3.1.41 \(\int \frac {a+b x^2}{(c+d x^2)^{7/2} \sqrt {e+f x^2}} \, dx\) [41]

Optimal. Leaf size=401 \[ -\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt {e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (b c \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+a d \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} \sqrt {d} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} \left (b c e (d e-9 c f)+a \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-1/15*(b*c*e*(-9*c*f+d*e)+a*(15*c^2*f^2-11*c*d*e*f+4*d^2*e^2))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*Ellipti
cF(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^3/(-c*f+d*e)^3/(e*
(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/5*(-a*d+b*c)*x*(f*x^2+e)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(5/2)+1/1
5*(4*a*d*(-2*c*f+d*e)+b*c*(3*c*f+d*e))*x*(f*x^2+e)^(1/2)/c^2/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+1/15*(b*c*(-3*c^2*f^
2-7*c*d*e*f+2*d^2*e^2)+a*d*(23*c^2*f^2-23*c*d*e*f+8*d^2*e^2))*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Elliptic
E(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)/(-c*f+d*e)^3/d^(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 = 0.28, antiderivative size = 401, 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 {e+f x^2} \left (a d \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b c \left (-3 c^2 f^2-7 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} \sqrt {d} \sqrt {c+d x^2} (d e-c f)^3 \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} \left (a \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )+b c e (d e-9 c f)\right ) 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} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {e+f x^2} (4 a d (d e-2 c f)+b c (3 c f+d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (d e-c f)^2}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c \left (c+d x^2\right )^{5/2} (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*((b*c - a*d)*x*Sqrt[e + f*x^2])/(c*(d*e - c*f)*(c + d*x^2)^(5/2)) + ((4*a*d*(d*e - 2*c*f) + b*c*(d*e + 3*
c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(d*e - c*f)^2*(c + d*x^2)^(3/2)) + ((b*c*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2)
+ a*d*(8*d^2*e^2 - 23*c*d*e*f + 23*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]*(d*e - c*f)^3*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (Sqrt[e]*Sq
rt[f]*(b*c*e*(d*e - 9*c*f) + a*(4*d^2*e^2 - 11*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f
]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(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 )^{7/2} \sqrt {e+f x^2}} \, dx &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}-\frac {\int \frac {-b c e-4 a d e+5 a c f+3 (b c-a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c (d e-c f)}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt {e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {2 b c e (d e-3 c f)+a \left (8 d^2 e^2-19 c d e f+15 c^2 f^2\right )+f (4 a d (d e-2 c f)+b c (d e+3 c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 (d e-c f)^2}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt {e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}-\frac {\left (f \left (b c e (d e-9 c f)+a \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 (d e-c f)^3}+\frac {\left (b c \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+a d \left (8 d^2 e^2-23 c d e f+23 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 e-c f)^3}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {(4 a d (d e-2 c f)+b c (d e+3 c f)) x \sqrt {e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (b c \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )+a d \left (8 d^2 e^2-23 c d e f+23 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} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {\sqrt {e} \sqrt {f} \left (b c e (d e-9 c f)+a \left (4 d^2 e^2-11 c d e f+15 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 )}{15 c^3 (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 = 10.92, size = 393, normalized size = 0.98 \begin {gather*} \frac {-\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) (4 a d (d e-2 c f)+b c (d e+3 c f)) \left (c+d x^2\right )+\left (a d \left (-8 d^2 e^2+23 c d e f-23 c^2 f^2\right )+b c \left (-2 d^2 e^2+7 c d e f+3 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 (e \left (a d \left (-8 d^2 e^2+23 c d e f-23 c^2 f^2\right )+b c \left (-2 d^2 e^2+7 c d e f+3 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 (2 b c e (d e-3 c f)+a \left (8 d^2 e^2-19 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 )\right )}{15 c^3 \sqrt {\frac {d}{c}} (d e-c f)^3 \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)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]

[Out]

(-(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)*(4*a*d*(d*e - 2*c*f) + b*c*(d*e
+ 3*c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2 + 23*c*d*e*f - 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*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]*(e*(a*d*(-8*d^2*e^2 + 23*c*d*e*f
- 23*c^2*f^2) + b*c*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*
e - c*f)*(2*b*c*e*(d*e - 3*c*f) + a*(8*d^2*e^2 - 19*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (
c*f)/(d*e)]))/(15*c^3*Sqrt[d/c]*(d*e - c*f)^3*(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. \(3038\) vs. \(2(435)=870\).
time = 0.13, size = 3039, normalized size = 7.58

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {x \left (a d -b c \right ) \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d^{3} c \left (c f -d e \right ) \left (x^{2}+\frac {c}{d}\right )^{3}}-\frac {\left (8 a c d f -4 a \,d^{2} e -3 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 c^{2} \left (c f -d e \right )^{2} d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {\left (d f \,x^{2}+d e \right ) x \left (23 a \,c^{2} d \,f^{2}-23 a c \,d^{2} e f +8 a \,d^{3} e^{2}-3 b \,c^{3} f^{2}-7 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d \,c^{3} \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 (8 a c d f -4 a \,d^{2} e -3 b \,c^{2} f -b c d e \right )}{15 d \,c^{2} \left (c f -d e \right )^{2}}+\frac {23 a \,c^{2} d \,f^{2}-23 a c \,d^{2} e f +8 a \,d^{3} e^{2}-3 b \,c^{3} f^{2}-7 b \,c^{2} d e f +2 b c \,d^{2} e^{2}}{15 \left (c f -d e \right )^{2} d \,c^{3}}+\frac {e \left (23 a \,c^{2} d \,f^{2}-23 a c \,d^{2} e f +8 a \,d^{3} e^{2}-3 b \,c^{3} f^{2}-7 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 c^{3} \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 (23 a \,c^{2} d \,f^{2}-23 a c \,d^{2} e f +8 a \,d^{3} e^{2}-3 b \,c^{3} f^{2}-7 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 c^{3} \left (c f -d e \right )^{3} \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}}\)	\(760\)
default	\(\text {Expression too large to display}\)	\(3039\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/15*(41*(-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+23*(-d/c)^(1/2)*a*c*d^4*e*f^2*x^7+7*(-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+35*(-d/c)^(1/2)*a*c^2*d^3*e*f^2*x^5+3*(-d/c)^(1/2
)*a*c*d^4*e^2*f*x^5+15*(-d/c)^(1/2)*b*c^3*d^2*e*f^2*x^5+2*(-d/c)^(1/2)*b*c^2*d^3*e^2*f*x^5-13*(-d/c)^(1/2)*a*c
^3*d^2*e*f^2*x^3+43*(-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+12*(-d/c)^(1/2)*b*c^3*d^
2*e^2*f*x^3-34*(-d/c)^(1/2)*a*c^4*d*e*f^2*x+15*((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^5*f^3-23*(-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+3*(-d/c)^(1/2)*b
*c^3*d^2*f^3*x^7-54*(-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
^5-34*(-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+9*(-d/c)^(1/2)*b*c^5*e*f^2*x+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^2*f+23*((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^2-23*((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-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^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^3-6*((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))*b*c^5*e*f^2-2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*E
llipticF(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)*EllipticE(x*(
-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d^3*e^3-3*((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^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+9*(-d/c)^(1/2)*b*c^5*f^3*x^3+15*((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*f^3*x^4+30*((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*f^3*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*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)*EllipticF(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^2-34*((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+27*((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-34*((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+27*((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^4-6*((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))*b*c^3*d^2*e*f^2*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))*b*c^2*d^3*e^2*f*x^4+23*((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^4-23*((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*d^4*e^2*f*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))*b*c^3*d^2*e*f^2*x^4-7*((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^3*e^2*f*x^4-68*((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+54*((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^2-12*((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+16*((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+46*((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^2-46*((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^2-6*((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^4*d*e*f
^2*x^2-14*((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*f-d*e)^3/c^3/(-d/c)^(1/2)/(d*x^2+c)^(5/2)

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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)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*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)^(7/2)/(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 {a + b x^{2}}{\left (c + d x^{2}\right )^{\frac {7}{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)**(7/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral((a + b*x**2)/((c + d*x**2)**(7/2)*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)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(7/2)*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 {b\,x^2+a}{{\left (d\,x^2+c\right )}^{7/2}\,\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)^(7/2)*(e + f*x^2)^(1/2)),x)

[Out]

int((a + b*x^2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(1/2)), x)

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