3.5.95 \(\int x^4 (c+d x+e x^2+f x^3) \sqrt {a+b x^4} \, dx\) [495]

Optimal. Leaf size=418 \[ \frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac {a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {2 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^{7/4} \left (5 \sqrt {b} c+7 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}} \]

[Out]

1/10*f*x^4*(b*x^4+a)^(3/2)/b-1/120*(-15*b*d*x^2+8*a*f)*(b*x^4+a)^(3/2)/b^2-1/16*a^2*d*arctanh(x^2*b^(1/2)/(b*x
^4+a)^(1/2))/b^(3/2)+2/21*a*c*x*(b*x^4+a)^(1/2)/b-1/16*a*d*x^2*(b*x^4+a)^(1/2)/b+2/45*a*e*x^3*(b*x^4+a)^(1/2)/
b+1/63*x^5*(7*e*x^2+9*c)*(b*x^4+a)^(1/2)-2/15*a^2*e*x*(b*x^4+a)^(1/2)/b^(3/2)/(a^(1/2)+x^2*b^(1/2))+2/15*a^(9/
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)-1/105*a^(7/4)*(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))*(7*e*a^(1/2)+5*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)

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Rubi [A]
time = 0.26, antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps used = 14, 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 {a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{7/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} 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 )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (a+b x^4\right )^{3/2} \left (8 a f-15 b d x^2\right )}{120 b^2}+\frac {1}{63} x^5 \sqrt {a+b x^4} \left (9 c+7 e x^2\right )+\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*c*x*Sqrt[a + b*x^4])/(21*b) - (a*d*x^2*Sqrt[a + b*x^4])/(16*b) + (2*a*e*x^3*Sqrt[a + b*x^4])/(45*b) - (2*
a^2*e*x*Sqrt[a + b*x^4])/(15*b^(3/2)*(Sqrt[a] + Sqrt[b]*x^2)) + (x^5*(9*c + 7*e*x^2)*Sqrt[a + b*x^4])/63 + (f*
x^4*(a + b*x^4)^(3/2))/(10*b) - ((8*a*f - 15*b*d*x^2)*(a + b*x^4)^(3/2))/(120*b^2) - (a^2*d*ArcTanh[(Sqrt[b]*x
^2)/Sqrt[a + b*x^4]])/(16*b^(3/2)) + (2*a^(9/4)*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])/(15*b^(7/4)*Sqrt[a + b*x^4]) - (a^(7/4)*(5*Sqrt[b]*c +
7*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])/(105*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 ) \sqrt {a+b x^4} \, dx &=\int \left (x^4 \left (c+e x^2\right ) \sqrt {a+b x^4}+x^5 \left (d+f x^2\right ) \sqrt {a+b x^4}\right ) \, dx\\ &=\int x^4 \left (c+e x^2\right ) \sqrt {a+b x^4} \, dx+\int x^5 \left (d+f x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{2} \text {Subst}\left (\int x^2 (d+f x) \sqrt {a+b x^2} \, dx,x,x^2\right )+\frac {1}{63} (2 a) \int \frac {x^4 \left (9 c+7 e x^2\right )}{\sqrt {a+b x^4}} \, dx\\ &=\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}+\frac {\text {Subst}\left (\int x (-2 a f+5 b d x) \sqrt {a+b x^2} \, dx,x,x^2\right )}{10 b}-\frac {(2 a) \int \frac {x^2 \left (21 a e-45 b c x^2\right )}{\sqrt {a+b x^4}} \, dx}{315 b}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}+\frac {(2 a) \int \frac {-45 a b c-63 a b e x^2}{\sqrt {a+b x^4}} \, dx}{945 b^2}-\frac {(a d) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )}{8 b}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac {\left (a^2 d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 b}+\frac {\left (2 a^{5/2} e\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 b^{3/2}}-\frac {\left (2 a^2 \left (5 \sqrt {b} c+7 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{105 b^{3/2}}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}+\frac {2 a^{9/4} 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 )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^{7/4} \left (5 \sqrt {b} c+7 \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 )}{105 b^{7/4} \sqrt {a+b x^4}}-\frac {\left (a^2 d\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{16 b}\\ &=\frac {2 a c x \sqrt {a+b x^4}}{21 b}-\frac {a d x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a e x^3 \sqrt {a+b x^4}}{45 b}-\frac {2 a^2 e x \sqrt {a+b x^4}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x^5 \left (9 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {f x^4 \left (a+b x^4\right )^{3/2}}{10 b}-\frac {\left (8 a f-15 b d x^2\right ) \left (a+b x^4\right )^{3/2}}{120 b^2}-\frac {a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {2 a^{9/4} 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 )}{15 b^{7/4} \sqrt {a+b x^4}}-\frac {a^{7/4} \left (5 \sqrt {b} c+7 \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 )}{105 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 = 10.45, size = 202, normalized size = 0.48 \begin {gather*} \frac {\sqrt {a+b x^4} \left (720 b c x \left (a+b x^4\right )+560 b e x^3 \left (a+b x^4\right )+315 b d x^2 \left (a+2 b x^4\right )+168 f \left (a+b x^4\right ) \left (-2 a+3 b x^4\right )-\frac {315 a^{3/2} \sqrt {b} d \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\frac {720 a b c x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}-\frac {560 a b e x^3 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right )}{5040 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^4]*(720*b*c*x*(a + b*x^4) + 560*b*e*x^3*(a + b*x^4) + 315*b*d*x^2*(a + 2*b*x^4) + 168*f*(a + b*x
^4)*(-2*a + 3*b*x^4) - (315*a^(3/2)*Sqrt[b]*d*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a] - (720*a*b*c
*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (560*a*b*e*x^3*Hypergeometric2F1[-1/
2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(5040*b^2)

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Maple [C] Result contains complex when optimal does not.
time = 0.37, size = 331, normalized size = 0.79

method	result	size
default	\(-\frac {f \left (b \,x^{4}+a \right )^{\frac {3}{2}} \left (-3 b \,x^{4}+2 a \right )}{30 b^{2}}+e \left (\frac {x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {2 a \,x^{3} \sqrt {b \,x^{4}+a}}{45 b}-\frac {2 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {x^{2} \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{8 b}-\frac {a \,x^{2} \sqrt {b \,x^{4}+a}}{16 b}-\frac {a^{2} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}\right )+c \left (\frac {x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {2 a x \sqrt {b \,x^{4}+a}}{21 b}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(331\)
risch	\(-\frac {\left (-504 b^{2} f \,x^{8}-560 b^{2} e \,x^{7}-630 b^{2} d \,x^{6}-720 b^{2} c \,x^{5}-168 a b f \,x^{4}-224 a b e \,x^{3}-315 a b d \,x^{2}-480 a b c x +336 a^{2} f \right ) \sqrt {b \,x^{4}+a}}{5040 b^{2}}-\frac {2 i a^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {2 i a^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{2} d \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}-\frac {2 a^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(349\)
elliptic	\(\frac {f \,x^{8} \sqrt {b \,x^{4}+a}}{10}+\frac {e \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {d \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {c \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {a f \,x^{4} \sqrt {b \,x^{4}+a}}{30 b}+\frac {2 a e \,x^{3} \sqrt {b \,x^{4}+a}}{45 b}+\frac {a d \,x^{2} \sqrt {b \,x^{4}+a}}{16 b}+\frac {2 a c x \sqrt {b \,x^{4}+a}}{21 b}-\frac {a^{2} f \sqrt {b \,x^{4}+a}}{15 b^{2}}-\frac {2 a^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {a^{2} d \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}-\frac {2 i a^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(358\)

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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/30*f*(b*x^4+a)^(3/2)*(-3*b*x^4+2*a)/b^2+e*(1/9*x^7*(b*x^4+a)^(1/2)+2/45*a/b*x^3*(b*x^4+a)^(1/2)-2/15*I/b^(3
/2)*a^(5/2)/(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)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)))+d*(1/8*x^2*(b*x^4
+a)^(3/2)/b-1/16*a/b*x^2*(b*x^4+a)^(1/2)-1/16*a^2/b^(3/2)*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2)))+c*(1/7*x^5*(b*x^4+a
)^(1/2)+2/21*a/b*x*(b*x^4+a)^(1/2)-2/21/b*a^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))

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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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.12, size = 214, normalized size = 0.51 \begin {gather*} -\frac {1344 \, a^{2} \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) - 315 \, a^{2} \sqrt {b} d x \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 192 \, {\left (5 \, a b c - 7 \, a^{2} 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 (504 \, b^{2} f x^{9} + 560 \, b^{2} e x^{8} + 630 \, b^{2} d x^{7} + 720 \, b^{2} c x^{6} + 168 \, a b f x^{5} + 224 \, a b e x^{4} + 315 \, a b d x^{3} + 480 \, a b c x^{2} - 336 \, a^{2} f x - 672 \, a^{2} e\right )} \sqrt {b x^{4} + a}}{10080 \, 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)^(1/2),x, algorithm="fricas")

[Out]

-1/10080*(1344*a^2*sqrt(b)*e*x*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4)/x), -1) - 315*a^2*sqrt(b)*d*x*log(-
2*b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(b)*x^2 - a) + 192*(5*a*b*c - 7*a^2*e)*sqrt(b)*x*(-a/b)^(3/4)*elliptic_f(arcsi
n((-a/b)^(1/4)/x), -1) - 2*(504*b^2*f*x^9 + 560*b^2*e*x^8 + 630*b^2*d*x^7 + 720*b^2*c*x^6 + 168*a*b*f*x^5 + 22
4*a*b*e*x^4 + 315*a*b*d*x^3 + 480*a*b*c*x^2 - 336*a^2*f*x - 672*a^2*e)*sqrt(b*x^4 + a))/(b^2*x)

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Sympy [A]
time = 3.71, size = 252, normalized size = 0.60 \begin {gather*} \frac {a^{\frac {3}{2}} d x^{2}}{16 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} 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 {3 \sqrt {a} d x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} 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 {a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + 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 ) + \frac {b d x^{10}}{8 \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)**(1/2),x)

[Out]

a**(3/2)*d*x**2/(16*b*sqrt(1 + b*x**4/a)) + sqrt(a)*c*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_po
lar(I*pi)/a)/(4*gamma(9/4)) + 3*sqrt(a)*d*x**6/(16*sqrt(1 + b*x**4/a)) + sqrt(a)*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)) - a**2*d*asinh(sqrt(b)*x**2/sqrt(a))/(16*b**(3/2))
+ 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, N
e(b, 0)), (sqrt(a)*x**8/8, True)) + b*d*x**10/(8*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)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^4 + a)*(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\,\sqrt {b\,x^4+a}\,\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)^(1/2)*(c + d*x + e*x^2 + f*x^3),x)

[Out]

int(x^4*(a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3), x)

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