3.1.76 \(\int \frac {(e+f x^2)^{3/2}}{(a+b x^2) (c+d x^2)^{7/2}} \, dx\) [76]

Optimal. Leaf size=639 \[ -\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {d} (b c-a d)^2 (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {e^{3/2} \sqrt {f} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 (b c-a d)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 e^{3/2} (b e-a f) \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

b^2*e^(3/2)*(-a*f+b*e)*(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/c/(-a*d+b*c)^3/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e
)^(1/2)+1/15*e^(3/2)*(3*b*c*(-2*c*f+3*d*e)-a*d*(-c*f+4*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))*f^(1/2)*(d*x^2+c)^(1/2)/c^3/(-a*d+b*c)^2/(-c*f+d*e)/(e
*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/5*(-c*f+d*e)*x*(f*x^2+e)^(1/2)/c/(-a*d+b*c)/(d*x^2+c)^(5/2)-1/
15*(3*b*c*(-c*f+3*d*e)-2*a*d*(c*f+2*d*e))*x*(f*x^2+e)^(1/2)/c^2/(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/15*(a*d*(-2*c^2
*f^2-3*c*d*e*f+8*d^2*e^2)-3*b*c*(c^2*f^2-6*c*d*e*f+6*d^2*e^2))*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ellipti
cE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*(f*x^2+e)^(1/2)/c^(5/2)/(-a*d+b*c)^2/(-c*f+d*e)/d^(1
/2)/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)-b*(-a*f+b*e)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ellip
ticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*d^(1/2)*(f*x^2+e)^(1/2)/(-a*d+b*c)^3/c^(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 = 0.51, antiderivative size = 639, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {558, 555, 553, 422, 540, 541, 539, 429} \begin {gather*} \frac {b^2 e^{3/2} \sqrt {c+d x^2} (b e-a f) \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 c \sqrt {f} \sqrt {e+f x^2} (b c-a d)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 \sqrt {e+f x^2} (b c-a d)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )-3 b c \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {b \sqrt {d} \sqrt {e+f x^2} (b e-a f) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {x \sqrt {e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {x \sqrt {e+f x^2} (d e-c f)}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]

[Out]

-1/5*((d*e - c*f)*x*Sqrt[e + f*x^2])/(c*(b*c - a*d)*(c + d*x^2)^(5/2)) - ((3*b*c*(3*d*e - c*f) - 2*a*d*(2*d*e
+ c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b*Sqrt[d]*(b*e - a*f)*Sqrt[e + f*x^2]*E
llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*
x^2))/(e*(c + d*x^2))]) + ((a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - 3*b*c*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))
*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(b*c - a*d)^2*(d
*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (e^(3/2)*Sqrt[f]*(3*b*c*(3*d*e - 2*c*f) - a
*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(b*c - a*d)
^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*(b*e - a*f)*Sqrt[c + d*x^
2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)^3*Sqrt[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 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 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]

Rule 540

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

Rule 541

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

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 555

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

Rule 558

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

Rubi steps

\begin {align*} \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx &=-\frac {\int \frac {\sqrt {e+f x^2} \left (2 b c d e-a d^2 e-b c^2 f+d^2 (b e-a f) x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx}{(b c-a d)^2}+\frac {(b (b e-a f)) \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^2}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {\int \frac {-d e (b c (9 d e-4 c f)-a d (4 d e+c f))-d f (b c (8 d e-3 c f)-a d (3 d e+2 c f)) x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c d (b c-a d)^2}+\frac {\left (b^2 (b e-a f)\right ) \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^3}-\frac {(b d (b e-a f)) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^3}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b^2 e^{3/2} (b e-a f) \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 c (b c-a d)^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 {\int \frac {d e (d e-c f) (9 b c (2 d e-c f)-a d (8 d e+c f))+d f (d e-c f) (3 b c (3 d e-c f)-2 a d (2 d e+c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 d (b c-a d)^2 (d e-c f)}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b^2 e^{3/2} (b e-a f) \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 c (b c-a d)^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 {\left (e f \left (a d (4 d e-c f)-b \left (9 c d e-6 c^2 f\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)}+\frac {\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)}\\ &=-\frac {(d e-c f) x \sqrt {e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac {(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt {e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b \sqrt {d} (b e-a f) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {d} (b c-a d)^2 (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} \left (a d (4 d e-c f)-b \left (9 c d e-6 c^2 f\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 )}{15 c^3 (b c-a d)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 e^{3/2} (b e-a f) \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 c (b c-a d)^3 \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 = 7.90, size = 570, normalized size = 0.89 \begin {gather*} \frac {-a \sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)^2+c (b c-a d) (-d e+c f) (3 b c (-3 d e+c f)+2 a d (2 d e+c f)) \left (c+d x^2\right )+\left (a^2 d^2 \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )+3 b^2 c^2 \left (11 d^2 e^2-11 c d e f+c^2 f^2\right )+2 a b c d \left (-13 d^2 e^2+3 c d e f+7 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )+i \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (a e \left (-3 b^2 c^2 \left (11 d^2 e^2-11 c d e f+c^2 f^2\right )+a^2 d^2 \left (-8 d^2 e^2+3 c d e f+2 c^2 f^2\right )-2 a b c d \left (-13 d^2 e^2+3 c d e f+7 c^2 f^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+(d e-c f) \left (a \left (3 b^2 c^2 e (11 d e-8 c f)+a^2 d^2 e (8 d e+c f)+a b c \left (-26 d^2 e^2-7 c d e f+15 c^2 f^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-15 b c^3 (b e-a f)^2 \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )\right )}{15 a c^3 \sqrt {\frac {d}{c}} (b c-a d)^3 (d e-c f) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]

[Out]

(-(a*Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 + c*(b*c - a*d)*(-(d*e) + c*f)*(3*b*c*(-3*d*e
+ c*f) + 2*a*d*(2*d*e + c*f))*(c + d*x^2) + (a^2*d^2*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) + 3*b^2*c^2*(11*d^2*e
^2 - 11*c*d*e*f + c^2*f^2) + 2*a*b*c*d*(-13*d^2*e^2 + 3*c*d*e*f + 7*c^2*f^2))*(c + d*x^2)^2)) + I*(c + d*x^2)^
2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*e*(-3*b^2*c^2*(11*d^2*e^2 - 11*c*d*e*f + c^2*f^2) + a^2*d^2*(-8*d
^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) - 2*a*b*c*d*(-13*d^2*e^2 + 3*c*d*e*f + 7*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/
c]*x], (c*f)/(d*e)] + (d*e - c*f)*(a*(3*b^2*c^2*e*(11*d*e - 8*c*f) + a^2*d^2*e*(8*d*e + c*f) + a*b*c*(-26*d^2*
e^2 - 7*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - 15*b*c^3*(b*e - a*f)^2*Ellipti
cPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*a*c^3*Sqrt[d/c]*(b*c - a*d)^3*(d*e - c*f)*(c + d*
x^2)^(5/2)*Sqrt[e + f*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6210\) vs. \(2(723)=1446\).
time = 0.16, size = 6211, normalized size = 9.72

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3112\)
default	\(\text {Expression too large to display}\)	\(6211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e)^(3/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((f*x^2+e)^(3/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)), 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((f*x^2+e)^(3/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((f*x**2+e)**(3/2)/(b*x**2+a)/(d*x**2+c)**(7/2),x)

[Out]

Timed out

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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((f*x^2+e)^(3/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f\,x^2+e\right )}^{3/2}}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x)

[Out]

int((e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)), x)

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