3.355 \(\int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=324 \[ \frac {\left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {b \sqrt {d+e x^2}}{c^2}+\frac {\left (d+e x^2\right )^{3/2}}{3 c e} \]

[Out]

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

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Rubi [A]  time = 3.53, antiderivative size = 324, 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 = {1251, 897, 1287, 1166, 208} \[ \frac {\left (-\frac {3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (\frac {3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {b \sqrt {d+e x^2}}{c^2}+\frac {\left (d+e x^2\right )^{3/2}}{3 c e} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]

[Out]

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

Rule 208

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

Rule 897

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 1166

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 1251

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 1287

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

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Mathematica [A]  time = 7.25, size = 591, normalized size = 1.82 \[ \frac {c \left (d+e x^2\right )^{7/2} \left (\frac {e^2 \left (\sqrt {2} \sqrt {\frac {c \left (d+e x^2\right )}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}} \left (5 b e \left (3 e \sqrt {b^2-4 a c}+c \left (d+e x^2\right )\right )+c \left (d+e x^2\right ) \left (-5 e \sqrt {b^2-4 a c}+4 c d-6 c e x^2\right )+30 a c e^2-15 b^2 e^2\right )-15 e^2 \left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {c \left (d+e x^2\right )}{e \sqrt {b^2-4 a c}-b e+2 c d}}\right )\right )}{\left (e \left (b-\sqrt {b^2-4 a c}\right )-2 c d\right ) \left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right )^2 \left (\frac {c \left (d+e x^2\right )}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}\right )^{7/2}}-\frac {e^2 \left (\sqrt {2} \sqrt {\frac {c \left (d+e x^2\right )}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (-5 b e \left (3 e \sqrt {b^2-4 a c}-c \left (d+e x^2\right )\right )+c \left (d+e x^2\right ) \left (5 e \sqrt {b^2-4 a c}+4 c d-6 c e x^2\right )+30 a c e^2-15 b^2 e^2\right )+15 e^2 \left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {c \left (d+e x^2\right )}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )\right )}{\left (e \left (\sqrt {b^2-4 a c}+b\right )-2 c d\right )^3 \left (\frac {c \left (d+e x^2\right )}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{7/2}}\right )}{30 \sqrt {2} e^4 \sqrt {b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]

[Out]

(c*(d + e*x^2)^(7/2)*((e^2*(Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(-15*b^2*e^2 +
30*a*c*e^2 + c*(d + e*x^2)*(4*c*d - 5*Sqrt[b^2 - 4*a*c]*e - 6*c*e*x^2) + 5*b*e*(3*Sqrt[b^2 - 4*a*c]*e + c*(d +
e*x^2))) - 15*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*e^2*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e)]]))/((-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^2*((c*(d + e
*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^(7/2)) - (e^2*(Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^
2 - 4*a*c])*e)]*(-15*b^2*e^2 + 30*a*c*e^2 + c*(d + e*x^2)*(4*c*d + 5*Sqrt[b^2 - 4*a*c]*e - 6*c*e*x^2) - 5*b*e*
(3*Sqrt[b^2 - 4*a*c]*e - c*(d + e*x^2))) + 15*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*e^2*ArcTanh[Sqrt[2]*Sqrt[(c*
(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]]))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)^3*((c*(d + e*x^2))/
(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(7/2))))/(30*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e^4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.82, size = 745, normalized size = 2.30 \[ \frac {{\left ({\left (x^{2} e + d\right )}^{\frac {3}{2}} c^{2} e^{2} - 3 \, \sqrt {x^{2} e + d} b c e^{3}\right )} e^{\left (-3\right )}}{3 \, c^{3}} + \frac {{\left ({\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d e - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2}\right )} c^{2} + 2 \, {\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{2} - {\left (3 \, b^{3} c^{3} - 8 \, a b c^{4}\right )} d e - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b^{2} c^{2} d e + \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{2}\right )} {\left | c \right |} + {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {{\left (2 \, c^{4} d e^{4} - b c^{3} e^{5} + \sqrt {-4 \, {\left (c^{4} d^{2} e^{4} - b c^{3} d e^{5} + a c^{3} e^{6}\right )} c^{4} e^{4} + {\left (2 \, c^{4} d e^{4} - b c^{3} e^{5}\right )}^{2}}\right )} e^{\left (-4\right )}}{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 ({\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d e - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2}\right )} c^{2} + 2 \, {\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{2} - {\left (3 \, b^{3} c^{3} - 8 \, a b c^{4}\right )} d e + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b^{2} c^{2} d e + \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{2}\right )} {\left | c \right |} + {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {{\left (2 \, c^{4} d e^{4} - b c^{3} e^{5} - \sqrt {-4 \, {\left (c^{4} d^{2} e^{4} - b c^{3} d e^{5} + a c^{3} e^{6}\right )} c^{4} e^{4} + {\left (2 \, c^{4} d e^{4} - b c^{3} e^{5}\right )}^{2}}\right )} e^{\left (-4\right )}}{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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/3*((x^2*e + d)^(3/2)*c^2*e^2 - 3*sqrt(x^2*e + d)*b*c*e^3)*e^(-3)/c^3 + (((b^3*c - 4*a*b*c^2)*d*e - (b^4 - 5*
a*b^2*c + 4*a^2*c^2)*e^2)*c^2 + 2*(b^2*c^4 - 2*a*c^5)*d^2 - (3*b^3*c^3 - 8*a*b*c^4)*d*e - 2*(sqrt(b^2 - 4*a*c)
*b*c^3*d^2 - sqrt(b^2 - 4*a*c)*b^2*c^2*d*e + sqrt(b^2 - 4*a*c)*a*b*c^2*e^2)*abs(c) + (b^4*c^2 - 3*a*b^2*c^3)*e
^2)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c^4*d*e^4 - b*c^3*e^5 + sqrt(-4*(c^4*d^2*e^4 - b*c^3*d*e^5 + a
*c^3*e^6)*c^4*e^4 + (2*c^4*d*e^4 - b*c^3*e^5)^2))*e^(-4)/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)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2) - (((b^3*c - 4*a*b*c^2)*
d*e - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^2)*c^2 + 2*(b^2*c^4 - 2*a*c^5)*d^2 - (3*b^3*c^3 - 8*a*b*c^4)*d*e + 2*(sq
rt(b^2 - 4*a*c)*b*c^3*d^2 - sqrt(b^2 - 4*a*c)*b^2*c^2*d*e + sqrt(b^2 - 4*a*c)*a*b*c^2*e^2)*abs(c) + (b^4*c^2 -
3*a*b^2*c^3)*e^2)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c^4*d*e^4 - b*c^3*e^5 - sqrt(-4*(c^4*d^2*e^4 -
b*c^3*d*e^5 + a*c^3*e^6)*c^4*e^4 + (2*c^4*d*e^4 - b*c^3*e^5)^2))*e^(-4)/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)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2)

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maple [C]  time = 0.03, size = 332, normalized size = 1.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d} x^{5}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x^2 + d)*x^5/(c*x^4 + b*x^2 + a), x)

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mupad [B]  time = 1.99, size = 8222, normalized size = 25.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(d + e*x^2)^(1/2))/(a + b*x^2 + c*x^4),x)

[Out]

(d + e*x^2)^(3/2)/(3*c*e) - atan(((((4*a*b^3*c^3*e^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2
- 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^3 - (2*(d + e*x^2)^(1/2)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c
- b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5
*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2
) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2)
- b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c
^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*
(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*
a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b
^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c^3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^
3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e +
8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a
*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 1
6*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^3 + (2*(d +
e*x^2)^(1/2)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^
3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c
- b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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)*(-
(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^
2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2)
+ 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^
4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2)
)/c^3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e
+ a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)
^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7 + b^4*
c^5 - 8*a*b^2*c^6)))^(1/2)*1i)/((2*(a^4*c*e^5 - a^3*b^2*e^5 + a^2*b^3*d*e^4 + a^3*c^2*d^2*e^3 + a^2*b*c^2*d^3*
e^2 - 2*a^2*b^2*c*d^2*e^3))/c^3 + (((4*a*b^3*c^3*e^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2
- 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^3 - (2*(d + e*x^2)^(1/2)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c
- b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5
*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2
) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2)
- b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c
^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*
(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*
a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b
^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c^3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^
3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e +
8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a
*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2) + (((4*a*b^3*c^3*e^4 - 16*a
^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^3 + (2*(d + e*
x^2)^(1/2)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c
^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c -
b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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)*(-(b^
7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e
*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) +
2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^4*c
^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c
^3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e +
a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)
^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7 + b^4*c^5
- 8*a*b^2*c^6)))^(1/2)))*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d
+ 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*
c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8
*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*2i - atan(((((4*a*b^3*c^3*e^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e
^3 + 4*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^3 - (2*(d + e*x^2)^(1/2)*((b^4*e*(-(4*a*c
- b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b
^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(
-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) -
8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*
a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)
^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2
*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c^3)*((b^4*e*(-(4*a*c - b^2)^3)
^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1
/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c -
b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*
d*e^3)/c^3 + (2*(d + e*x^2)^(1/2)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^
2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e
- b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1
/2))/(8*(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)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*
b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a
*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^
2*c^4*d^2*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*
a*b^2*c^3*d^2*e^2))/c^3)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d -
25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*
d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(
16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*1i)/((2*(a^4*c*e^5 - a^3*b^2*e^5 + a^2*b^3*d*e^4 + a^3*c^2*d^2*e^3
+ a^2*b*c^2*d^3*e^2 - 2*a^2*b^2*c*d^2*e^3))/c^3 + (((4*a*b^3*c^3*e^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4
*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^3 - (2*(d + e*x^2)^(1/2)*((b^4*e*(-(4*a*c - b^2)
^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)
^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*
c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3
*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*
c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2)
- 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c
*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c^3)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2)
- 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) +
9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^
3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2) + (((4*a*b^3*
c^3*e^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d*e^3)/c^
3 + (2*(d + e*x^2)^(1/2)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d -
25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*
d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(
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)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b^3*c^2*e
+ a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)
^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(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^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2
*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3
*d^2*e^2))/c^3)*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2*b^2*c^3*d - 25*a^2*b
^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*
c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^
7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)))*((b^4*e*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c^4*d - b^7*e + b^6*c*d + 18*a^2
*b^2*c^3*d - 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c*e - 8*a*b^4*c^2*d + 20*a^3*b*c^
3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)
^(1/2))/(8*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2)*2i - ((2*d)/(c*e) + (b*e^2 - 2*c*d*e)/(c^2*e^2))*(d +
e*x^2)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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