3.1.35 \(\int \frac {(a+b x^2) (c+d x^2)^{5/2}}{\sqrt {e+f x^2}} \, dx\) [35]

Optimal. Leaf size=551 \[ \frac {\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d f^3 \sqrt {e+f x^2}}-\frac {\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 f^3}-\frac {(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}-\frac {\sqrt {e} \left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d f^{7/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 f^{7/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

1/105*(7*a*d*f*(23*c^2*f^2-23*c*d*e*f+8*d^2*e^2)-b*(-15*c^3*f^3+103*c^2*d*e*f^2-128*c*d^2*e^2*f+48*d^3*e^3))*x
*(d*x^2+c)^(1/2)/d/f^3/(f*x^2+e)^(1/2)-1/105*(7*a*d*f*(23*c^2*f^2-23*c*d*e*f+8*d^2*e^2)-b*(-15*c^3*f^3+103*c^2
*d*e*f^2-128*c*d^2*e^2*f+48*d^3*e^3))*(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)/d/f^(7/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(
1/2)+1/105*(7*a*f*(15*c^2*f^2-11*c*d*e*f+4*d^2*e^2)-b*e*(45*c^2*f^2-61*c*d*e*f+24*d^2*e^2))*(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)
/f^(7/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/35*(-7*a*d*f-5*b*c*f+6*b*d*e)*x*(d*x^2+c)^(3/2)*(f*
x^2+e)^(1/2)/f^2+1/7*b*x*(d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/f-1/105*(28*a*d*f*(-2*c*f+d*e)-b*(15*c^2*f^2-43*c*d*e
*f+24*d^2*e^2))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/f^3

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Rubi [A]
time = 0.43, antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {542, 545, 429, 506, 422} \begin {gather*} \frac {\sqrt {e} \sqrt {c+d x^2} \left (7 a f \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )-b e \left (45 c^2 f^2-61 c d e f+24 d^2 e^2\right )\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 f^{7/2} \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} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (-15 c^3 f^3+103 c^2 d e f^2-128 c d^2 e^2 f+48 d^3 e^3\right )\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 d f^{7/2} \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} \sqrt {e+f x^2} \left (28 a d f (d e-2 c f)-b \left (15 c^2 f^2-43 c d e f+24 d^2 e^2\right )\right )}{105 f^3}+\frac {x \sqrt {c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (-15 c^3 f^3+103 c^2 d e f^2-128 c d^2 e^2 f+48 d^3 e^3\right )\right )}{105 d f^3 \sqrt {e+f x^2}}-\frac {x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 - 128*c*d^2*e^2*f + 103*c^2*d*e*f^2 - 15*c^3*f
^3))*x*Sqrt[c + d*x^2])/(105*d*f^3*Sqrt[e + f*x^2]) - ((28*a*d*f*(d*e - 2*c*f) - b*(24*d^2*e^2 - 43*c*d*e*f +
15*c^2*f^2))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(105*f^3) - ((6*b*d*e - 5*b*c*f - 7*a*d*f)*x*(c + d*x^2)^(3/2)
*Sqrt[e + f*x^2])/(35*f^2) + (b*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*f) - (Sqrt[e]*(7*a*d*f*(8*d^2*e^2 - 23
*c*d*e*f + 23*c^2*f^2) - b*(48*d^3*e^3 - 128*c*d^2*e^2*f + 103*c^2*d*e*f^2 - 15*c^3*f^3))*Sqrt[c + d*x^2]*Elli
pticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*d*f^(7/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt
[e + f*x^2]) + (Sqrt[e]*(7*a*f*(4*d^2*e^2 - 11*c*d*e*f + 15*c^2*f^2) - b*e*(24*d^2*e^2 - 61*c*d*e*f + 45*c^2*f
^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*f^(7/2)*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 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\sqrt {e+f x^2}} \, dx &=\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}+\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (-c (b e-7 a f)+(-6 b d e+5 b c f+7 a d f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{7 f}\\ &=-\frac {(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}+\frac {\int \frac {\sqrt {c+d x^2} \left (-c (7 a f (d e-5 c f)-2 b e (3 d e-5 c f))+\left (-28 a d f (d e-2 c f)+b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x^2\right )}{\sqrt {e+f x^2}} \, dx}{35 f^2}\\ &=-\frac {\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 f^3}-\frac {(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}+\frac {\int \frac {c \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\right )+\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 f^3}\\ &=-\frac {\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 f^3}-\frac {(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}+\frac {\left (c \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 f^3}+\frac {\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 f^3}\\ &=\frac {\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d f^3 \sqrt {e+f x^2}}-\frac {\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 f^3}-\frac {(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}+\frac {\sqrt {e} \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\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 )}{105 f^{7/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (e \left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{105 d f^3}\\ &=\frac {\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 d f^3 \sqrt {e+f x^2}}-\frac {\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 f^3}-\frac {(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{35 f^2}+\frac {b x \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{7 f}-\frac {\sqrt {e} \left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\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 )}{105 d f^{7/2} \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 (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\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 )}{105 f^{7/2} \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 = 4.27, size = 386, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (-4 d e+11 c f+3 d f x^2\right )+b \left (45 c^2 f^2+c d f \left (-61 e+45 f x^2\right )+3 d^2 \left (8 e^2-6 e f x^2+5 f^2 x^4\right )\right )\right )-i e \left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )+b \left (-48 d^3 e^3+128 c d^2 e^2 f-103 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}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i (-d e+c f) \left (4 b e \left (12 d^2 e^2-26 c d e f+15 c^2 f^2\right )-7 a f \left (8 d^2 e^2-19 c d e f+15 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 )}{105 \sqrt {\frac {d}{c}} f^4 \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(7*a*d*f*(-4*d*e + 11*c*f + 3*d*f*x^2) + b*(45*c^2*f^2 + c*d*f*(-61*e +
45*f*x^2) + 3*d^2*(8*e^2 - 6*e*f*x^2 + 5*f^2*x^4))) - I*e*(7*a*d*f*(8*d^2*e^2 - 23*c*d*e*f + 23*c^2*f^2) + b*
(-48*d^3*e^3 + 128*c*d^2*e^2*f - 103*c^2*d*e*f^2 + 15*c^3*f^3))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ellipt
icE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(-(d*e) + c*f)*(4*b*e*(12*d^2*e^2 - 26*c*d*e*f + 15*c^2*f^2) - 7*
a*f*(8*d^2*e^2 - 19*c*d*e*f + 15*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)])/(105*Sqrt[d/c]*f^4*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. \(1385\) vs. \(2(573)=1146\).
time = 0.14, size = 1386, normalized size = 2.52

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b \,d^{2} x^{5} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{7 f}+\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {b \,d^{2} \left (6 c f +6 d e \right )}{7 f}\right ) x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d f}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d -\frac {5 c \,d^{2} e b}{7 f}-\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {b \,d^{2} \left (6 c f +6 d e \right )}{7 f}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (c^{3} a -\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d -\frac {5 c \,d^{2} e b}{7 f}-\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {b \,d^{2} \left (6 c f +6 d e \right )}{7 f}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (3 a \,c^{2} d +b \,c^{3}-\frac {3 \left (a \,d^{3}+3 b c \,d^{2}-\frac {b \,d^{2} \left (6 c f +6 d e \right )}{7 f}\right ) c e}{5 d f}-\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d -\frac {5 c \,d^{2} e b}{7 f}-\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {b \,d^{2} \left (6 c f +6 d e \right )}{7 f}\right ) \left (4 c f +4 d e \right )}{5 d f}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(691\)
risch	\(\frac {x \left (15 b \,x^{4} d^{2} f^{2}+21 a \,d^{2} f^{2} x^{2}+45 b c d \,f^{2} x^{2}-18 b \,d^{2} e f \,x^{2}+77 a c d \,f^{2}-28 a \,d^{2} e f +45 b \,c^{2} f^{2}-61 b c d e f +24 b \,d^{2} e^{2}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{105 f^{3}}+\frac {\left (-\frac {\left (161 a \,c^{2} d \,f^{3}-161 a c \,d^{2} e \,f^{2}+56 a \,d^{3} e^{2} f +15 b \,c^{3} f^{3}-103 b \,c^{2} d e \,f^{2}+128 b c \,d^{2} e^{2} f -48 b \,d^{3} e^{3}\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 {105 c^{3} a \,f^{3} \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 {77 a \,c^{2} d e \,f^{2} \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 {28 a c \,d^{2} e^{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 {45 b \,c^{3} e \,f^{2} \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 {61 b \,c^{2} d \,e^{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 {24 b c \,d^{2} e^{3} \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 ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{105 f^{3} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(922\)
default	\(\text {Expression too large to display}\)	\(1386\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(128*((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))*b*c*d^2*e^3*f-103*((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))*b*c^2*d*e^2*f^2-152*((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))*b*c
*d^2*e^3*f+161*((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*c^2*d*e*f^3
-161*((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*c*d^2*e^2*f^2-238*((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*c^2*d*e*f^3+189*((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*c*d^2*e^2*f^2+164*((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))*b*c^2*d*e^2*f^2+45*(-d/c)^(1/2)*b*c^3*e*f^3*x-19*
(-d/c)^(1/2)*b*c*d^2*e*f^3*x^5+70*(-d/c)^(1/2)*a*c*d^2*e*f^3*x^3+29*(-d/c)^(1/2)*b*c^2*d*e*f^3*x^3-55*(-d/c)^(
1/2)*b*c*d^2*e^2*f^2*x^3+60*(-d/c)^(1/2)*b*c*d^2*f^4*x^7-3*(-d/c)^(1/2)*b*d^3*e*f^3*x^7+98*(-d/c)^(1/2)*a*c*d^
2*f^4*x^5-7*(-d/c)^(1/2)*a*d^3*e*f^3*x^5+90*(-d/c)^(1/2)*b*c^2*d*f^4*x^5+6*(-d/c)^(1/2)*b*d^3*e^2*f^2*x^5-56*(
(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*d^3*e^3*f-60*((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))*b*c^3*e*f^3+56*((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*d^3*e^3*f+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))*b*c^3*e*f^3+77*(-d/c)^(1/2)*a*c^2*d*e*f^3*x-28*(-d/c)^(1/2)*a*c*d^2*
e^2*f^2*x-61*(-d/c)^(1/2)*b*c^2*d*e^2*f^2*x+24*(-d/c)^(1/2)*b*c*d^2*e^3*f*x+77*(-d/c)^(1/2)*a*c^2*d*f^4*x^3-28
*(-d/c)^(1/2)*a*d^3*e^2*f^2*x^3+24*(-d/c)^(1/2)*b*d^3*e^3*f*x^3+15*(-d/c)^(1/2)*b*d^3*f^4*x^9+21*(-d/c)^(1/2)*
a*d^3*f^4*x^7+45*(-d/c)^(1/2)*b*c^3*f^4*x^3+48*((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))*b*d^3*e^4-48*((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))*b*d^3*e^4+105*((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*c^3*f^4)
/f^4/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/(-d/c)^(1/2)

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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