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

Optimal. Leaf size=476 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]

[Out]

(4*a^2*c*x*Sqrt[a + b*x^4])/(77*b) - (a^2*d*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2*e*x^3*Sqrt[a + b*x^4])/(195*b
) - (4*a^3*e*x*Sqrt[a + b*x^4])/(65*b^(3/2)*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^5*(117*c + 77*e*x^2)*Sqrt[a + b*
x^4])/3003 - (a*d*x^2*(a + b*x^4)^(3/2))/(48*b) + (x^5*(13*c + 11*e*x^2)*(a + b*x^4)^(3/2))/143 + (f*x^4*(a +
b*x^4)^(5/2))/(14*b) - ((12*a*f - 35*b*d*x^2)*(a + b*x^4)^(5/2))/(420*b^2) - (a^3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt
[a + b*x^4]])/(32*b^(3/2)) + (4*a^(13/4)*e*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]
*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(65*b^(7/4)*Sqrt[a + b*x^4]) - (2*a^(11/4)*(65*Sqrt[b]*c + 77*
Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/
a^(1/4)], 1/2])/(5005*b^(7/4)*Sqrt[a + b*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.439691, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1833, 1274, 1280, 1198, 220, 1196, 1252, 833, 780, 195, 217, 206} \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (77 \sqrt{a} e+65 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left (a+b x^4\right )^{5/2} \left (12 a f-35 b d x^2\right )}{420 b^2}+\frac{1}{143} x^5 \left (a+b x^4\right )^{3/2} \left (13 c+11 e x^2\right )+\frac{2 a x^5 \sqrt{a+b x^4} \left (117 c+77 e x^2\right )}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*c*x*Sqrt[a + b*x^4])/(77*b) - (a^2*d*x^2*Sqrt[a + b*x^4])/(32*b) + (4*a^2*e*x^3*Sqrt[a + b*x^4])/(195*b
) - (4*a^3*e*x*Sqrt[a + b*x^4])/(65*b^(3/2)*(Sqrt[a] + Sqrt[b]*x^2)) + (2*a*x^5*(117*c + 77*e*x^2)*Sqrt[a + b*
x^4])/3003 - (a*d*x^2*(a + b*x^4)^(3/2))/(48*b) + (x^5*(13*c + 11*e*x^2)*(a + b*x^4)^(3/2))/143 + (f*x^4*(a +
b*x^4)^(5/2))/(14*b) - ((12*a*f - 35*b*d*x^2)*(a + b*x^4)^(5/2))/(420*b^2) - (a^3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt
[a + b*x^4]])/(32*b^(3/2)) + (4*a^(13/4)*e*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]
*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(65*b^(7/4)*Sqrt[a + b*x^4]) - (2*a^(11/4)*(65*Sqrt[b]*c + 77*
Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/
a^(1/4)], 1/2])/(5005*b^(7/4)*Sqrt[a + b*x^4])

Rule 1833

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

Rule 1274

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

Rule 1280

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

Rule 1198

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

Rule 220

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

Rule 1196

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

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

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

Rule 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 x^4 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx &=\int \left (x^4 \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}+x^5 \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}\right ) \, dx\\ &=\int x^4 \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2} \, dx+\int x^5 \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx\\ &=\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{1}{2} \operatorname{Subst}\left (\int x^2 (d+f x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )+\frac{1}{143} (6 a) \int x^4 \left (13 c+11 e x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}+\frac{\left (4 a^2\right ) \int \frac{x^4 \left (117 c+77 e x^2\right )}{\sqrt{a+b x^4}} \, dx}{3003}+\frac{\operatorname{Subst}\left (\int x (-2 a f+7 b d x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )}{14 b}\\ &=\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}-\frac{\left (4 a^2\right ) \int \frac{x^2 \left (231 a e-585 b c x^2\right )}{\sqrt{a+b x^4}} \, dx}{15015 b}-\frac{(a d) \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )}{12 b}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}+\frac{\left (4 a^2\right ) \int \frac{-585 a b c-693 a b e x^2}{\sqrt{a+b x^4}} \, dx}{45045 b^2}-\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^2\right )}{16 b}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}-\frac{\left (a^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{32 b}+\frac{\left (4 a^{7/2} e\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{65 b^{3/2}}-\frac{\left (4 a^3 \left (65 \sqrt{b} c+77 \sqrt{a} e\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{5005 b^{3/2}}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{2 a^{11/4} \left (65 \sqrt{b} c+77 \sqrt{a} e\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}-\frac{\left (a^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{32 b}\\ &=\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a x^5 \left (117 c+77 e x^2\right ) \sqrt{a+b x^4}}{3003}-\frac{a d x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac{1}{143} x^5 \left (13 c+11 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f x^4 \left (a+b x^4\right )^{5/2}}{14 b}-\frac{\left (12 a f-35 b d x^2\right ) \left (a+b x^4\right )^{5/2}}{420 b^2}-\frac{a^3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{32 b^{3/2}}+\frac{4 a^{13/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{2 a^{11/4} \left (65 \sqrt{b} c+77 \sqrt{a} e\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.864564, size = 225, normalized size = 0.47 \[ \frac{\sqrt{a+b x^4} \left (5005 b d x^2 \left (3 a^2+14 a b x^4+8 b^2 x^8\right )-\frac{43680 a^2 b c x \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}-\frac{15015 a^{5/2} \sqrt{b} d \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{\frac{b x^4}{a}+1}}-\frac{36960 a^2 b e x^3 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+43680 b c x \left (a+b x^4\right )^2+36960 b e x^3 \left (a+b x^4\right )^2+6864 f \left (a+b x^4\right )^2 \left (5 b x^4-2 a\right )\right )}{480480 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^4]*(43680*b*c*x*(a + b*x^4)^2 + 36960*b*e*x^3*(a + b*x^4)^2 + 6864*f*(a + b*x^4)^2*(-2*a + 5*b*x
^4) + 5005*b*d*x^2*(3*a^2 + 14*a*b*x^4 + 8*b^2*x^8) - (15015*a^(5/2)*Sqrt[b]*d*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])
/Sqrt[1 + (b*x^4)/a] - (43680*a^2*b*c*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] -
(36960*a^2*b*e*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(480480*b^2)

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Maple [C]  time = 0.039, size = 462, normalized size = 1. \begin{align*} -{\frac{f \left ( -5\,b{x}^{4}+2\,a \right ) \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{70\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{be{x}^{11}}{13}\sqrt{b{x}^{4}+a}}+{\frac{5\,ae{x}^{7}}{39}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}e{x}^{3}}{195\,b}\sqrt{b{x}^{4}+a}}-{{\frac{4\,i}{65}}e{a}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{4\,i}{65}}e{a}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{bd{x}^{10}}{12}\sqrt{b{x}^{4}+a}}+{\frac{7\,ad{x}^{6}}{48}\sqrt{b{x}^{4}+a}}+{\frac{{a}^{2}d{x}^{2}}{32\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{3}d}{32}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{bc{x}^{9}}{11}\sqrt{b{x}^{4}+a}}+{\frac{13\,ac{x}^{5}}{77}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}cx}{77\,b}\sqrt{b{x}^{4}+a}}-{\frac{4\,{a}^{3}c}{77\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/70*f*(b*x^4+a)^(1/2)*(-5*b*x^4+2*a)*(b^2*x^8+2*a*b*x^4+a^2)/b^2+1/13*e*b*x^11*(b*x^4+a)^(1/2)+5/39*e*a*x^7*
(b*x^4+a)^(1/2)+4/195*a^2*e*x^3*(b*x^4+a)^(1/2)/b-4/65*I*e/b^(3/2)*a^(7/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1
/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I
)+4/65*I*e/b^(3/2)*a^(7/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)
^(1/2)/(b*x^4+a)^(1/2)*EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+1/12*d*b*x^10*(b*x^4+a)^(1/2)+7/48*d*a*x^6*(b*
x^4+a)^(1/2)+1/32*a^2*d*x^2*(b*x^4+a)^(1/2)/b-1/32*d/b^(3/2)*a^3*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2))+1/11*c*b*x^9*
(b*x^4+a)^(1/2)+13/77*c*a*x^5*(b*x^4+a)^(1/2)+4/77*a^2*c*x*(b*x^4+a)^(1/2)/b-4/77*c/b*a^3/(I/a^(1/2)*b^(1/2))^
(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b
^(1/2))^(1/2),I)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b f x^{11} + b e x^{10} + b d x^{9} + b c x^{8} + a f x^{7} + a e x^{6} + a d x^{5} + a c x^{4}\right )} \sqrt{b x^{4} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*f*x^11 + b*e*x^10 + b*d*x^9 + b*c*x^8 + a*f*x^7 + a*e*x^6 + a*d*x^5 + a*c*x^4)*sqrt(b*x^4 + a), x)

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Sympy [A]  time = 18.0418, size = 462, normalized size = 0.97 \begin{align*} \frac{a^{\frac{5}{2}} d x^{2}}{32 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{17 a^{\frac{3}{2}} d x^{6}}{96 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{\sqrt{a} b c x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{11 \sqrt{a} b d x^{10}}{48 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b e x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} - \frac{a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 b^{\frac{3}{2}}} + a f \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + b f \left (\begin{cases} \frac{4 a^{3} \sqrt{a + b x^{4}}}{105 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{4}}}{105 b^{2}} + \frac{a x^{8} \sqrt{a + b x^{4}}}{70 b} + \frac{x^{12} \sqrt{a + b x^{4}}}{14} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} d x^{14}}{12 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(5/2)*d*x**2/(32*b*sqrt(1 + b*x**4/a)) + a**(3/2)*c*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_p
olar(I*pi)/a)/(4*gamma(9/4)) + 17*a**(3/2)*d*x**6/(96*sqrt(1 + b*x**4/a)) + a**(3/2)*e*x**7*gamma(7/4)*hyper((
-1/2, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(11/4)) + sqrt(a)*b*c*x**9*gamma(9/4)*hyper((-1/2, 9/4)
, (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(13/4)) + 11*sqrt(a)*b*d*x**10/(48*sqrt(1 + b*x**4/a)) + sqrt(a)*
b*e*x**11*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(15/4)) - a**3*d*asinh(sq
rt(b)*x**2/sqrt(a))/(32*b**(3/2)) + a*f*Piecewise((-a**2*sqrt(a + b*x**4)/(15*b**2) + a*x**4*sqrt(a + b*x**4)/
(30*b) + x**8*sqrt(a + b*x**4)/10, Ne(b, 0)), (sqrt(a)*x**8/8, True)) + b*f*Piecewise((4*a**3*sqrt(a + b*x**4)
/(105*b**3) - 2*a**2*x**4*sqrt(a + b*x**4)/(105*b**2) + a*x**8*sqrt(a + b*x**4)/(70*b) + x**12*sqrt(a + b*x**4
)/14, Ne(b, 0)), (sqrt(a)*x**12/12, True)) + b**2*d*x**14/(12*sqrt(a)*sqrt(1 + b*x**4/a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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