3.1.47 \(\int \frac {a+b x^2}{(c+d x^2)^{5/2} (e+f x^2)^{3/2}} \, dx\) [47]

Optimal. Leaf size=375 \[ -\frac {(b c-a d) x}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {(2 a d (d e-3 c f)+b c (d e+3 c f)) x}{3 c^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {f} \left (b c e (d e+7 c f)+a \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {f} (a d (d e-9 c f)+b c (5 d e+3 c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 375, 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 = {541, 539, 429, 422} \begin {gather*} \frac {\sqrt {f} \sqrt {c+d x^2} \left (a \left (-3 c^2 f^2-7 c d e f+2 d^2 e^2\right )+b c e (7 c f+d e)\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 \sqrt {e} \sqrt {e+f x^2} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (a d (d e-9 c f)+b c (3 c f+5 d e)) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 \sqrt {e+f x^2} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x (2 a d (d e-3 c f)+b c (3 c f+d e))}{3 c^2 \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)^2}-\frac {x (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]

[Out]

-1/3*((b*c - a*d)*x)/(c*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) + ((2*a*d*(d*e - 3*c*f) + b*c*(d*e + 3*
c*f))*x)/(3*c^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[f]*(b*c*e*(d*e + 7*c*f) + a*(2*d^2*e^2
- 7*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*Sqrt
[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[f]*(a*d*(d*e - 9*c*f)
+ b*c*(5*d*e + 3*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(d*e -
c*f)^3*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 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 {a+b x^2}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx &=-\frac {(b c-a d) x}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {\int \frac {-b c e-2 a d e+3 a c f+3 (b c-a d) f x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx}{3 c (d e-c f)}\\ &=-\frac {(b c-a d) x}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {(2 a d (d e-3 c f)+b c (d e+3 c f)) x}{3 c^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\int \frac {-c f (4 b c e-a d e-3 a c f)+f (2 a d (d e-3 c f)+b c (d e+3 c f)) x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 (d e-c f)^2}\\ &=-\frac {(b c-a d) x}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {(2 a d (d e-3 c f)+b c (d e+3 c f)) x}{3 c^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {(f (a d (d e-9 c f)+b c (5 d e+3 c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c (d e-c f)^3}+\frac {\left (f \left (b c e (d e+7 c f)+a \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 (d e-c f)^3}\\ &=-\frac {(b c-a d) x}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {(2 a d (d e-3 c f)+b c (d e+3 c f)) x}{3 c^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {f} \left (b c e (d e+7 c f)+a \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\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 )}{3 c^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {f} (a d (d e-9 c f)+b c (5 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 c^2 (d e-c f)^3 \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 = 11.49, size = 428, normalized size = 1.14 \begin {gather*} \frac {\sqrt {\frac {d}{c}} x \left (-b c e \left (3 c^3 f^2+d^3 e x^2 \left (e+f x^2\right )+c d^2 f x^2 \left (4 e+7 f x^2\right )+c^2 d f \left (5 e+11 f x^2\right )\right )+a \left (3 c^4 f^3+6 c^3 d f^3 x^2-2 d^4 e^2 x^2 \left (e+f x^2\right )+c^2 d^2 f \left (8 e^2+8 e f x^2+3 f^2 x^4\right )+c d^3 e \left (-3 e^2+4 e f x^2+7 f^2 x^4\right )\right )\right )-i d e \left (b c e (d e+7 c f)+a \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \left (c+d x^2\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 e (-d e+c f) (2 a d (d e-3 c f)+b c (d e+3 c f)) \left (c+d x^2\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 )}{3 c^2 \sqrt {\frac {d}{c}} e (-d e+c f)^3 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1741\) vs. \(2(409)=818\).
time = 0.14, size = 1742, normalized size = 4.65

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+c f \right ) f x \left (a f -b e \right )}{e \left (c f -d e \right )^{3} \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {x \left (a d -b c \right ) \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d c \left (c f -d e \right )^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) x \left (7 a c d f -2 a \,d^{2} e -4 b \,c^{2} f -b c d e \right )}{3 c^{2} \left (c f -d e \right )^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {f \left (a f -b e \right )}{\left (c f -d e \right )^{2} e}-\frac {c \,f^{2} \left (a f -b e \right )}{e \left (c f -d e \right )^{3}}+\frac {\left (a d -b c \right ) f}{3 c \left (c f -d e \right )^{2}}-\frac {7 a c d f -2 a \,d^{2} e -4 b \,c^{2} f -b c d e}{3 \left (c f -d e \right )^{2} c^{2}}-\frac {d e \left (7 a c d f -2 a \,d^{2} e -4 b \,c^{2} f -b c d e \right )}{3 c^{2} \left (c f -d e \right )^{3}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (-\frac {f^{2} d \left (a f -b e \right )}{e \left (c f -d e \right )^{3}}-\frac {d f \left (7 a c d f -2 a \,d^{2} e -4 b \,c^{2} f -b c d e \right )}{3 c^{2} \left (c f -d e \right )^{3}}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(673\)
default	\(\text {Expression too large to display}\)	\(1742\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(7*(-d/c)^(1/2)*b*c^2*d^2*e*f^2*x^5+(-d/c)^(1/2)*b*c*d^3*e^2*f*x^5-8*(-d/c)^(1/2)*a*c^2*d^2*e*f^2*x^3-4*(
-d/c)^(1/2)*a*c*d^3*e^2*f*x^3+11*(-d/c)^(1/2)*b*c^3*d*e*f^2*x^3+4*(-d/c)^(1/2)*b*c^2*d^2*e^2*f*x^3-8*(-d/c)^(1
/2)*a*c^2*d^2*e^2*f*x+5*(-d/c)^(1/2)*b*c^3*d*e^2*f*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*d^4*e^3*x^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))*a*d^4*e^3*x^2+2*(-d/c)^(1/2)*a*d^4*e^3*x^3-3*(-d/c)^(1/2)*a*c^4*f^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*d^3*e^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))*b*c^4*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))*b*c^2*d^2*e^3-2*((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*d^3*e^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^2*d^2*e^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))*b*c*d^3*e^3*x
^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^3*e^3*x^2+6*((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*e*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*c^2*d^2*e^2*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))*b*c^3*d*e^2*f+3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*E
llipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*d*e*f^2+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*c^2*d^2*e^2*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))*b*c^3*d*e^2*f-6*(-d/c)^(1/2)*a*c^3*d*f^3*x^3+(-d/c)^(1/2)*b*c*d^3*e^3*x^3+3*(-d/c)^(1/2)*a*
c*d^3*e^3*x+3*(-d/c)^(1/2)*b*c^4*e*f^2*x-7*(-d/c)^(1/2)*a*c*d^3*e*f^2*x^5-3*(-d/c)^(1/2)*a*c^2*d^2*f^3*x^5+2*(
-d/c)^(1/2)*a*d^4*e^2*f*x^5+6*((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^2*e*f^2*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))*a*c*
d^3*e^2*f*x^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))*b*c^3*d*e*f^
2*x^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^2*d^2*e^2*f*x^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^2*d^2*e*f^2*x^2+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*c*d^3*e^2*f*x^2-7*((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))*b*c^2*d^2*e^2*f*x^2)/(f*x^2+e)^(1/2)/(c*
f-d*e)^3/e/(-d/c)^(1/2)/c^2/(d*x^2+c)^(3/2)

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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)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), 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)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),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 {a + b x^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{\frac {3}{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)**(3/2),x)

[Out]

Integral((a + b*x**2)/((c + d*x**2)**(5/2)*(e + f*x**2)**(3/2)), 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)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {b\,x^2+a}{{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,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)^(3/2)),x)

[Out]

int((a + b*x^2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)), x)

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