3.6.10 \(\int x^4 (c+d x+e x^2+f x^3) (a+b x^4)^{3/2} \, dx\) [510]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1847, 1288, 1294, 1212, 226, 1210, 1266, 847, 794, 201, 223, 212} \begin {gather*} -\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 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5005 b^{7/4} \sqrt {a+b x^4}}+\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 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 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^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} \end {gather*}

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 201

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 212

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

Rule 223

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 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 794

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 847

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 1210

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)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

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 1266

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 1288

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 1294

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 1847

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)/c^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, {
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)]

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} \text {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 {\text {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) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.54, size = 225, normalized size = 0.47 \begin {gather*} \frac {\sqrt {a+b x^4} \left (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 (-2 a+5 b x^4\right )+5005 b d x^2 \left (3 a^2+14 a b x^4+8 b^2 x^8\right )-\frac {15015 a^{5/2} \sqrt {b} d \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\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 {1+\frac {b x^4}{a}}}-\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 {1+\frac {b x^4}{a}}}\right )}{480480 b^2} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.37, size = 400, normalized size = 0.84

method	result	size
risch	\(-\frac {\left (-34320 f \,x^{12} b^{3}-36960 b^{3} e \,x^{11}-40040 b^{3} d \,x^{10}-43680 b^{3} c \,x^{9}-54912 a \,b^{2} f \,x^{8}-61600 a \,b^{2} e \,x^{7}-70070 a \,b^{2} d \,x^{6}-81120 a \,b^{2} c \,x^{5}-6864 a^{2} b f \,x^{4}-9856 a^{2} b e \,x^{3}-15015 a^{2} b d \,x^{2}-24960 a^{2} b c x +13728 a^{3} f \right ) \sqrt {b \,x^{4}+a}}{480480 b^{2}}-\frac {4 i a^{\frac {7}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{65 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i a^{\frac {7}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{65 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{3} d \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}-\frac {4 a^{3} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(397\)
default	\(-\frac {f \sqrt {b \,x^{4}+a}\, \left (-5 b \,x^{4}+2 a \right ) \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )}{70 b^{2}}+e \left (\frac {b \,x^{11} \sqrt {b \,x^{4}+a}}{13}+\frac {5 a \,x^{7} \sqrt {b \,x^{4}+a}}{39}+\frac {4 a^{2} x^{3} \sqrt {b \,x^{4}+a}}{195 b}-\frac {4 i a^{\frac {7}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {b \,x^{10} \sqrt {b \,x^{4}+a}}{12}+\frac {7 a \,x^{6} \sqrt {b \,x^{4}+a}}{48}+\frac {a^{2} x^{2} \sqrt {b \,x^{4}+a}}{32 b}-\frac {a^{3} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}\right )+c \left (\frac {b \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {4 a^{2} x \sqrt {b \,x^{4}+a}}{77 b}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(400\)
elliptic	\(\frac {b f \,x^{12} \sqrt {b \,x^{4}+a}}{14}+\frac {b e \,x^{11} \sqrt {b \,x^{4}+a}}{13}+\frac {b d \,x^{10} \sqrt {b \,x^{4}+a}}{12}+\frac {b c \,x^{9} \sqrt {b \,x^{4}+a}}{11}+\frac {4 a f \,x^{8} \sqrt {b \,x^{4}+a}}{35}+\frac {5 a e \,x^{7} \sqrt {b \,x^{4}+a}}{39}+\frac {7 a d \,x^{6} \sqrt {b \,x^{4}+a}}{48}+\frac {13 a c \,x^{5} \sqrt {b \,x^{4}+a}}{77}+\frac {a^{2} f \,x^{4} \sqrt {b \,x^{4}+a}}{70 b}+\frac {4 a^{2} e \,x^{3} \sqrt {b \,x^{4}+a}}{195 b}+\frac {a^{2} d \,x^{2} \sqrt {b \,x^{4}+a}}{32 b}+\frac {4 a^{2} c x \sqrt {b \,x^{4}+a}}{77 b}-\frac {a^{3} f \sqrt {b \,x^{4}+a}}{35 b^{2}}-\frac {4 a^{3} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{3} d \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}-\frac {4 i a^{\frac {7}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(434\)

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,method=_RETURNVERBOSE)

[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+e*(1/13*b*x^11*(b*x^4+a)^(1/2)+5/39*a*x^7*(
b*x^4+a)^(1/2)+4/195/b*a^2*x^3*(b*x^4+a)^(1/2)-4/65*I/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)-El
lipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)))+d*(1/12*b*x^10*(b*x^4+a)^(1/2)+7/48*a*x^6*(b*x^4+a)^(1/2)+1/32/b*a^2*
x^2*(b*x^4+a)^(1/2)-1/32/b^(3/2)*a^3*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2)))+c*(1/11*b*x^9*(b*x^4+a)^(1/2)+13/77*a*x^
5*(b*x^4+a)^(1/2)+4/77/b*a^2*x*(b*x^4+a)^(1/2)-4/77/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 = 0.00, 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^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 + x^2*e + d*x + c)*x^4, x)

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Fricas [A]
time = 0.15, size = 264, normalized size = 0.55 \begin {gather*} -\frac {59136 \, a^{3} \sqrt {b} e x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 15015 \, a^{3} \sqrt {b} d x \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 768 \, {\left (65 \, a^{2} b c - 77 \, a^{3} e\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, {\left (34320 \, b^{3} f x^{13} + 36960 \, b^{3} e x^{12} + 40040 \, b^{3} d x^{11} + 43680 \, b^{3} c x^{10} + 54912 \, a b^{2} f x^{9} + 61600 \, a b^{2} e x^{8} + 70070 \, a b^{2} d x^{7} + 81120 \, a b^{2} c x^{6} + 6864 \, a^{2} b f x^{5} + 9856 \, a^{2} b e x^{4} + 15015 \, a^{2} b d x^{3} + 24960 \, a^{2} b c x^{2} - 13728 \, a^{3} f x - 29568 \, a^{3} e\right )} \sqrt {b x^{4} + a}}{960960 \, b^{2} x} \end {gather*}

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]

-1/960960*(59136*a^3*sqrt(b)*e*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4)/x), -1) - 15015*a^3*sqrt(b)*d*x*l
og(-2*b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(b)*x^2 - a) + 768*(65*a^2*b*c - 77*a^3*e)*sqrt(b)*x*(-a/b)^(3/4)*elliptic
_f(arcsin((-a/b)^(1/4)/x), -1) - 2*(34320*b^3*f*x^13 + 36960*b^3*e*x^12 + 40040*b^3*d*x^11 + 43680*b^3*c*x^10
+ 54912*a*b^2*f*x^9 + 61600*a*b^2*e*x^8 + 70070*a*b^2*d*x^7 + 81120*a*b^2*c*x^6 + 6864*a^2*b*f*x^5 + 9856*a^2*
b*e*x^4 + 15015*a^2*b*d*x^3 + 24960*a^2*b*c*x^2 - 13728*a^3*f*x - 29568*a^3*e)*sqrt(b*x^4 + a))/(b^2*x)

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Sympy [A]
time = 10.23, size = 462, normalized size = 0.97 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 + x^2*e + d*x + c)*x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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