3.103 \(\int (d+e x+f x^2+g x^3) (a+b x^2+c x^4)^{3/2} \, dx\)

Optimal. Leaf size=717 \[ -\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (24 a b c f-180 a c^2 d+9 b^2 c d-4 b^3 f\right )+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x \sqrt{a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d+9 b^2 c d-4 b^3 f\right )}{315 c^2}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]

[Out]

-((18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*x*Sqrt[a + b*x^2 + c*x^4])/(315*c^(5/2)
*(Sqrt[a] + Sqrt[c]*x^2)) - (3*(b^2 - 4*a*c)*(2*c*e - b*g)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^3) +
(x*(9*b^2*c*d + 90*a*c^2*d - 4*b^3*f + 9*a*b*c*f + 3*c*(9*b*c*d - 4*b^2*f + 14*a*c*f)*x^2)*Sqrt[a + b*x^2 + c*
x^4])/(315*c^2) + ((2*c*e - b*g)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*c^2) + (x*(3*(3*c*d + b*f) + 7*c
*f*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c) + (g*(a + b*x^2 + c*x^4)^(5/2))/(10*c) + (3*(b^2 - 4*a*c)^2*(2*c*e -
b*g)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)) + (a^(1/4)*(18*b^3*c*d - 144*a
*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] +
Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(315*c^(11/4)*Sqrt[a +
b*x^2 + c*x^4]) - (a^(1/4)*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f + Sqrt[a]*Sqrt[
c]*(9*b^2*c*d - 180*a*c^2*d - 4*b^3*f + 24*a*b*c*f))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11/4)*Sqrt[a
+ b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.595828, antiderivative size = 717, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1673, 1176, 1197, 1103, 1195, 1247, 640, 612, 621, 206} \[ -\frac{x \sqrt{a+b x^2+c x^4} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (24 a b c f-180 a c^2 d+9 b^2 c d-4 b^3 f\right )+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x \sqrt{a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d+9 b^2 c d-4 b^3 f\right )}{315 c^2}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*x*Sqrt[a + b*x^2 + c*x^4])/(315*c^(5/2)
*(Sqrt[a] + Sqrt[c]*x^2)) - (3*(b^2 - 4*a*c)*(2*c*e - b*g)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^3) +
(x*(9*b^2*c*d + 90*a*c^2*d - 4*b^3*f + 9*a*b*c*f + 3*c*(9*b*c*d - 4*b^2*f + 14*a*c*f)*x^2)*Sqrt[a + b*x^2 + c*
x^4])/(315*c^2) + ((2*c*e - b*g)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*c^2) + (x*(3*(3*c*d + b*f) + 7*c
*f*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c) + (g*(a + b*x^2 + c*x^4)^(5/2))/(10*c) + (3*(b^2 - 4*a*c)^2*(2*c*e -
b*g)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)) + (a^(1/4)*(18*b^3*c*d - 144*a
*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] +
Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(315*c^(11/4)*Sqrt[a +
b*x^2 + c*x^4]) - (a^(1/4)*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f + Sqrt[a]*Sqrt[
c]*(9*b^2*c*d - 180*a*c^2*d - 4*b^3*f + 24*a*b*c*f))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11/4)*Sqrt[a
+ b*x^2 + c*x^4])

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(2*b*e*p + c*d*(4*p
+ 3) + c*e*(4*p + 1)*x^2)*(a + b*x^2 + c*x^4)^p)/(c*(4*p + 1)*(4*p + 3)), x] + Dist[(2*p)/(c*(4*p + 1)*(4*p +
3)), Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a +
b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]

Rule 1197

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

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[
(d*x*Sqrt[a + b*x^2 + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q
^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0
]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(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]

Rule 640

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\int \left (d+f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx+\int x \left (e+g x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx\\ &=\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{1}{2} \operatorname{Subst}\left (\int (e+g x) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )+\frac{\int \left (a (18 c d-b f)+\left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4} \, dx}{21 c}\\ &=\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\int \frac{-a \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )+\left (-18 b^3 c d+144 a b c^2 d+8 b^4 f-57 a b^2 c f+84 a^2 c^2 f\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{315 c^2}+\frac{(2 c e-b g) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{4 c}\\ &=\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\left (\sqrt{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{315 c^{5/2}}-\frac{\left (\sqrt{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{315 c^{5/2}}-\frac{\left (3 \left (b^2-4 a c\right ) (2 c e-b g)\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{64 c^2}\\ &=-\frac{\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}+\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c e-b g)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^3}\\ &=-\frac{\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}+\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c e-b g)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{256 c^3}\\ &=-\frac{\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}+\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}

Mathematica [F]  time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [B]  time = 0.095, size = 3038, normalized size = 4.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*d*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(
1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*
(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*a^2-2/15*f*a^3*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^
(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*El
lipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+2/15*f*a
^3*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2
))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a
)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/35*d*a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*
((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+
b^2)^(1/2))*b^3/c*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/
a/c)^(1/2))+19/210*f*a^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+
2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))/c*b^2*EllipticF(1/2*x*2^(1/
2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-19/210*f*a^2*2^(1/2)/(((-4*
a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(
c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))/c*b^2*EllipticE(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2
*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-4/315*f*a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b
^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2)
)*b^4/c^2*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/
2))+4/315*f*a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*
c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*b^4/c^2*EllipticE(1/2*x*2^(1/2)*(((-4*
a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/35*d*a*2^(1/2)/(((-4*a*c+b^2)^(1/2
)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a
)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*b^3/c*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(
-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/140*d*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*
x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^
2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*a/c*b^2-8/35*d*a^2*2^(1/2)/(((-4*a*c+b^2)^
(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x
^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*b*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(
-4*a*c+b^2)^(1/2))/a/c)^(1/2))+8/35*d*a^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)
/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*b*Elliptic
E(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-2/105*f/c*a^2*
2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/
a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4
*a*c+b^2)^(1/2))/a/c)^(1/2))*b+1/315*f/c^2*a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4-2*((-4*a*c+b^2)^(1/2)
-b)/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*(((-4*
a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b^3+1/8*e*c*x^6*(c*x^4+b*x^2+a)^(1/2
)+3/16*e*b*x^4*(c*x^4+b*x^2+a)^(1/2)+5/16*e*a*x^2*(c*x^4+b*x^2+a)^(1/2)-3/128*e*b^3/c^2*(c*x^4+b*x^2+a)^(1/2)+
3/256*e*b^4/c^(5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+3/16*e*a^2*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4
+b*x^2+a)^(1/2))/c^(1/2)+1/7*d*c*x^5*(c*x^4+b*x^2+a)^(1/2)+8/35*d*b*x^3*(c*x^4+b*x^2+a)^(1/2)+3/7*d*x*(c*x^4+b
*x^2+a)^(1/2)*a+1/10*g*c*x^8*(c*x^4+b*x^2+a)^(1/2)+11/80*g*b*x^6*(c*x^4+b*x^2+a)^(1/2)+1/5*g*a*x^4*(c*x^4+b*x^
2+a)^(1/2)+3/256*g*b^4/c^3*(c*x^4+b*x^2+a)^(1/2)-3/512*g*b^5/c^(7/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^
(1/2))+7/160*g*a*b*x^2/c*(c*x^4+b*x^2+a)^(1/2)+8/105*f/c*x*(c*x^4+b*x^2+a)^(1/2)*a*b-1/128*g*b^3/c^2*x^2*(c*x^
4+b*x^2+a)^(1/2)-5/64*g*a*b^2/c^2*(c*x^4+b*x^2+a)^(1/2)+3/64*g*a*b^3/c^(5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b
*x^2+a)^(1/2))-3/32*g*a^2*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/105*f/c*x^3*(c*x^4+b*x^2
+a)^(1/2)*b^2-4/315*f/c^2*x*(c*x^4+b*x^2+a)^(1/2)*b^3+1/64*e*b^2*x^2/c*(c*x^4+b*x^2+a)^(1/2)+5/32*e*a*b/c*(c*x
^4+b*x^2+a)^(1/2)-3/32*e*a*b^2/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/35*d/c*x*(c*x^4+b*x^2
+a)^(1/2)*b^2+1/160*g*b^2*x^4/c*(c*x^4+b*x^2+a)^(1/2)+1/10*g*a^2/c*(c*x^4+b*x^2+a)^(1/2)+1/9*f*c*x^7*(c*x^4+b*
x^2+a)^(1/2)+10/63*f*b*x^5*(c*x^4+b*x^2+a)^(1/2)+11/45*f*x^3*(c*x^4+b*x^2+a)^(1/2)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (g x^{3} + f x^{2} + e x + d\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c g x^{7} + c f x^{6} +{\left (c e + b g\right )} x^{5} +{\left (c d + b f\right )} x^{4} +{\left (b e + a g\right )} x^{3} + a e x +{\left (b d + a f\right )} x^{2} + a d\right )} \sqrt{c x^{4} + b x^{2} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*g*x^7 + c*f*x^6 + (c*e + b*g)*x^5 + (c*d + b*f)*x^4 + (b*e + a*g)*x^3 + a*e*x + (b*d + a*f)*x^2 +
a*d)*sqrt(c*x^4 + b*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (g x^{3} + f x^{2} + e x + d\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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