3.1.33 \(\int \frac {(a+b x^2) (e+f x^2)^{3/2}}{(c+d x^2)^{7/2}} \, dx\) [33]

Optimal. Leaf size=376 \[ \frac {(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt {e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (2 d^2 e^2+3 c d e f-8 c^2 f^2\right )+a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} d^{5/2} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} (b c (d e-4 c f)+a d (4 d e-c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 d^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-1/5*(-a*d+b*c)*x*(f*x^2+e)^(3/2)/c/d/(d*x^2+c)^(5/2)-1/15*e^(3/2)*(b*c*(-4*c*f+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/d^2/(-c*f+d*e)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/15*(d*(4*a*d+b*c)*e-c*(a*d+4*
b*c)*f)*x*(f*x^2+e)^(1/2)/c^2/d^2/(d*x^2+c)^(3/2)+1/15*(b*c*(-8*c^2*f^2+3*c*d*e*f+2*d^2*e^2)+a*d*(-2*c^2*f^2-3
*c*d*e*f+8*d^2*e^2))*(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^(5/2)/d^(5/2)/(-c*f+d*e)/(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 = 0.27, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {540, 539, 429, 422} \begin {gather*} -\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 d^2 \sqrt {e+f x^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 )+b c \left (-8 c^2 f^2+3 c d e f+2 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} d^{5/2} \sqrt {c+d x^2} (d e-c f) \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 (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(b*c + 4*a*d)*e - c*(4*b*c + a*d)*f)*x*Sqrt[e + f*x^2])/(15*c^2*d^2*(c + d*x^2)^(3/2)) - ((b*c - a*d)*x*(e
+ f*x^2)^(3/2))/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c*(2*d^2*e^2 + 3*c*d*e*f - 8*c^2*f^2) + a*d*(8*d^2*e^2 - 3*c*
d*e*f - 2*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)*d^(5/
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]*(b*c*(d*e - 4*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*d^2*(d*e -
c*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]

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 )^{7/2}} \, dx &=-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac {\int \frac {\sqrt {e+f x^2} \left (-(b c+4 a d) e-(4 b c+a d) f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx}{5 c d}\\ &=\frac {(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt {e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {\int \frac {e (a d (8 d e+c f)+2 b c (d e+2 c f))+f (2 a d (2 d e+c f)+b c (d e+8 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 d^2}\\ &=\frac {(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt {e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac {(e f (b c (d e-4 c f)+a d (4 d e-c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 d^2 (d e-c f)}+\frac {\left (b c \left (2 d^2 e^2+3 c d e f-8 c^2 f^2\right )+a d \left (8 d^2 e^2-3 c d e f-2 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 d^2 (d e-c f)}\\ &=\frac {(d (b c+4 a d) e-c (4 b c+a d) f) x \sqrt {e+f x^2}}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (2 d^2 e^2+3 c d e f-8 c^2 f^2\right )+a d \left (8 d^2 e^2-3 c d e f-2 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} d^{5/2} (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} (b c (d e-4 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 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}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.24, size = 382, normalized size = 1.02 \begin {gather*} \frac {\sqrt {\frac {d}{c}} \left (-\sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (3 c^2 (b c-a d) (d e-c f)^2-c (d e-c f) (b c (d e-7 c f)+2 a d (2 d e+c f)) \left (c+d x^2\right )+\left (a d \left (-8 d^2 e^2+3 c d e f+2 c^2 f^2\right )+b c \left (-2 d^2 e^2-3 c d e f+8 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )-i e \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (\left (a d \left (-8 d^2 e^2+3 c d e f+2 c^2 f^2\right )+b c \left (-2 d^2 e^2-3 c d e f+8 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) (a d (8 d e+c f)+2 b c (d e+2 c f)) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )\right )}{15 c^2 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[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]

[Out]

(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e - c*f)*(b*c*(d*e - 7*c*f) + 2*
a*d*(2*d*e + c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c
^2*f^2))*(c + d*x^2)^2)) - I*e*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((a*d*(-8*d^2*e^2 + 3*c*d
*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] +
(d*e - c*f)*(a*d*(8*d*e + c*f) + 2*b*c*(d*e + 2*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*c
^2*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. \(2859\) vs. \(2(410)=820\).
time = 0.14, size = 2860, normalized size = 7.61

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d^{5} c \left (x^{2}+\frac {c}{d}\right )^{3}}+\frac {\left (2 a c d f +4 a \,d^{2} e -7 b \,c^{2} f +b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{15 d^{4} c^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) x \left (2 a \,c^{2} d \,f^{2}+3 a c \,d^{2} e f -8 a \,d^{3} e^{2}+8 b \,c^{3} f^{2}-3 b \,c^{2} d e f -2 b c \,d^{2} e^{2}\right )}{15 d^{3} c^{3} \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {b \,f^{2}}{d^{3}}+\frac {f \left (2 a c d f +4 a \,d^{2} e -7 b \,c^{2} f +b c d e \right )}{15 d^{3} c^{2}}-\frac {2 a \,c^{2} d \,f^{2}+3 a c \,d^{2} e f -8 a \,d^{3} e^{2}+8 b \,c^{3} f^{2}-3 b \,c^{2} d e f -2 b c \,d^{2} e^{2}}{15 d^{3} c^{3}}-\frac {e \left (2 a \,c^{2} d \,f^{2}+3 a c \,d^{2} e f -8 a \,d^{3} e^{2}+8 b \,c^{3} f^{2}-3 b \,c^{2} d e f -2 b c \,d^{2} e^{2}\right )}{15 d^{2} c^{3} \left (c f -d e \right )}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (2 a \,c^{2} d \,f^{2}+3 a c \,d^{2} e f -8 a \,d^{3} e^{2}+8 b \,c^{3} f^{2}-3 b \,c^{2} d e f -2 b c \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{15 d^{2} c^{3} \left (c f -d e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(749\)
default	\(\text {Expression too large to display}\)	\(2860\)

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)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/15*(-11*(-d/c)^(1/2)*a*c^3*d^2*e^2*f*x+(-d/c)^(1/2)*b*c^4*d*e^2*f*x-3*(-d/c)^(1/2)*a*c*d^4*e*f^2*x^7+3*(-d/
c)^(1/2)*b*c^2*d^3*e*f^2*x^7+2*(-d/c)^(1/2)*b*c*d^4*e^2*f*x^7-10*(-d/c)^(1/2)*a*c^2*d^3*e*f^2*x^5+17*(-d/c)^(1
/2)*a*c*d^4*e^2*f*x^5-10*(-d/c)^(1/2)*b*c^3*d^2*e*f^2*x^5+8*(-d/c)^(1/2)*b*c^2*d^3*e^2*f*x^5-17*(-d/c)^(1/2)*a
*c^3*d^2*e*f^2*x^3+7*(-d/c)^(1/2)*a*c^2*d^3*e^2*f*x^3-8*(-d/c)^(1/2)*b*c^4*d*e*f^2*x^3-2*(-d/c)^(1/2)*b*c^3*d^
2*e^2*f*x^3-(-d/c)^(1/2)*a*c^4*d*e*f^2*x-2*(-d/c)^(1/2)*a*c^2*d^3*f^3*x^7+8*(-d/c)^(1/2)*a*d^5*e^2*f*x^7-8*(-d
/c)^(1/2)*b*c^3*d^2*f^3*x^7-6*(-d/c)^(1/2)*a*c^3*d^2*f^3*x^5-9*(-d/c)^(1/2)*b*c^4*d*f^3*x^5+2*(-d/c)^(1/2)*b*c
*d^4*e^3*x^5-(-d/c)^(1/2)*a*c^4*d*f^3*x^3+20*(-d/c)^(1/2)*a*c*d^4*e^3*x^3+5*(-d/c)^(1/2)*b*c^2*d^3*e^3*x^3+15*
(-d/c)^(1/2)*a*c^2*d^3*e^3*x-4*(-d/c)^(1/2)*b*c^5*e*f^2*x+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))*b*c^4*d*e^2*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*c^4*d*e*f^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*c^3*d^2*e^2*f-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))*b*
c^4*d*e^2*f+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*d^5*e^3*x^4-
8*((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^5*e^3*x^4+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*c^2*d^3*e^3-4*((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^5*e*f^2+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))*b*c^3*d^2*e^3-8*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elliptic
E(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d^3*e^3+8*((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^5*e*f^2-2*((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))*b*c^3*d^2*e^3+8*(-d/c)^(1/2)*a*d^5*e^3*x^5-4*(-d/c)^(1/2)*b*c^5*f^3*x^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))*b*c*d^4*e^3*x^4-2*((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))*b*c*d^4*e^3*x^4+16*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipti
cF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c*d^4*e^3*x^2+4*((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^3*e^3*x^2-16*((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^4*e^3*x^2-4*((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^3*e^3*x^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*
c^4*d*e*f^2-7*((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*d^2*e^2*
f-((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^3*e*f^2*x^4-7*((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^4*e^2*f*x^4-4*((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*d^2*e*f^2*x^4+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))*b*c^2*d^3*e^2*f*x^4+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*c^2*d^3*e*f^2*x^4+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*c*d^4*e^2*f*x^4+8*((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*d^2*e*f^2*x^4-3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^2*d^3*e^2*f*x^4-2*((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*c^3*d^2*e*f^2*x^2-14*((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^3*e^2*f*x^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))*b*c^4*d*e*f^2*x^2+4*((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*d^2*e^2*f*x^2+4*((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^3*d^2*e*f^2*x^2+6*((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^3*e^2*f*x^2+16*((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^
4*d*e*f^2*x^2-6*((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*d^2*e^
2*f*x^2)/(f*x^2+e)^(1/2)/c^3/(c*f-d*e)/(-d/c)^(1/2)/(d*x^2+c)^(5/2)/d^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)*(f*x^2+e)^(3/2)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2), 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)^(7/2),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 (e + f x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \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)**(7/2),x)

[Out]

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

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(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 (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{7/2}} \,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)^(7/2),x)

[Out]

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

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