3.1.28 \(\int \frac {(a+b x^2) \sqrt {e+f x^2}}{(c+d x^2)^{7/2}} \, dx\) [28]

Optimal. Leaf size=385 \[ -\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} d^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} (2 a d (2 d e-3 c f)+b c (d e+c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 d (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-1/15*e^(3/2)*(2*a*d*(-3*c*f+2*d*e)+b*c*(c*f+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/(-c*f+d*e)^2/(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/d/(d*x^2+c)^(5/2)+1/15*(a*d*(-3*c*f+4*d*e)+b*c*(
-2*c*f+d*e))*x*(f*x^2+e)^(1/2)/c^2/d/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/15*(2*b*c*(c^2*f^2-c*d*e*f+d^2*e^2)+a*d*(3*c
^2*f^2-13*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^(3/2)/(-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 = 0.26, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 5, 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} (2 a d (2 d e-3 c f)+b c (c f+d e)) F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c^3 d \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 (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+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^{3/2} \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} (a d (4 d e-3 c f)+b c (d e-2 c f))}{15 c^2 d \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac {x \sqrt {e+f x^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)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),x]

[Out]

-1/5*((b*c - a*d)*x*Sqrt[e + f*x^2])/(c*d*(c + d*x^2)^(5/2)) + ((a*d*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*x*Sq
rt[e + f*x^2])/(15*c^2*d*(d*e - c*f)*(c + d*x^2)^(3/2)) + ((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e
^2 - 13*c*d*e*f + 3*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^(3/2)*(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]*(2*a*d*(2
*d*e - 3*c*f) + b*c*(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*(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 ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac {\int \frac {-(b c+4 a d) e-(2 b c+3 a d) f x^2}{\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx}{5 c d}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {e (a d (8 d e-9 c f)+b c (2 d e-c f))+f (d (b c+4 a d) e-c (2 b c+3 a d) f) x^2}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{15 c^2 d (d e-c f)}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {(e f (2 a d (2 d e-3 c f)+b c (d e+c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 c^2 d (d e-c f)^2}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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 (d e-c f)^2}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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^{3/2} (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} (2 a d (2 d e-3 c f)+b c (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 (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 = 4.14, size = 379, 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) (a d (4 d e-3 c f)+b c (d e-2 c f)) \left (c+d x^2\right )-\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 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 (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 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) (b c (-2 d e+c f)+a d (-8 d e+9 c f)) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{15 c^4 \left (\frac {d}{c}\right )^{3/2} (d e-c f)^2 \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)*Sqrt[e + f*x^2])/(c + d*x^2)^(7/2),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)*(a*d*(4*d*e - 3*c*f) + b*c*(d*e -
2*c*f))*(c + d*x^2) - (2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*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]*((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a
*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(b*c*
(-2*d*e + c*f) + a*d*(-8*d*e + 9*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*c^4*(d/c)^(3/2)*(d
*e - c*f)^2*(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. \(2860\) vs. \(2(419)=838\).
time = 0.15, size = 2861, normalized size = 7.43

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (a d -b c \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d^{4} c \left (x^{2}+\frac {c}{d}\right )^{3}}+\frac {\left (3 a c d f -4 a \,d^{2} e +2 b \,c^{2} f -b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{15 d^{3} c^{2} \left (c f -d e \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) x \left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d^{2} c^{3} \left (c f -d e \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {f \left (3 a c d f -4 a \,d^{2} e +2 b \,c^{2} f -b c d e \right )}{15 c^{2} \left (c f -d e \right ) d^{2}}-\frac {3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}}{15 d^{2} \left (c f -d e \right ) c^{3}}-\frac {e \left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d \,c^{3} \left (c f -d e \right )^{2}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{15 d \,c^{3} \left (c f -d e \right )^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(756\)
default	\(\text {Expression too large to display}\)	\(2861\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(26*(-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+13*(-d/c)^(1/2)*a*c*d^4*e*f^2*x^7+2*(-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+30*(-d/c)^(1/2)*a*c^2*d^3*e*f^2*x^5-7*(-d/c)^(1/
2)*a*c*d^4*e^2*f*x^5+5*(-d/c)^(1/2)*b*c^3*d^2*e*f^2*x^5-3*(-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+18*(-d/c)^(1/2)*a*c^2*d^3*e^2*f*x^3-7*(-d/c)^(1/2)*b*c^4*d*e*f^2*x^3+7*(-d/c)^(1/2)*b*c^3*d^2
*e^2*f*x^3-9*(-d/c)^(1/2)*a*c^4*d*e*f^2*x-3*(-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^7-2*(-
d/c)^(1/2)*b*c^3*d^2*f^3*x^7-9*(-d/c)^(1/2)*a*c^3*d^2*f^3*x^5-6*(-d/c)^(1/2)*b*c^4*d*f^3*x^5-2*(-d/c)^(1/2)*b*
c*d^4*e^3*x^5-9*(-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-(-d/c)^(1/2)*b*c^5*e*f^2*x+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*d*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))*a*c^4*d*e*f^2-13*((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-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^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-((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+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^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-(-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)*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^2-9*((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+17*((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-9*((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+17*
((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-((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+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^2*d^3*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))*a*c^2*d^3*e*f^2*x^4-13*((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+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*f^2*x^4-2*((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))*b*c^2*d^3*e^2*f*x^4-18*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipt
icF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*d^2*e*f^2*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^2*d^3*e^2*f*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^4*d*e*f^2*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))*b*c^3*d^2*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))*a*c^3*d^2*e*f^2*x^2-26*((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+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^4*d*e*f^2*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^3*d^
2*e^2*f*x^2)/(f*x^2+e)^(1/2)/(-d/c)^(1/2)/(c*f-d*e)^2/c^3/d/(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)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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