3.4.54 \(\int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) [354]

Optimal. Leaf size=406 \[ \frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac {\left (d+e x^2\right )^{5/2}}{5 c e^2}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/3*(b*e+c*d)*(e*x^2+d)^(3/2)/c^2/e^2+1/5*(e*x^2+d)^(5/2)/c/e^2+(-a*c+b^2)*(e*x^2+d)^(1/2)/c^3-1/2*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^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e+(2*a^2*
c^2*e-4*a*b^2*c*e+3*a*b*c^2*d+b^4*e-b^3*c*d)/(-4*a*c+b^2)^(1/2))/c^(7/2)*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^2*c*d-a*c^2*d
-b^3*e+2*a*b*c*e+(-2*a^2*c^2*e+4*a*b^2*c*e-3*a*b*c^2*d-b^4*e+b^3*c*d)/(-4*a*c+b^2)^(1/2))/c^(7/2)*2^(1/2)/(2*c
*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 5.44, antiderivative size = 406, 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, 911, 1301, 1180, 214} \begin {gather*} -\frac {\left (-\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\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^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\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^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {\left (d+e x^2\right )^{3/2} (b e+c d)}{3 c^2 e^2}+\frac {\left (d+e x^2\right )^{5/2}}{5 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^7*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]

[Out]

((b^2 - a*c)*Sqrt[d + e*x^2])/c^3 - ((c*d + b*e)*(d + e*x^2)^(3/2))/(3*c^2*e^2) + (d + e*x^2)^(5/2)/(5*c*e^2)
- ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b
^2 - 4*a*c])*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^(7
/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*
d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*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^(7/2)*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 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

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]

Rule 1301

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

Rubi steps

\begin {align*} \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {x^2 \left (-\frac {d}{e}+\frac {x^2}{e}\right )^3}{\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}} \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\left (b^2-a c\right ) e}{c^3}-\frac {(c d+b e) x^2}{c^2 e}+\frac {x^4}{c e}-\frac {\left (b^2-a c\right ) \left (c d^2-b d e+a e^2\right )-\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x^2}{c^3 e \left (\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}\right )}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac {\left (d+e x^2\right )^{5/2}}{5 c e^2}-\frac {\text {Subst}\left (\int \frac {\left (b^2-a c\right ) \left (c d^2-b d e+a e^2\right )+\left (-b^2 c d+a c^2 d+b^3 e-2 a b c e\right ) x^2}{\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}} \, dx,x,\sqrt {d+e x^2}\right )}{c^3 e^2}\\ &=\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac {\left (d+e x^2\right )^{5/2}}{5 c e^2}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x^2}\right )}{2 c^3 e^2}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x^2}\right )}{2 c^3 e^2}\\ &=\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac {\left (d+e x^2\right )^{5/2}}{5 c e^2}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\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^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\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^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]
time = 1.44, size = 475, normalized size = 1.17 \begin {gather*} \frac {\frac {2 \sqrt {c} \sqrt {d+e x^2} \left (15 b^2 e^2+c^2 \left (-2 d^2+d e x^2+3 e^2 x^4\right )-5 c e \left (3 a e+b \left (d+e x^2\right )\right )\right )}{e^2}-\frac {15 \sqrt {2} \left (-b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 c \left (-\sqrt {b^2-4 a c} d+4 a e\right )+b^3 \left (c d+\sqrt {b^2-4 a c} e\right )-a b c \left (3 c d+2 \sqrt {b^2-4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )-b^2 c \left (\sqrt {b^2-4 a c} d+4 a e\right )+a b c \left (3 c d-2 \sqrt {b^2-4 a c} e\right )+b^3 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \tan ^{-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 {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{30 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^7*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]

[Out]

((2*Sqrt[c]*Sqrt[d + e*x^2]*(15*b^2*e^2 + c^2*(-2*d^2 + d*e*x^2 + 3*e^2*x^4) - 5*c*e*(3*a*e + b*(d + e*x^2))))
/e^2 - (15*Sqrt[2]*(-(b^4*e) + a*c^2*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) +
b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^
2])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) -
(15*Sqrt[2]*(b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d
- 2*Sqrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-
2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]))/(30*c^(7/2)
)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.17, size = 457, normalized size = 1.13

method	result	size
risch	\(-\frac {\left (-3 c^{2} e^{2} x^{4}+5 b c \,e^{2} x^{2}-c^{2} d e \,x^{2}+15 a c \,e^{2}-15 e^{2} b^{2}+5 b c d e +2 c^{2} d^{2}\right ) \sqrt {e \,x^{2}+d}}{15 e^{2} c^{3}}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (\left (2 a b c e -a \,c^{2} d -b^{3} e +b^{2} c d \right ) \textit {\_R}^{6}+\left (4 a^{2} c \,e^{2}-4 a \,b^{2} e^{2}-2 a b c d e +3 a \,c^{2} d^{2}+3 b^{3} d e -3 b^{2} c \,d^{2}\right ) \textit {\_R}^{4}+d \left (-4 a^{2} c \,e^{2}+4 a \,b^{2} e^{2}+2 a b c d e -3 a \,c^{2} d^{2}-3 b^{3} d e +3 b^{2} c \,d^{2}\right ) \textit {\_R}^{2}-2 a b c \,d^{3} e +a \,c^{2} d^{4}+b^{3} d^{3} e -b^{2} c \,d^{4}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 c^{3}}\)	\(412\)
default	\(\frac {\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}}{c}-\frac {b \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} e}+\frac {-\frac {\left (a c -b^{2}\right ) \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{2 c}-\frac {d \left (a c -b^{2}\right )}{2 c \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (\left (-2 a b c e +a \,c^{2} d +b^{3} e -b^{2} c d \right ) \textit {\_R}^{6}+\left (-4 a^{2} c \,e^{2}+4 a \,b^{2} e^{2}+2 a b c d e -3 a \,c^{2} d^{2}-3 b^{3} d e +3 b^{2} c \,d^{2}\right ) \textit {\_R}^{4}+d \left (4 a^{2} c \,e^{2}-4 a \,b^{2} e^{2}-2 a b c d e +3 a \,c^{2} d^{2}+3 b^{3} d e -3 b^{2} c \,d^{2}\right ) \textit {\_R}^{2}+2 a b c \,d^{3} e -a \,c^{2} d^{4}-b^{3} d^{3} e +b^{2} c \,d^{4}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 c}}{c^{2}}\)	\(457\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/c*(1/5*x^2*(e*x^2+d)^(3/2)/e-2/15*d/e^2*(e*x^2+d)^(3/2))-1/3*b/c^2*(e*x^2+d)^(3/2)/e+1/c^2*(-1/2*(a*c-b^2)/c
*((e*x^2+d)^(1/2)-e^(1/2)*x)-1/2/c*d*(a*c-b^2)/((e*x^2+d)^(1/2)-e^(1/2)*x)-1/4/c*sum(((-2*a*b*c*e+a*c^2*d+b^3*
e-b^2*c*d)*_R^6+(-4*a^2*c*e^2+4*a*b^2*e^2+2*a*b*c*d*e-3*a*c^2*d^2-3*b^3*d*e+3*b^2*c*d^2)*_R^4+d*(4*a^2*c*e^2-4
*a*b^2*e^2-2*a*b*c*d*e+3*a*c^2*d^2+3*b^3*d*e-3*b^2*c*d^2)*_R^2+2*a*b*c*d^3*e-a*c^2*d^4-b^3*d^3*e+b^2*c*d^4)/(_
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)
))

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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^7*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2*e + d)*x^7/(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^7*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} \sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**7*sqrt(d + e*x**2)/(a + b*x**2 + c*x**4), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (374) = 748\).
time = 4.07, size = 928, normalized size = 2.29 \begin {gather*} -\frac {{\left ({\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e^{2}\right )} c^{2} + 2 \, {\left (b^{3} c^{4} - 3 \, a b c^{5}\right )} d^{2} - {\left (3 \, b^{4} c^{3} - 11 \, a b^{2} c^{4} + 4 \, a^{2} c^{5}\right )} d e - 2 \, {\left ({\left (b^{2} c^{3} - a c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} c^{2} - a b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} c^{2} - a^{2} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | c \right |} + {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3} + 2 \, a^{2} b c^{4}\right )} e^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {{\left (2 \, c^{6} d e^{12} - b c^{5} e^{13} + \sqrt {-4 \, {\left (c^{6} d^{2} e^{12} - b c^{5} d e^{13} + a c^{5} e^{14}\right )} c^{6} e^{12} + {\left (2 \, c^{6} d e^{12} - b c^{5} e^{13}\right )}^{2}}\right )} e^{\left (-12\right )}}{c^{6}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d + {\left (b^{2} c^{3} - 4 \, a c^{4} - \sqrt {b^{2} - 4 \, a c} b c^{3}\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 ({\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e^{2}\right )} c^{2} + 2 \, {\left (b^{3} c^{4} - 3 \, a b c^{5}\right )} d^{2} - {\left (3 \, b^{4} c^{3} - 11 \, a b^{2} c^{4} + 4 \, a^{2} c^{5}\right )} d e + 2 \, {\left ({\left (b^{2} c^{3} - a c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} c^{2} - a b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} c^{2} - a^{2} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | c \right |} + {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3} + 2 \, a^{2} b c^{4}\right )} e^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {{\left (2 \, c^{6} d e^{12} - b c^{5} e^{13} - \sqrt {-4 \, {\left (c^{6} d^{2} e^{12} - b c^{5} d e^{13} + a c^{5} e^{14}\right )} c^{6} e^{12} + {\left (2 \, c^{6} d e^{12} - b c^{5} e^{13}\right )}^{2}}\right )} e^{\left (-12\right )}}{c^{6}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d - {\left (b^{2} c^{3} - 4 \, a c^{4} + \sqrt {b^{2} - 4 \, a c} b c^{3}\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 (3 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} c^{4} e^{8} - 5 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} c^{4} d e^{8} - 5 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} b c^{3} e^{9} + 15 \, \sqrt {x^{2} e + d} b^{2} c^{2} e^{10} - 15 \, \sqrt {x^{2} e + d} a c^{3} e^{10}\right )} e^{\left (-10\right )}}{15 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-(((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*e - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e^2)*c^2 + 2*(b^3*c^4 - 3*a*b*c^5)*
d^2 - (3*b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d*e - 2*((b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c^2 - a*b
*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^2)*abs(c) + (b^5*c^2 - 4*a*b^3*c^3 + 2
*a^2*b*c^4)*e^2)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c^6*d*e^12 - b*c^5*e^13 + sqrt(-4*(c^6*d^2*e^12 -
b*c^5*d*e^13 + a*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13)^2))*e^(-12)/c^6))/((2*sqrt(b^2 - 4*a*c)*c^4
*d + (b^2*c^3 - 4*a*c^4 - sqrt(b^2 - 4*a*c)*b*c^3)*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2) +
(((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*e - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e^2)*c^2 + 2*(b^3*c^4 - 3*a*b*c^5)*d
^2 - (3*b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d*e + 2*((b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c^2 - a*b*
c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^2)*abs(c) + (b^5*c^2 - 4*a*b^3*c^3 + 2*
a^2*b*c^4)*e^2)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c^6*d*e^12 - b*c^5*e^13 - sqrt(-4*(c^6*d^2*e^12 -
b*c^5*d*e^13 + a*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13)^2))*e^(-12)/c^6))/((2*sqrt(b^2 - 4*a*c)*c^4*
d - (b^2*c^3 - 4*a*c^4 + sqrt(b^2 - 4*a*c)*b*c^3)*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2) + 1
/15*(3*(x^2*e + d)^(5/2)*c^4*e^8 - 5*(x^2*e + d)^(3/2)*c^4*d*e^8 - 5*(x^2*e + d)^(3/2)*b*c^3*e^9 + 15*sqrt(x^2
*e + d)*b^2*c^2*e^10 - 15*sqrt(x^2*e + d)*a*c^3*e^10)*e^(-10)/c^5

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Mupad [B]
time = 2.48, size = 2500, normalized size = 6.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(d + e*x^2)^(1/2))/(a + b*x^2 + c*x^4),x)

[Out]

(d + e*x^2)^(1/2)*((3*d^2)/(c*e^2) - (a*e^4 + c*d^2*e^2 - b*d*e^3)/(c^2*e^4) + (((3*d)/(c*e^2) + (b*e^3 - 2*c*
d*e^2)/(c^2*e^4))*(b*e^3 - 2*c*d*e^2))/(c*e^2)) - (d + e*x^2)^(3/2)*(d/(c*e^2) + (b*e^3 - 2*c*d*e^2)/(3*c^2*e^
4)) + atan(((((16*a^3*c^6*e^4 + 4*a*b^4*c^4*e^4 - 4*b^5*c^4*d*e^3 - 20*a^2*b^2*c^5*e^4 + 16*a^2*c^7*d^2*e^2 +
4*b^4*c^5*d^2*e^2 + 20*a*b^3*c^5*d*e^3 - 16*a^2*b*c^6*d*e^3 - 20*a*b^2*c^6*d^2*e^2)/c^5 - (2*(d + e*x^2)^(1/2)
*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*
a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4
*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c -
b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*
c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(4*b^3*c^7*e^3 - 8*b^2*c^8*d*e^2 - 16*a*b*c^8*e^3 + 32*a*c^9*d*e^2))/c^5)
*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*
a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4
*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c -
b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*
c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (2*(d + e*x^2)^(1/2)*(b^8*e^4 + 2*a^4*c^4*e^4 + 20*a^2*b^4*c^2*e^4 - 16
*a^3*b^2*c^3*e^4 - 2*a^3*c^5*d^2*e^2 + b^6*c^2*d^2*e^2 - 8*a*b^6*c*e^4 - 2*b^7*c*d*e^3 + 9*a^2*b^2*c^4*d^2*e^2
+ 14*a*b^5*c^2*d*e^3 + 14*a^3*b*c^4*d*e^3 - 6*a*b^4*c^3*d^2*e^2 - 28*a^2*b^3*c^3*d*e^3))/c^5)*(-(b^9*e - 8*a^
4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*a^2*b^5*c^2*e -
63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4*e + b^5*c
*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) +
3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^9 + b^4*c^7 -
8*a*b^2*c^8)))^(1/2)*1i - (((16*a^3*c^6*e^4 + 4*a*b^4*c^4*e^4 - 4*b^5*c^4*d*e^3 - 20*a^2*b^2*c^5*e^4 + 16*a^2*
c^7*d^2*e^2 + 4*b^4*c^5*d^2*e^2 + 20*a*b^3*c^5*d*e^3 - 16*a^2*b*c^6*d*e^3 - 20*a*b^2*c^6*d^2*e^2)/c^5 + (2*(d
+ e*x^2)^(1/2)*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b
^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*
c^2*d + 28*a^4*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2
*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2
))/(8*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(4*b^3*c^7*e^3 - 8*b^2*c^8*d*e^2 - 16*a*b*c^8*e^3 + 32*a*c^
9*d*e^2))/c^5)*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b
^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*
c^2*d + 28*a^4*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2
*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2
))/(8*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (2*(d + e*x^2)^(1/2)*(b^8*e^4 + 2*a^4*c^4*e^4 + 20*a^2*b^
4*c^2*e^4 - 16*a^3*b^2*c^3*e^4 - 2*a^3*c^5*d^2*e^2 + b^6*c^2*d^2*e^2 - 8*a*b^6*c*e^4 - 2*b^7*c*d*e^3 + 9*a^2*b
^2*c^4*d^2*e^2 + 14*a*b^5*c^2*d*e^3 + 14*a^3*b*c^4*d*e^3 - 6*a*b^4*c^3*d^2*e^2 - 28*a^2*b^3*c^3*d*e^3))/c^5)*(
-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*a^
2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b
*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c - b^
2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^
9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i)/((((16*a^3*c^6*e^4 + 4*a*b^4*c^4*e^4 - 4*b^5*c^4*d*e^3 - 20*a^2*b^2*c^5
*e^4 + 16*a^2*c^7*d^2*e^2 + 4*b^4*c^5*d^2*e^2 + 20*a*b^3*c^5*d*e^3 - 16*a^2*b*c^6*d*e^3 - 20*a*b^2*c^6*d^2*e^2
)/c^5 - (2*(d + e*x^2)^(1/2)*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^
3*d + 38*a^3*b^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c
*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2)
- 4*a*b^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c
- b^2)^3)^(1/2))/(8*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(4*b^3*c^7*e^3 - 8*b^2*c^8*d*e^2 - 16*a*b*c^8
*e^3 + 32*a*c^9*d*e^2))/c^5)*(-(b^9*e - 8*a^4*c...

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