3.4.87 \(\int \frac {x^6}{\sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\) [387]

Optimal. Leaf size=366 \[ \frac {x \sqrt {d+e x^2}}{2 c e}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c e^{3/2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2 \sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.82, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1317, 223, 212, 327, 1706, 385, 211} \begin {gather*} \frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {ArcTan}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {ArcTan}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2 \sqrt {e}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c e^{3/2}}+\frac {x \sqrt {d+e x^2}}{2 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

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

Rule 211

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1317

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2
- 4*a*c, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 10.73, size = 355, normalized size = 0.97 \begin {gather*} \frac {\frac {c x \sqrt {d+e x^2}}{e}+\frac {2 \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {2 \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}-\frac {c d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 267, normalized size = 0.73

method	result	size
default	\(\frac {\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}}{c}-\frac {b \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{c^{2} \sqrt {e}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (-a c +b^{2}\right ) \textit {\_R}^{2}+2 \left (2 a b e +a c d -b^{2} d \right ) \textit {\_R} -a c \,d^{2}+b^{2} d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 c^{2}}\)	\(267\)
risch	\(\frac {x \sqrt {e \,x^{2}+d}}{2 c e}-\frac {b \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{c^{2} \sqrt {e}}-\frac {\ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) d}{2 c \,e^{\frac {3}{2}}}+\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (a c -b^{2}\right ) \textit {\_R}^{2}+2 \left (-2 a b e -a c d +b^{2} d \right ) \textit {\_R} +a c \,d^{2}-b^{2} d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 c^{2}}\)	\(269\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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^6/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^6/((c*x^4 + b*x^2 + a)*sqrt(x^2*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7362 vs. \(2 (312) = 624\).
time = 55.00, size = 7362, normalized size = 20.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

-1/4*(sqrt(1/2)*c^2*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + ((b^2*c^5
- 4*a*c^6)*d^2 - (b^3*c^4 - 4*a*b*c^5)*d*e + (a*b^2*c^4 - 4*a^2*c^5)*e^2)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4
*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4
*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/((b^2*c^10 - 4*a*c^11)*d^4 - 2*(b^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*
b^2*c^9 - 8*a^2*c^10)*d^2*e^2 - 2*(a*b^3*c^8 - 4*a^2*b*c^9)*d*e^3 + (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))/((b^2*c^5
- 4*a*c^6)*d^2 - (b^3*c^4 - 4*a*b*c^5)*d*e + (a*b^2*c^4 - 4*a^2*c^5)*e^2))*e^2*log(-((a^2*b^5 - 3*a^3*b^3*c +
a^4*b*c^2)*d^2*x^2 + 4*(a^4*b^3 - 2*a^5*b*c)*x^2*e^2 - 2*(a^3*b^4 - 3*a^4*b^2*c + a^5*c^2)*d^2 + 2*sqrt(1/2)*
((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 17*a^3*b^2*c^3 + 4*a^4*c^4)*d^2*x - (2*a*b^7 - 14*a^2*b^5*c + 27*a^3*b^3*
c^2 - 12*a^4*b*c^3)*d*x*e + (a^2*b^6 - 6*a^3*b^4*c + 8*a^4*b^2*c^2)*x*e^2 - ((b^5*c^5 - 7*a*b^3*c^6 + 12*a^2*b
*c^7)*d^3*x - (b^6*c^4 - 6*a*b^4*c^5 + 6*a^2*b^2*c^6 + 8*a^3*c^7)*d^2*x*e + (2*a*b^5*c^4 - 13*a^2*b^3*c^5 + 20
*a^3*b*c^6)*d*x*e^2 - (a^2*b^4*c^4 - 6*a^3*b^2*c^5 + 8*a^4*c^6)*x*e^3)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2
- 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3
*b^4*c + 4*a^4*b^2*c^2)*e^2)/((b^2*c^10 - 4*a*c^11)*d^4 - 2*(b^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*b^2*
c^9 - 8*a^2*c^10)*d^2*e^2 - 2*(a*b^3*c^8 - 4*a^2*b*c^9)*d*e^3 + (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))*sqrt(x^2*e +
d)*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + ((b^2*c^5 - 4*a*c^6)*d^2 -
(b^3*c^4 - 4*a*b*c^5)*d*e + (a*b^2*c^4 - 4*a^2*c^5)*e^2)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*
c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^
4*b^2*c^2)*e^2)/((b^2*c^10 - 4*a*c^11)*d^4 - 2*(b^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*b^2*c^9 - 8*a^2*c
^10)*d^2*e^2 - 2*(a*b^3*c^8 - 4*a^2*b*c^9)*d*e^3 + (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))/((b^2*c^5 - 4*a*c^6)*d^2 -
(b^3*c^4 - 4*a*b*c^5)*d*e + (a*b^2*c^4 - 4*a^2*c^5)*e^2)) - ((5*a^3*b^4 - 14*a^4*b^2*c + 4*a^5*c^2)*d*x^2 - 2
*(a^4*b^3 - 2*a^5*b*c)*d)*e - ((a^2*b^2*c^5 - 4*a^3*c^6)*d^3*x^2 - (a^2*b^3*c^4 - 4*a^3*b*c^5)*d^2*x^2*e + (a^
3*b^2*c^4 - 4*a^4*c^5)*d*x^2*e^2)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(
a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/((b^2*c^
10 - 4*a*c^11)*d^4 - 2*(b^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*b^2*c^9 - 8*a^2*c^10)*d^2*e^2 - 2*(a*b^3*
c^8 - 4*a^2*b*c^9)*d*e^3 + (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))/x^2) - sqrt(1/2)*c^2*sqrt(-((b^5 - 5*a*b^3*c + 5*a
^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + ((b^2*c^5 - 4*a*c^6)*d^2 - (b^3*c^4 - 4*a*b*c^5)*d*e + (a*
b^2*c^4 - 4*a^2*c^5)*e^2)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 -
5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/((b^2*c^10 - 4*a
*c^11)*d^4 - 2*(b^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*b^2*c^9 - 8*a^2*c^10)*d^2*e^2 - 2*(a*b^3*c^8 - 4*
a^2*b*c^9)*d*e^3 + (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))/((b^2*c^5 - 4*a*c^6)*d^2 - (b^3*c^4 - 4*a*b*c^5)*d*e + (a*
b^2*c^4 - 4*a^2*c^5)*e^2))*e^2*log(-((a^2*b^5 - 3*a^3*b^3*c + a^4*b*c^2)*d^2*x^2 + 4*(a^4*b^3 - 2*a^5*b*c)*x^2
*e^2 - 2*(a^3*b^4 - 3*a^4*b^2*c + a^5*c^2)*d^2 - 2*sqrt(1/2)*((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 17*a^3*b^2*c
^3 + 4*a^4*c^4)*d^2*x - (2*a*b^7 - 14*a^2*b^5*c + 27*a^3*b^3*c^2 - 12*a^4*b*c^3)*d*x*e + (a^2*b^6 - 6*a^3*b^4*
c + 8*a^4*b^2*c^2)*x*e^2 - ((b^5*c^5 - 7*a*b^3*c^6 + 12*a^2*b*c^7)*d^3*x - (b^6*c^4 - 6*a*b^4*c^5 + 6*a^2*b^2*
c^6 + 8*a^3*c^7)*d^2*x*e + (2*a*b^5*c^4 - 13*a^2*b^3*c^5 + 20*a^3*b*c^6)*d*x*e^2 - (a^2*b^4*c^4 - 6*a^3*b^2*c^
5 + 8*a^4*c^6)*x*e^3)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^
2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/((b^2*c^10 - 4*a*c^1
1)*d^4 - 2*(b^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*b^2*c^9 - 8*a^2*c^10)*d^2*e^2 - 2*(a*b^3*c^8 - 4*a^2*
b*c^9)*d*e^3 + (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))*sqrt(x^2*e + d)*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*
b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + ((b^2*c^5 - 4*a*c^6)*d^2 - (b^3*c^4 - 4*a*b*c^5)*d*e + (a*b^2*c^4 - 4*a^2*c
^5)*e^2)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a
^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/((b^2*c^10 - 4*a*c^11)*d^4 - 2*(b
^3*c^9 - 4*a*b*c^10)*d^3*e + (b^4*c^8 - 2*a*b^2*c^9 - 8*a^2*c^10)*d^2*e^2 - 2*(a*b^3*c^8 - 4*a^2*b*c^9)*d*e^3
+ (a^2*b^2*c^8 - 4*a^3*c^9)*e^4)))/((b^2*c^5 - 4*a*c^6)*d^2 - (b^3*c^4 - 4*a*b*c^5)*d*e + (a*b^2*c^4 - 4*a^2*c
^5)*e^2)) - ((5*a^3*b^4 - 14*a^4*b^2*c + 4*a^5*c^2)*d*x^2 - 2*(a^4*b^3 - 2*a^5*b*c)*d)*e - ((a^2*b^2*c^5 - 4*a
^3*c^6)*d^3*x^2 - (a^2*b^3*c^4 - 4*a^3*b*c^5)*d...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Integral(x**6/(sqrt(d + e*x**2)*(a + b*x**2 + c*x**4)), x)

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Giac [A]
time = 6.63, size = 55, normalized size = 0.15 \begin {gather*} \frac {\sqrt {x^{2} e + d} x e^{\left (-1\right )}}{2 \, c} + \frac {{\left (c d + 2 \, b e\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2}\right )}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6}{\sqrt {e\,x^2+d}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)),x)

[Out]

int(x^6/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)), x)

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