3.1.64 \(\int \frac {(c+d x^2)^{5/2} \sqrt {e+f x^2}}{a+b x^2} \, dx\) [64]

Optimal. Leaf size=608 \[ \frac {d \left (7 c e-\frac {2 d e^2}{f}+\frac {3 c^2 f}{d}\right ) x \sqrt {c+d x^2}}{15 b \sqrt {e+f x^2}}+\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {\sqrt {e} \left (15 a^2 d^2 f^2-5 a b d f (d e+7 c f)+b^2 \left (-2 d^2 e^2+12 c d e f+23 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d e^{3/2} \left (-40 a b c d f+15 a^2 d^2 f+b^2 c (-d e+34 c f)\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b^3 c f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

1/5*d^2*x*(f*x^2+e)^(3/2)*(d*x^2+c)^(1/2)/b/f+1/15*d*(7*c*e-2*d*e^2/f+3*c^2*f/d)*x*(d*x^2+c)^(1/2)/b/(f*x^2+e)
^(1/2)+1/3*(-a*d+b*c)*(-3*a*d*f+4*b*c*f+b*d*e)*x*(d*x^2+c)^(1/2)/b^3/(f*x^2+e)^(1/2)+1/15*d*e^(3/2)*(-40*a*b*c
*d*f+15*a^2*d^2*f+b^2*c*(34*c*f-d*e))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f
*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/b^3/c/f^(3/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
-1/15*(15*a^2*d^2*f^2-5*a*b*d*f*(7*c*f+d*e)+b^2*(23*c^2*f^2+12*c*d*e*f-2*d^2*e^2))*(1/(1+f*x^2/e))^(1/2)*(1+f*
x^2/e)^(1/2)*EllipticE(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*(d*x^2+c)^(1/2)/b^3/f^(3
/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+(-a*d+b*c)^3*e^(3/2)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/
2)*EllipticPi(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),1-b*e/a/f,(1-d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/a/b^3/c/f^(1/2)
/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d*(-a*d+b*c)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^2-2/15*d
*(-3*c*f+d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b/f

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 776, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {559, 427, 542, 545, 429, 506, 422, 557, 553} \begin {gather*} \frac {d e^{3/2} \sqrt {c+d x^2} (5 b c-3 a d) (b c-a d) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {e^{3/2} \sqrt {c+d x^2} (b c-a d)^3 \Pi \left (1-\frac {b e}{a f};\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} (b c-a d) (-3 a d f+4 b c f+b d e)}{3 b^3 \sqrt {e+f x^2}}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)}{3 b^2}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {d e^{3/2} \sqrt {c+d x^2} (d e-9 c f) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {d x \sqrt {c+d x^2} \left (\frac {3 c^2 f}{d}+7 c e-\frac {2 d e^2}{f}\right )}{15 b \sqrt {e+f x^2}}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {2 d x \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-3 c f)}{15 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(7*c*e - (2*d*e^2)/f + (3*c^2*f)/d)*x*Sqrt[c + d*x^2])/(15*b*Sqrt[e + f*x^2]) + ((b*c - a*d)*(b*d*e + 4*b*c
*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(
3*b^2) - (2*d*(d*e - 3*c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b*f) + (d^2*x*Sqrt[c + d*x^2]*(e + f*x^2)^(
3/2))/(5*b*f) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/
Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqrt[e]*(
2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(1
5*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*(b*c - a*d)*e^(3/2)*Sq
rt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(
c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*e^(3/2)*(d*e - 9*c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[
e]], 1 - (d*e)/(c*f)])/(15*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^3*e
^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b^3*c*Sqr
t[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

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 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 542

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

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 553

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[c*(Sqrt[e +
f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), Ar
cTan[Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 557

Int[(((c_) + (d_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[(b*c - a*
d)^2/b^2, Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] + Dist[d/b^2, Int[(2*b*c - a*d + b*d*x^2)*
(Sqrt[e + f*x^2]/Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]

Rule 559

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/b, Int[
(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Dist[(b*c - a*d)/b, Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*
x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{a+b x^2} \, dx &=\frac {d \int \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} \, dx}{b}+\frac {(b c-a d) \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{a+b x^2} \, dx}{b}\\ &=\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {(d (b c-a d)) \int \frac {\left (2 b c-a d+b d x^2\right ) \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx}{b^3}+\frac {(b c-a d)^3 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}+\frac {d \int \frac {\sqrt {e+f x^2} \left (-c (d e-5 c f)-2 d (d e-3 c f) x^2\right )}{\sqrt {c+d x^2}} \, dx}{5 b f}\\ &=\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d) \int \frac {d (5 b c-3 a d) e+d (b d e+4 b c f-3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}+\frac {\int \frac {-c d e (d e-9 c f)-d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f}\\ &=\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(d (5 b c-3 a d) (b c-a d) e) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}-\frac {(c d e (d e-9 c f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f}+\frac {(d (b c-a d) (b d e+4 b c f-3 a d f)) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b^3}-\frac {\left (d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 b f}\\ &=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {e+f x^2}}-\frac {\left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b f \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d e^{3/2} (d e-9 c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {((b c-a d) e (b d e+4 b c f-3 a d f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b^3}+\frac {\left (e \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 b f}\\ &=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {e+f x^2}}-\frac {\left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) x \sqrt {c+d x^2}}{15 b f \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b^2}-\frac {2 d (d e-3 c f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 b f}+\frac {d^2 x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 b f}-\frac {(b c-a d) \sqrt {e} (b d e+4 b c f-3 a d f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d e^{3/2} (d e-9 c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b^3 c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 5.62, size = 456, normalized size = 0.75 \begin {gather*} \frac {-i a b d e \left (15 a^2 d^2 f^2-5 a b d f (d e+7 c f)+b^2 \left (-2 d^2 e^2+12 c d e f+23 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a \left (45 a^2 b c d^2 f^3-15 a^3 d^3 f^3+5 a b^2 d f \left (d^2 e^2-c d e f-9 c^2 f^2\right )+b^3 \left (2 d^3 e^3-13 c d^2 e^2 f+11 c^2 d e f^2+15 c^3 f^3\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+f \left (a b^2 d \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (11 b c f-5 a d f+b d \left (e+3 f x^2\right )\right )-15 i (b c-a d)^3 f (b e-a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{15 a b^4 \sqrt {\frac {d}{c}} f^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*a*b*d*e*(15*a^2*d^2*f^2 - 5*a*b*d*f*(d*e + 7*c*f) + b^2*(-2*d^2*e^2 + 12*c*d*e*f + 23*c^2*f^2))*Sqrt[1 +
(d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(45*a^2*b*c*d^2*f^3 - 15*
a^3*d^3*f^3 + 5*a*b^2*d*f*(d^2*e^2 - c*d*e*f - 9*c^2*f^2) + b^3*(2*d^3*e^3 - 13*c*d^2*e^2*f + 11*c^2*d*e*f^2 +
15*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^
2*d*Sqrt[d/c]*x*(c + d*x^2)*(e + f*x^2)*(11*b*c*f - 5*a*d*f + b*d*(e + 3*f*x^2)) - (15*I)*(b*c - a*d)^3*f*(b*e
- a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))
/(15*a*b^4*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1890\) vs. \(2(649)=1298\).
time = 0.23, size = 1891, normalized size = 3.11

method	result	size
risch	\(-\frac {d x \left (-3 b d \,x^{2} f +5 a d f -11 b c f -b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 f \,b^{2}}+\frac {\left (-\frac {d \left (15 a^{2} d^{2} f^{2}-35 a b c d \,f^{2}-5 a b \,d^{2} e f +23 b^{2} c^{2} f^{2}+12 b^{2} c d e f -2 b^{2} d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}-\frac {\left (15 a^{3} d^{3} f^{2}-45 a^{2} b c \,d^{2} f^{2}-15 a^{2} b \,d^{3} e f +45 a \,b^{2} c^{2} d \,f^{2}+40 a \,b^{2} c \,d^{2} e f -15 b^{3} c^{3} f^{2}-34 b^{3} c^{2} d e f +b^{3} c \,d^{2} e^{2}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {15 \left (a^{4} d^{3} f -3 a^{3} b c \,d^{2} f -a^{3} b \,d^{3} e +3 a^{2} b^{2} c^{2} d f +3 a^{2} b^{2} c \,d^{2} e -a \,b^{3} c^{3} f -3 a \,b^{3} c^{2} d e +c^{3} e \,b^{4}\right ) f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{b^{2} a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 f \,b^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(657\)
default	\(\text {Expression too large to display}\)	\(1891\)
elliptic	\(\text {Expression too large to display}\)	\(2436\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(5*(-d/c)^(1/2)*a^2*b^2*d^3*f^3*x^5-15*(-d/c)^(1/2)*a*b^3*c*d^2*e*f^2*x^
3-45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*b*c*d^2*f^3+45*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*c^2*d*f^3-5*((d*x^2+c)/c
)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*d^3*e^2*f-23*((d*x^2+c)/c)^(1/2)
*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*c^2*d*e*f^2-14*(-d/c)^(1/2)*a*b^3*c*d^2*f
^3*x^5-4*(-d/c)^(1/2)*a*b^3*d^3*e*f^2*x^5+5*(-d/c)^(1/2)*a^2*b^2*c*d^2*f^3*x^3+5*(-d/c)^(1/2)*a^2*b^2*d^3*e*f^
2*x^3-11*(-d/c)^(1/2)*a*b^3*c^2*d*f^3*x^3-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*a*b^3*c^3*f^3-2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1
/2))*a*b^3*d^3*e^3+2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*d
^3*e^3+15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))
*a*b^3*c^3*f^3-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c
)^(1/2))*b^4*c^3*e*f^2-(-d/c)^(1/2)*a*b^3*d^3*e^2*f*x^3+15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^4*d^3*f^3-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2
),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^4*d^3*f^3-11*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)
^(1/2),(c*f/d/e)^(1/2))*a*b^3*c^2*d*e*f^2+13*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*a*b^3*c*d^2*e^2*f+35*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/
e)^(1/2))*a^2*b^2*c*d^2*e*f^2-11*(-d/c)^(1/2)*a*b^3*c^2*d*e*f^2*x-(-d/c)^(1/2)*a*b^3*c*d^2*e^2*f*x+5*((d*x^2+c
)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*d^3*e^2*f+45*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^3*b*c*d^2*f^3+15*((d*x^
2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^3*b*d^3*e*f^2
-45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*b
^2*c^2*d*f^3-3*(-d/c)^(1/2)*a*b^3*d^3*f^3*x^7+5*(-d/c)^(1/2)*a^2*b^2*c*d^2*e*f^2*x-15*((d*x^2+c)/c)^(1/2)*((f*
x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*b*d^3*e*f^2+5*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^
(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*c*d^2*e*f^2-12*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)
*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*c*d^2*e^2*f-45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipt
icPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*b^2*c*d^2*e*f^2+45*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b^3*c^2*d*e*f^2)/(d*f*x^4+c*f*x^2+d*e*x
^2+c*e)/b^4/f^2/(-d/c)^(1/2)/a

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}} \sqrt {e + f x^{2}}}{a + b x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}}{b\,x^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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