\(\int \frac {(a+b x^2) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx\) [37]

Optimal result

Integrand size = 30, antiderivative size = 282 \[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=-\frac {(2 b d e-b c f-3 a d f) x \sqrt {c+d x^2}}{3 d f \sqrt {e+f x^2}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}+\frac {\sqrt {e} (2 b d e-b c f-3 a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 d f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-3 a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {542, 545, 429, 506, 422} \[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=-\frac {\sqrt {e} \sqrt {c+d x^2} (b e-3 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {e} \sqrt {c+d x^2} (-3 a d f-b c f+2 b d e) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 d f^{3/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {x \sqrt {c+d x^2} (-3 a d f-b c f+2 b d e)}{3 d f \sqrt {e+f x^2}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f} \]

[In]

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

[Out]

-1/3*((2*b*d*e - b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(d*f*Sqrt[e + f*x^2]) + (b*x*Sqrt[c + d*x^2]*Sqrt[e + f*x
^2])/(3*f) + (Sqrt[e]*(2*b*d*e - b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (
d*e)/(c*f)])/(3*d*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(b*e - 3*a*f)*Sqrt
[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(3/2)*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 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}+\frac {\int \frac {-c (b e-3 a f)+(-2 b d e+b c f+3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 f} \\ & = \frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {(c (b e-3 a f)) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 f}+\frac {(-2 b d e+b c f+3 a d f) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 f} \\ & = -\frac {(2 b d e-b c f-3 a d f) x \sqrt {c+d x^2}}{3 d f \sqrt {e+f x^2}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\sqrt {e} (b e-3 a 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 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {(e (-2 b d e+b c f+3 a d f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 d f} \\ & = -\frac {(2 b d e-b c f-3 a d f) x \sqrt {c+d x^2}}{3 d f \sqrt {e+f x^2}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}+\frac {\sqrt {e} (2 b d e-b c f-3 a d f) \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 d f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-3 a 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 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.37 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\frac {b \sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right )-i e (-2 b d e+b c f+3 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i (2 b e-3 a f) (-d e+c f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{3 \sqrt {\frac {d}{c}} f^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

(b*Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2) - I*e*(-2*b*d*e + b*c*f + 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^
2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*(2*b*e - 3*a*f)*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sq
rt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*
x^2])

Maple [A] (verified)

Time = 5.26 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.11

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 f}+\frac {\left (a c -\frac {c e b}{3 f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (a d +b c -\frac {b \left (2 c f +2 d e \right )}{3 f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(312\)
risch	\(\frac {b x \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{3 f}+\frac {\left (\frac {3 a c f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b c e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (3 a d f +b c f -2 b d e \right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{3 f \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(386\)
default	\(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, \left (\sqrt {-\frac {d}{c}}\, b d \,f^{2} x^{5}+\sqrt {-\frac {d}{c}}\, b c \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, b d e f \,x^{3}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c \,f^{2}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e f -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c e f +2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b d \,e^{2}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c e f -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b d \,e^{2}+\sqrt {-\frac {d}{c}}\, b c e f x \right )}{3 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) f^{2} \sqrt {-\frac {d}{c}}}\)	\(501\)

[In]

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

[Out]

((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(1/3*b/f*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+(a*
c-1/3*c*e/f*b)/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(
x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-(a*d+b*c-1/3*b/f*(2*c*f+2*d*e))*e/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f
*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/f*(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-Ellipt
icE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\frac {{\left (2 \, b d e^{3} - {\left (b c + 3 \, a d\right )} e^{2} f\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (2 \, b d e^{3} + b c e f^{2} - 3 \, a c f^{3} - {\left (b c + 3 \, a d\right )} e^{2} f\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) + {\left (b d e f^{2} x^{2} - 2 \, b d e^{2} f + {\left (b c + 3 \, a d\right )} e f^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{3 \, d e f^{3} x} \]

[In]

integrate((b*x^2+a)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="fricas")

[Out]

1/3*((2*b*d*e^3 - (b*c + 3*a*d)*e^2*f)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (2
*b*d*e^3 + b*c*e*f^2 - 3*a*c*f^3 - (b*c + 3*a*d)*e^2*f)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_f(arcsin(sqrt(-e/f)/x)
, c*f/(d*e)) + (b*d*e*f^2*x^2 - 2*b*d*e^2*f + (b*c + 3*a*d)*e*f^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(d*e*f^3*x
)

Sympy [F]

\[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\int \frac {\left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}{\sqrt {e + f x^{2}}}\, dx \]

[In]

integrate((b*x**2+a)*(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral((a + b*x**2)*sqrt(c + d*x**2)/sqrt(e + f*x**2), x)

Maxima [F]

\[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c}}{\sqrt {f x^{2} + e}} \,d x } \]

[In]

integrate((b*x^2+a)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c}}{\sqrt {f x^{2} + e}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right ) \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}}{\sqrt {f\,x^2+e}} \,d x \]

[In]

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

[Out]

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