3.1.34 \(\int \frac {(a+b x^2) (e+f x^2)^{3/2}}{(c+d x^2)^{9/2}} \, dx\) [34]

Optimal. Leaf size=531 \[ \frac {(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt {e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt {e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}+\frac {\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{105 c^{7/2} d^{5/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} \left (3 a d \left (8 d^2 e^2-11 c d e f+c^2 f^2\right )+2 b c \left (2 d^2 e^2-c d e f+2 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{105 c^4 d^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-1/7*(-a*d+b*c)*x*(f*x^2+e)^(3/2)/c/d/(d*x^2+c)^(7/2)-1/105*e^(3/2)*(3*a*d*(c^2*f^2-11*c*d*e*f+8*d^2*e^2)+2*b*
c*(2*c^2*f^2-c*d*e*f+2*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))*f^(1/2)*(d*x^2+c)^(1/2)/c^4/d^2/(-c*f+d*e)^2/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^
2+e)^(1/2)+1/35*(d*(6*a*d+b*c)*e-c*(3*a*d+4*b*c)*f)*x*(f*x^2+e)^(1/2)/c^2/d^2/(d*x^2+c)^(5/2)+1/105*(b*c*(-8*c
^2*f^2+c*d*e*f+4*d^2*e^2)+3*a*d*(-2*c^2*f^2-5*c*d*e*f+8*d^2*e^2))*x*(f*x^2+e)^(1/2)/c^3/d^2/(-c*f+d*e)/(d*x^2+
c)^(3/2)+1/105*(6*a*d*(c^3*f^3+2*c^2*d*e*f^2-12*c*d^2*e^2*f+8*d^3*e^3)+b*c*(8*c^3*f^3-5*c^2*d*e*f^2-5*c*d^2*e^
2*f+8*d^3*e^3))*(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-c*f/d
/e)^(1/2))*(f*x^2+e)^(1/2)/c^(7/2)/d^(5/2)/(-c*f+d*e)^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 = 0.42, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {540, 541, 539, 429, 422} \begin {gather*} -\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} \left (3 a d \left (c^2 f^2-11 c d e f+8 d^2 e^2\right )+2 b c \left (2 c^2 f^2-c d e f+2 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 c^4 d^2 \sqrt {e+f x^2} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e+f x^2} \left (6 a d \left (c^3 f^3+2 c^2 d e f^2-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (8 c^3 f^3-5 c^2 d e f^2-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{105 c^{7/2} d^{5/2} \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {x \sqrt {e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac {x \sqrt {e+f x^2} \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )}{105 c^3 d^2 \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{7 c d \left (c+d x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]

[Out]

((d*(b*c + 6*a*d)*e - c*(4*b*c + 3*a*d)*f)*x*Sqrt[e + f*x^2])/(35*c^2*d^2*(c + d*x^2)^(5/2)) + ((b*c*(4*d^2*e^
2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^2 - 5*c*d*e*f - 2*c^2*f^2))*x*Sqrt[e + f*x^2])/(105*c^3*d^2*(d*e - c
*f)*(c + d*x^2)^(3/2)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(7*c*d*(c + d*x^2)^(7/2)) + ((6*a*d*(8*d^3*e^3 - 12
*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[e
+ f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(105*c^(7/2)*d^(5/2)*(d*e - c*f)^2*Sqrt[c +
d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(3*a*d*(8*d^2*e^2 - 11*c*d*e*f + c^2*f^2) + 2
*b*c*(2*d^2*e^2 - c*d*e*f + 2*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)
])/(105*c^4*d^2*(d*e - c*f)^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 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx &=-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac {\int \frac {\sqrt {e+f x^2} \left (-(b c+6 a d) e-(4 b c+3 a d) f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx}{7 c d}\\ &=\frac {(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt {e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}+\frac {\int \frac {e (4 b c (d e+c f)+3 a d (8 d e+c f))+f (6 a d (3 d e+c f)+b c (3 d e+8 c f)) x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{35 c^2 d^2}\\ &=\frac {(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt {e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt {e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac {\int \frac {-e \left (b c \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )+3 a d \left (16 d^2 e^2-16 c d e f-c^2 f^2\right )\right )-f \left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{105 c^3 d^2 (d e-c f)}\\ &=\frac {(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt {e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt {e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac {\left (e f \left (3 a d \left (8 d^2 e^2-11 c d e f+c^2 f^2\right )+2 b c \left (2 d^2 e^2-c d e f+2 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{105 c^3 d^2 (d e-c f)^2}+\frac {\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{105 c^3 d^2 (d e-c f)^2}\\ &=\frac {(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt {e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt {e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}+\frac {\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\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 )}{105 c^{7/2} d^{5/2} (d e-c f)^2 \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 (3 a d \left (8 d^2 e^2-11 c d e f+c^2 f^2\right )+2 b c \left (2 d^2 e^2-c d e f+2 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 c^4 d^2 (d e-c f)^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 = 6.71, size = 545, normalized size = 1.03 \begin {gather*} \frac {\sqrt {\frac {d}{c}} \left (-\sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (15 c^3 (b c-a d) (d e-c f)^3-3 c^2 (d e-c f)^2 (b c (d e-9 c f)+2 a d (3 d e+c f)) \left (c+d x^2\right )-c (d e-c f) \left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) \left (c+d x^2\right )^2-\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \left (c+d x^2\right )^3\right )+i e \left (c+d x^2\right )^3 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\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 (3 a d \left (-16 d^2 e^2+16 c d e f+c^2 f^2\right )+b c \left (-8 d^2 e^2+c d e f+4 c^2 f^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )\right )}{105 c^3 d^3 (d e-c f)^2 \left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]

[Out]

(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(15*c^3*(b*c - a*d)*(d*e - c*f)^3 - 3*c^2*(d*e - c*f)^2*(b*c*(d*e - 9*c*
f) + 2*a*d*(3*d*e + c*f))*(c + d*x^2) - c*(d*e - c*f)*(b*c*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^
2 - 5*c*d*e*f - 2*c^2*f^2))*(c + d*x^2)^2 - (6*a*d*(8*d^3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*
c*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*(c + d*x^2)^3)) + I*e*(c + d*x^2)^3*Sqrt[1 + (d*x^2
)/c]*Sqrt[1 + (f*x^2)/e]*((6*a*d*(8*d^3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c
*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(3*a*
d*(-16*d^2*e^2 + 16*c*d*e*f + c^2*f^2) + b*c*(-8*d^2*e^2 + c*d*e*f + 4*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]
*x], (c*f)/(d*e)])))/(105*c^3*d^3*(d*e - c*f)^2*(c + d*x^2)^(7/2)*Sqrt[e + f*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5112\) vs. \(2(559)=1118\).
time = 0.16, size = 5113, normalized size = 9.63

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{7 d^{6} c \left (x^{2}+\frac {c}{d}\right )^{4}}+\frac {\left (2 a c d f +6 a \,d^{2} e -9 b \,c^{2} f +b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{35 d^{5} c^{2} \left (x^{2}+\frac {c}{d}\right )^{3}}+\frac {\left (6 a \,c^{2} d \,f^{2}+15 a c \,d^{2} e f -24 a \,d^{3} e^{2}+8 b \,c^{3} f^{2}-b \,c^{2} d e f -4 b c \,d^{2} e^{2}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{105 d^{4} c^{3} \left (c f -d e \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) x \left (6 a \,c^{3} d \,f^{3}+12 a \,c^{2} d^{2} e \,f^{2}-72 a c \,d^{3} e^{2} f +48 a \,d^{4} e^{3}+8 b \,c^{4} f^{3}-5 b \,c^{3} d e \,f^{2}-5 b \,c^{2} d^{2} e^{2} f +8 b c \,d^{3} e^{3}\right )}{105 d^{3} c^{4} \left (c f -d e \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {f \left (6 a \,c^{2} d \,f^{2}+15 a c \,d^{2} e f -24 a \,d^{3} e^{2}+8 b \,c^{3} f^{2}-b \,c^{2} d e f -4 b c \,d^{2} e^{2}\right )}{105 d^{3} c^{3} \left (c f -d e \right )}-\frac {6 a \,c^{3} d \,f^{3}+12 a \,c^{2} d^{2} e \,f^{2}-72 a c \,d^{3} e^{2} f +48 a \,d^{4} e^{3}+8 b \,c^{4} f^{3}-5 b \,c^{3} d e \,f^{2}-5 b \,c^{2} d^{2} e^{2} f +8 b c \,d^{3} e^{3}}{105 d^{3} \left (c f -d e \right ) c^{4}}-\frac {e \left (6 a \,c^{3} d \,f^{3}+12 a \,c^{2} d^{2} e \,f^{2}-72 a c \,d^{3} e^{2} f +48 a \,d^{4} e^{3}+8 b \,c^{4} f^{3}-5 b \,c^{3} d e \,f^{2}-5 b \,c^{2} d^{2} e^{2} f +8 b c \,d^{3} e^{3}\right )}{105 d^{2} c^{4} \left (c f -d e \right )^{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 )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (6 a \,c^{3} d \,f^{3}+12 a \,c^{2} d^{2} e \,f^{2}-72 a c \,d^{3} e^{2} f +48 a \,d^{4} e^{3}+8 b \,c^{4} f^{3}-5 b \,c^{3} d e \,f^{2}-5 b \,c^{2} d^{2} e^{2} f +8 b c \,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 )}{105 d^{2} c^{4} \left (c f -d e \right )^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(1023\)
default	\(\text {Expression too large to display}\)	\(5113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),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((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), 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)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

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

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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((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(9/2),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((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), 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 (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x)

[Out]

int(((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2), x)

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