3.1.77 \(\int \frac {(c+d x^2)^{5/2}}{(a+b x^2) \sqrt {e+f x^2}} \, dx\) [77]

Optimal. Leaf size=621 \[ \frac {d (b c-a d) x \sqrt {c+d x^2}}{b^2 \sqrt {e+f x^2}}-\frac {2 d (d e-2 c f) x \sqrt {c+d x^2}}{3 b f \sqrt {e+f x^2}}+\frac {d^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b f}-\frac {d (b c-a d) \sqrt {e} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{b^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {2 d \sqrt {e} (d e-2 c f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (b c-a d) \sqrt {e} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{b^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {e} (d e-3 c f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {c^{3/2} (b c-a d)^2 \sqrt {e+f x^2} \Pi \left (1-\frac {b c}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a b^2 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {559, 427, 545, 429, 506, 422, 433, 553} \begin {gather*} \frac {c^{3/2} \sqrt {e+f x^2} (b c-a d)^2 \Pi \left (1-\frac {b c}{a d};\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{a b^2 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d \sqrt {e} \sqrt {c+d x^2} (b c-a d) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{b^2 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {d \sqrt {e} \sqrt {c+d x^2} (b c-a d) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{b^2 \sqrt {f} \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} (b c-a d)}{b^2 \sqrt {e+f x^2}}-\frac {d \sqrt {e} \sqrt {c+d x^2} (d e-3 c f) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 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 {2 d \sqrt {e} \sqrt {c+d x^2} (d e-2 c f) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 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^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b f}-\frac {2 d x \sqrt {c+d x^2} (d e-2 c f)}{3 b f \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*c - a*d)*x*Sqrt[c + d*x^2])/(b^2*Sqrt[e + f*x^2]) - (2*d*(d*e - 2*c*f)*x*Sqrt[c + d*x^2])/(3*b*f*Sqrt[e
+ f*x^2]) + (d^2*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b*f) - (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticE
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*
x^2]) + (2*d*Sqrt[e]*(d*e - 2*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3
*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*Ell
ipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[
e + f*x^2]) - (d*Sqrt[e]*(d*e - 3*c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)]
)/(3*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)^2*Sqrt[e + f*x^2]
*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^2*Sqrt[d]*e*Sqrt[c + d*x^2]*S
qrt[(c*(e + f*x^2))/(e*(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 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 433

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 4.98, size = 350, normalized size = 0.56 \begin {gather*} \frac {-i a b d^2 e (-2 b d e+7 b c f-3 a d f) \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 d \left (3 a^2 d^2 f^2+3 a b d f (d e-3 c f)+b^2 \left (2 d^2 e^2-8 c d e f+9 c^2 f^2\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 c d \left (\frac {d}{c}\right )^{3/2} x \left (c+d x^2\right ) \left (e+f x^2\right )-3 i (b c-a d)^3 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 )}{3 a b^3 \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)/((a + b*x^2)*Sqrt[e + f*x^2]),x]

[Out]

((-I)*a*b*d^2*e*(-2*b*d*e + 7*b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqr
t[d/c]*x], (c*f)/(d*e)] - I*a*d*(3*a^2*d^2*f^2 + 3*a*b*d*f*(d*e - 3*c*f) + b^2*(2*d^2*e^2 - 8*c*d*e*f + 9*c^2*
f^2))*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*c*d*(d
/c)^(3/2)*x*(c + d*x^2)*(e + f*x^2) - (3*I)*(b*c - a*d)^3*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)]))/(3*a*b^3*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]
time = 0.19, size = 988, normalized size = 1.59

method	result	size
risch	\(\frac {d^{2} x \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{3 b f}-\frac {\left (\frac {d \left (-\frac {\left (3 a b \,d^{2} f -7 b^{2} c d f +2 b^{2} d^{2} e \right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}-\frac {3 a^{2} d^{2} f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {9 a b c d f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {9 b^{2} c^{2} f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {b^{2} d c e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{b^{2}}+\frac {3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{b^{2} a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{3 f b \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(741\)
default	\(\frac {\left (\sqrt {-\frac {d}{c}}\, a \,b^{2} d^{3} f^{2} x^{5}+\sqrt {-\frac {d}{c}}\, a \,b^{2} c \,d^{2} f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, a \,b^{2} d^{3} e f \,x^{3}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{3} d^{3} f^{2}-9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b c \,d^{2} f^{2}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b \,d^{3} e f +9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} c^{2} d \,f^{2}-8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} c \,d^{2} e f +2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d^{3} e^{2}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b \,d^{3} e f +7 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} c \,d^{2} e f -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d^{3} e^{2}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{3} d^{3} f^{2}+9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{2} b c \,d^{2} f^{2}-9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a \,b^{2} c^{2} d \,f^{2}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b^{3} c^{3} f^{2}+\sqrt {-\frac {d}{c}}\, a \,b^{2} c \,d^{2} e f x \right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{3 a \sqrt {-\frac {d}{c}}\, f^{2} b^{3} \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\)	\(988\)
elliptic	\(\text {Expression too large to display}\)	\(1356\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-d/c)^(1/2)*a*b^2*d^3*f^2*x^5+(-d/c)^(1/2)*a*b^2*c*d^2*f^2*x^3+(-d/c)^(1/2)*a*b^2*d^3*e*f*x^3+3*((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*d^3*f^2-9*((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*c*d^2*f^2+3*((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*d^3*e*f+9*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*c^2*d*f^2-8*((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^2*c*d^2*e*f+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^2*d^3*e^2-3*((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*d^3*e*f+7*((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^2*
c*d^2*e*f-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^2*d^3*e^2-3*
((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*d^3*f
^2+9*((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*c*d^2*f^2-9*((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^2*c^2*d*f^2+3*((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^3*c^3*f^2+(-d/c)^(1/2)*a*b^2*c*d^2*e*f*x)*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/a/(-d/c)^(1/2)/f^2/b
^3/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*sqrt(f*x^2 + e)), 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)/(b*x^2+a)/(f*x^2+e)^(1/2),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}}}{\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*sqrt(f*x^2 + e)), 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}}{\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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