\(\int \frac {x^4 (a+b x^2)^{5/2}}{\sqrt {c+d x^2}} \, dx\) [960]

Optimal result

Integrand size = 26, antiderivative size = 553 \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {a+b x^2}}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\sqrt {c} \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b^2 d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{315 b d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

1/315*(-10*a^4*d^4-25*a^3*b*c*d^3+243*a^2*b^2*c^2*d^2-328*a*b^3*c^3*d+128*b^4*c^4)*x*(b*x^2+a)^(1/2)/b^2/d^4/(
d*x^2+c)^(1/2)+1/315*c^(3/2)*(-5*a^3*d^3+105*a^2*b*c*d^2-156*a*b^2*c^2*d+64*b^3*c^3)*(1/(1+d*x^2/c))^(1/2)*(1+
d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*(b*x^2+a)^(1/2)/b/d^(9/2)/(c*(
b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/315*(-10*a^4*d^4-25*a^3*b*c*d^3+243*a^2*b^2*c^2*d^2-328*a*b^3*c^
3*d+128*b^4*c^4)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/
a/d)^(1/2))*c^(1/2)*(b*x^2+a)^(1/2)/b^2/d^(9/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/9*b*x^5*(b*x
^2+a)^(3/2)*(d*x^2+c)^(1/2)/d-1/315*(-5*a^3*d^3+105*a^2*b*c*d^2-156*a*b^2*c^2*d+64*b^3*c^3)*x*(b*x^2+a)^(1/2)*
(d*x^2+c)^(1/2)/b/d^4+1/315*(75*a^2*d^2-115*a*b*c*d+48*b^2*c^2)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3-4/63*b
*(-3*a*d+2*b*c)*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^2

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {488, 595, 596, 545, 429, 506, 422} \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{315 d^3}+\frac {c^{3/2} \sqrt {a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{315 b d^{9/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{315 b d^4}-\frac {\sqrt {c} \sqrt {a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b^2 d^{9/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right )}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d} \]

[In]

Int[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

((128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*x*Sqrt[a + b*x^2])/(315*b
^2*d^4*Sqrt[c + d*x^2]) - ((64*b^3*c^3 - 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2]*Sqrt
[c + d*x^2])/(315*b*d^4) + ((48*b^2*c^2 - 115*a*b*c*d + 75*a^2*d^2)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*
d^3) - (4*b*(2*b*c - 3*a*d)*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*d^2) + (b*x^5*(a + b*x^2)^(3/2)*Sqrt[c +
d*x^2])/(9*d) - (Sqrt[c]*(128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*S
qrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(315*b^2*d^(9/2)*Sqrt[(c*(a + b*x^2))/
(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(64*b^3*c^3 - 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt
[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(315*b*d^(9/2)*Sqrt[(c*(a + b*x^2))/(a*(c
+ d*x^2))]*Sqrt[c + d*x^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 595

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rubi steps \begin{align*} \text {integral}& = \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\int \frac {x^4 \sqrt {a+b x^2} \left (-a (5 b c-9 a d)-4 b (2 b c-3 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{9 d} \\ & = -\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\int \frac {x^4 \left (a \left (40 b^2 c^2-95 a b c d+63 a^2 d^2\right )+b \left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{63 d^2} \\ & = \frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\int \frac {x^2 \left (3 a b c \left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right )+3 b \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{315 b d^3} \\ & = -\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\int \frac {3 a b c \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right )+3 b \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{945 b^2 d^4} \\ & = -\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\left (a c \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{315 b d^4}+\frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{315 b d^4} \\ & = \frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {a+b x^2}}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {c^{3/2} \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (c \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{315 b^2 d^4} \\ & = \frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {a+b x^2}}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\sqrt {c} \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b^2 d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.17 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 a^3 d^3+15 a^2 b d^2 \left (-7 c+5 d x^2\right )+a b^2 d \left (156 c^2-115 c d x^2+95 d^2 x^4\right )+b^3 \left (-64 c^3+48 c^2 d x^2-40 c d^2 x^4+35 d^3 x^6\right )\right )+i c \left (-128 b^4 c^4+328 a b^3 c^3 d-243 a^2 b^2 c^2 d^2+25 a^3 b c d^3+10 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (-128 b^4 c^4+392 a b^3 c^3 d-399 a^2 b^2 c^2 d^2+130 a^3 b c d^3+5 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b \sqrt {\frac {b}{a}} d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

Integrate[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(5*a^3*d^3 + 15*a^2*b*d^2*(-7*c + 5*d*x^2) + a*b^2*d*(156*c^2 - 115*c*d
*x^2 + 95*d^2*x^4) + b^3*(-64*c^3 + 48*c^2*d*x^2 - 40*c*d^2*x^4 + 35*d^3*x^6)) + I*c*(-128*b^4*c^4 + 328*a*b^3
*c^3*d - 243*a^2*b^2*c^2*d^2 + 25*a^3*b*c*d^3 + 10*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[
I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-128*b^4*c^4 + 392*a*b^3*c^3*d - 399*a^2*b^2*c^2*d^2 + 130*a^3*b*c
*d^3 + 5*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315
*b*Sqrt[b/a]*d^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 8.35 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.33

method	result	size
risch	\(\frac {x \left (35 b^{3} d^{3} x^{6}+95 a \,b^{2} d^{3} x^{4}-40 b^{3} c \,d^{2} x^{4}+75 x^{2} a^{2} b \,d^{3}-115 x^{2} a \,b^{2} c \,d^{2}+48 x^{2} b^{3} c^{2} d +5 a^{3} d^{3}-105 a^{2} b c \,d^{2}+156 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{315 b \,d^{4}}-\frac {\left (\frac {5 a^{4} c \,d^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {64 a \,b^{3} c^{4} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {105 a^{3} b \,c^{2} d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {156 a^{2} b^{2} c^{3} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (10 a^{4} d^{4}+25 a^{3} b c \,d^{3}-243 a^{2} b^{2} c^{2} d^{2}+328 a \,b^{3} c^{3} d -128 b^{4} c^{4}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{315 d^{4} b \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(738\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{2} x^{7} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{9 d}+\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{7 b d}+\frac {\left (3 a^{2} b -\frac {7 a \,b^{2} c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 b d}+\frac {\left (a^{3}-\frac {5 \left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) a c}{7 b d}-\frac {\left (3 a^{2} b -\frac {7 a \,b^{2} c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}-\frac {\left (a^{3}-\frac {5 \left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) a c}{7 b d}-\frac {\left (3 a^{2} b -\frac {7 a \,b^{2} c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (-\frac {3 \left (3 a^{2} b -\frac {7 a \,b^{2} c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) a c}{5 b d}-\frac {\left (a^{3}-\frac {5 \left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) a c}{7 b d}-\frac {\left (3 a^{2} b -\frac {7 a \,b^{2} c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(901\)
default	\(\text {Expression too large to display}\)	\(1047\)

[In]

int(x^4*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/315/b*x*(35*b^3*d^3*x^6+95*a*b^2*d^3*x^4-40*b^3*c*d^2*x^4+75*a^2*b*d^3*x^2-115*a*b^2*c*d^2*x^2+48*b^3*c^2*d*
x^2+5*a^3*d^3-105*a^2*b*c*d^2+156*a*b^2*c^2*d-64*b^3*c^3)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^4-1/315/d^4/b*(5*a
^4*c*d^3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/
a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-64*a*b^3*c^4/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*
x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-105*a^3*b*c^2*d^2/(-b/a)^(1/2)*(1+b*
x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)
^(1/2))+156*a^2*b^2*c^3*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(10*a^4*d^4+25*a^3*b*c*d^3-243*a^2*b^2*c^2*d^2+328*a*b^3*c
^3*d-128*b^4*c^4)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(El
lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))*((b*x^2+
a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (128 \, b^{4} c^{5} - 328 \, a b^{3} c^{4} d + 243 \, a^{2} b^{2} c^{3} d^{2} - 25 \, a^{3} b c^{2} d^{3} - 10 \, a^{4} c d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (128 \, b^{4} c^{5} - 328 \, a b^{3} c^{4} d - 5 \, a^{4} d^{5} + {\left (243 \, a^{2} b^{2} + 64 \, a b^{3}\right )} c^{3} d^{2} - {\left (25 \, a^{3} b + 156 \, a^{2} b^{2}\right )} c^{2} d^{3} - 5 \, {\left (2 \, a^{4} - 21 \, a^{3} b\right )} c d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (35 \, b^{4} d^{5} x^{8} + 128 \, b^{4} c^{4} d - 328 \, a b^{3} c^{3} d^{2} + 243 \, a^{2} b^{2} c^{2} d^{3} - 25 \, a^{3} b c d^{4} - 10 \, a^{4} d^{5} - 5 \, {\left (8 \, b^{4} c d^{4} - 19 \, a b^{3} d^{5}\right )} x^{6} + {\left (48 \, b^{4} c^{2} d^{3} - 115 \, a b^{3} c d^{4} + 75 \, a^{2} b^{2} d^{5}\right )} x^{4} - {\left (64 \, b^{4} c^{3} d^{2} - 156 \, a b^{3} c^{2} d^{3} + 105 \, a^{2} b^{2} c d^{4} - 5 \, a^{3} b d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{315 \, b^{2} d^{6} x} \]

[In]

integrate(x^4*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

-1/315*((128*b^4*c^5 - 328*a*b^3*c^4*d + 243*a^2*b^2*c^3*d^2 - 25*a^3*b*c^2*d^3 - 10*a^4*c*d^4)*sqrt(b*d)*x*sq
rt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (128*b^4*c^5 - 328*a*b^3*c^4*d - 5*a^4*d^5 + (243*a^2*b
^2 + 64*a*b^3)*c^3*d^2 - (25*a^3*b + 156*a^2*b^2)*c^2*d^3 - 5*(2*a^4 - 21*a^3*b)*c*d^4)*sqrt(b*d)*x*sqrt(-c/d)
*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (35*b^4*d^5*x^8 + 128*b^4*c^4*d - 328*a*b^3*c^3*d^2 + 243*a^2*b
^2*c^2*d^3 - 25*a^3*b*c*d^4 - 10*a^4*d^5 - 5*(8*b^4*c*d^4 - 19*a*b^3*d^5)*x^6 + (48*b^4*c^2*d^3 - 115*a*b^3*c*
d^4 + 75*a^2*b^2*d^5)*x^4 - (64*b^4*c^3*d^2 - 156*a*b^3*c^2*d^3 + 105*a^2*b^2*c*d^4 - 5*a^3*b*d^5)*x^2)*sqrt(b
*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^6*x)

Sympy [F]

\[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{4} \left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}}}\, dx \]

[In]

integrate(x**4*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**4*(a + b*x**2)**(5/2)/sqrt(c + d*x**2), x)

Maxima [F]

\[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{4}}{\sqrt {d x^{2} + c}} \,d x } \]

[In]

integrate(x^4*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c), x)

Giac [F]

\[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{4}}{\sqrt {d x^{2} + c}} \,d x } \]

[In]

integrate(x^4*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^4\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \]

[In]

int((x^4*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2),x)

[Out]

int((x^4*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2), x)