e/((c*d^2 - b
*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2)) + ((b*c^2*d^3 - b^2*c*d^2*e - 2*a*c^2*d^2*e + 3*a*b*c*d*e^2 - 2*a^
2*c*e^3)*x^2/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 -
8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4) + (2*a*c^2*d^3 - 3*a*b*c*d^2*
e + a*b^2*d*e^2 + 2*a^2*c*d*e^2 - a^2*b*e^3)/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^
4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^
4))/sqrt(c*x^4 + b*x^2 + a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{\left (e\,x^2+d\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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