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

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

[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])

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Rubi [A]  time = 8.59299, antiderivative size = 406, normalized size of antiderivative = 1., 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} \[ -\frac{\left (-\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{\left (d+e x^2\right )^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2} \]

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 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 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 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]

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 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]

Rubi steps

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

Mathematica [B]  time = 10.8281, size = 943, normalized size = 2.32 \[ \frac{\frac{c \left (105 \left (-b^3+\sqrt{b^2-4 a c} b^2+3 a c b-a c \sqrt{b^2-4 a c}\right ) \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}}\right ) e^3+\sqrt{2} \sqrt{\frac{c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}} \left (105 b^3 e^3-35 b^2 \left (3 \sqrt{b^2-4 a c} e+c \left (e x^2+d\right )\right ) e^2+7 b c \left (\left (e x^2+d\right ) \left (-5 c d+5 \sqrt{b^2-4 a c} e+3 c \left (e x^2+d\right )\right )-45 a e^2\right ) e+c \left (35 a \left (3 \sqrt{b^2-4 a c} e+2 c \left (e x^2+d\right )\right ) e^2+c \left (e x^2+d\right ) \left (7 \sqrt{b^2-4 a c} e \left (5 d-3 \left (e x^2+d\right )\right )+c \left (-70 d^2+84 \left (e x^2+d\right ) d-30 \left (e x^2+d\right )^2\right )\right )\right )\right )\right ) \left (e x^2+d\right )^{9/2}}{210 \sqrt{2} \sqrt{b^2-4 a c} e^4 \left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right )^3 \left (-\frac{2 c d-b e}{e^2}-\frac{\sqrt{b^2-4 a c}}{e}\right ) \left (\frac{c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )^{9/2}}+\frac{2 c d^3 \left (\frac{3 \left (b^3+\sqrt{b^2-4 a c} b^2-3 a c b-a c \sqrt{b^2-4 a c}\right ) e^3 \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right ) \left (e x^2+d\right )^3}{4 \sqrt{2} d^3 \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right )^3 \left (\frac{c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{9/2}}+\frac{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (-105 b^3 e^3+35 b^2 \left (c \left (e x^2+d\right )-3 \sqrt{b^2-4 a c} e\right ) e^2-7 b c \left (\left (e x^2+d\right ) \left (-5 c d-5 \sqrt{b^2-4 a c} e+3 c \left (e x^2+d\right )\right )-45 a e^2\right ) e+c \left (35 a \left (3 \sqrt{b^2-4 a c} e-2 c \left (e x^2+d\right )\right ) e^2+c \left (e x^2+d\right ) \left (7 \sqrt{b^2-4 a c} e \left (5 d-3 \left (e x^2+d\right )\right )+c \left (70 d^2-84 \left (e x^2+d\right ) d+30 \left (e x^2+d\right )^2\right )\right )\right )\right )}{140 c^4 d^3 \left (e x^2+d\right )}\right ) \left (e x^2+d\right )^{3/2}}{3 \sqrt{b^2-4 a c} e^4 \left (\frac{\sqrt{b^2-4 a c}}{e}-\frac{2 c d-b e}{e^2}\right )}}{e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((c*(d + e*x^2)^(9/2)*(Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)]*(105*b^3*e^3 - 35*b^2
*e^2*(3*Sqrt[b^2 - 4*a*c]*e + c*(d + e*x^2)) + 7*b*c*e*(-45*a*e^2 + (d + e*x^2)*(-5*c*d + 5*Sqrt[b^2 - 4*a*c]*
e + 3*c*(d + e*x^2))) + c*(35*a*e^2*(3*Sqrt[b^2 - 4*a*c]*e + 2*c*(d + e*x^2)) + c*(d + e*x^2)*(7*Sqrt[b^2 - 4*
a*c]*e*(5*d - 3*(d + e*x^2)) + c*(-70*d^2 + 84*d*(d + e*x^2) - 30*(d + e*x^2)^2)))) + 105*(-b^3 + 3*a*b*c + b^
2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^3*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2
- 4*a*c]*e)]]))/(210*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e^4*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*(-(Sqrt[b^2 - 4*a*c]
/e) - (2*c*d - b*e)/e^2)*((c*(d + e*x^2))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e))^(9/2)) + (2*c*d^3*(d + e*x^2)^(
3/2)*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(-105*b^3*e^3 + 35*b^2*e^2*(-3*Sqrt[b^2 - 4*a*c]*e + c*(d + e*x^2))
- 7*b*c*e*(-45*a*e^2 + (d + e*x^2)*(-5*c*d - 5*Sqrt[b^2 - 4*a*c]*e + 3*c*(d + e*x^2))) + c*(35*a*e^2*(3*Sqrt[
b^2 - 4*a*c]*e - 2*c*(d + e*x^2)) + c*(d + e*x^2)*(7*Sqrt[b^2 - 4*a*c]*e*(5*d - 3*(d + e*x^2)) + c*(70*d^2 - 8
4*d*(d + e*x^2) + 30*(d + e*x^2)^2)))))/(140*c^4*d^3*(d + e*x^2)) + (3*(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c]
- a*c*Sqrt[b^2 - 4*a*c])*e^3*(d + e*x^2)^3*ArcTanh[Sqrt[2]*Sqrt[(c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c
])*e)]])/(4*Sqrt[2]*d^3*(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))^(9/2))))/(3*Sqrt[b^2 - 4*a*c]*e^4*(Sqrt[b^2 - 4*a*c]/e - (2*c*d - b*e)/e^2)))/e

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Maple [C]  time = 0.049, size = 496, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{5\,ce} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,d}{15\,c{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{b}{3\,{c}^{2}e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{xa}{2\,{c}^{2}}\sqrt{e}}-{\frac{x{b}^{2}}{2\,{c}^{3}}\sqrt{e}}-{\frac{a}{2\,{c}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{{b}^{2}}{2\,{c}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{ad}{2\,{c}^{2}} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-1}}+{\frac{{b}^{2}d}{2\,{c}^{3}} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-1}}-{\frac{1}{4\,{c}^{3}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{ \left ( -2\,abce+a{c}^{2}d+{b}^{3}e-{b}^{2}cd \right ){{\it \_R}}^{6}+ \left ( -4\,{a}^{2}c{e}^{2}+4\,a{b}^{2}{e}^{2}+2\,abcde-3\,a{c}^{2}{d}^{2}-3\,{b}^{3}de+3\,{b}^{2}c{d}^{2} \right ){{\it \_R}}^{4}+d \left ( 4\,{a}^{2}c{e}^{2}-4\,a{b}^{2}{e}^{2}-2\,abcde+3\,a{c}^{2}{d}^{2}+3\,{b}^{3}de-3\,{b}^{2}c{d}^{2} \right ){{\it \_R}}^{2}+2\,abc{d}^{3}e-a{c}^{2}{d}^{4}-{b}^{3}{d}^{3}e+{b}^{2}c{d}^{4}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x-{\it \_R} \right ) }} \end{align*}

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)

[Out]

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

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

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7} \sqrt{d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \end{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out