\(\int \frac {\sqrt {c+d x^2}}{(a+b x^2)^{5/2}} \, dx\) [139]

Optimal result

Integrand size = 23, antiderivative size = 235 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 b c-a d) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} \sqrt {b} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*x*(d*x^2+c)^(1/2)/a/(b*x^2+a)^(3/2)+1/3*(-a*d+2*b*c)*(d*x^2+c)^(1/2)*E 
llipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(3/2)/b^ 
(1/2)/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*d*(d* 
x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/ 
a^(1/2)/b^(1/2)/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.60 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (2 a^2 d-2 b^2 c x^2+a b \left (-3 c+d x^2\right )\right )+i c (-2 b c+a d) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i c (-b c+a d) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} (-b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*x*(c + d*x^2)*(2*a^2*d - 2*b^2*c*x^2 + a*b*(-3*c + d*x^2)) + I* 
c*(-2*b*c + a*d)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip 
ticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*c*(-(b*c) + a*d)*(a + b* 
x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a] 
*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*(-(b*c) + a*d)*(a + b*x^2)^(3/2)*Sqrt[ 
c + d*x^2])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {314, 25, 400, 313, 320}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 314

\(\displaystyle \frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {d x^2+2 c}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{3 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {d x^2+2 c}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{3 a}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {\frac {(2 b c-a d) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {c d \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\sqrt {c+d x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c d \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {\sqrt {c+d x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 x]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(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] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 400

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2]), x], x] - Simp[(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]
 

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.78

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2} a \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) x \left (a d -2 b c \right )}{3 b \,a^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d}{3 a b}-\frac {a d -2 b c}{3 b \,a^{2}}-\frac {c \left (a d -2 b c \right )}{3 a^{2} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\left (a d -2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 \left (a d -b c \right ) a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(418\)
default	\(\frac {\sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{5}-2 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{5}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{2}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{3}-2 \sqrt {-\frac {b}{a}}\, a b c d \,x^{3}-2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}+2 \sqrt {-\frac {b}{a}}\, a^{2} c d x -3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x}{3 \sqrt {x^{2} d +c}\, \sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(617\)

Input:

int((d*x^2+c)^(1/2)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/3/b^2/a*x*( 
b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2+1/3*(b*d*x^2+b*c)/b/a^2/(a* 
d-b*c)*x*(a*d-2*b*c)/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(1/3/a/b*d-1/3*(a*d-2 
*b*c)/b/a^2-1/3*c/a^2/(a*d-b*c)*(a*d-2*b*c))/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2 
)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a) 
^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1/3*(a*d-2*b*c)/(a*d-b*c)/a^2*c/(-b/a)^(1 
/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2 
)*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^( 
1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left (2 \, a^{2} b^{2} c - a^{3} b d + {\left (2 \, b^{4} c - a b^{3} d\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, a^{2} b^{2} c + {\left (2 \, b^{4} c + {\left (a^{2} b^{2} - a b^{3}\right )} d\right )} x^{4} + 2 \, {\left (2 \, a b^{3} c + {\left (a^{3} b - a^{2} b^{2}\right )} d\right )} x^{2} + {\left (a^{4} - a^{3} b\right )} d\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{3} + {\left (3 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{5} b^{2} c - a^{6} b d + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} x^{4} + 2 \, {\left (a^{4} b^{3} c - a^{5} b^{2} d\right )} x^{2}\right )}} \] Input:

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

-1/3*((2*a^2*b^2*c - a^3*b*d + (2*b^4*c - a*b^3*d)*x^4 + 2*(2*a*b^3*c - a^ 
2*b^2*d)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b 
*c)) - (2*a^2*b^2*c + (2*b^4*c + (a^2*b^2 - a*b^3)*d)*x^4 + 2*(2*a*b^3*c + 
 (a^3*b - a^2*b^2)*d)*x^2 + (a^4 - a^3*b)*d)*sqrt(a*c)*sqrt(-b/a)*elliptic 
_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((2*a*b^3*c - a^2*b^2*d)*x^3 + (3*a^ 
2*b^2*c - 2*a^3*b*d)*x)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^5*b^2*c - a^6* 
b*d + (a^3*b^4*c - a^4*b^3*d)*x^4 + 2*(a^4*b^3*c - a^5*b^2*d)*x^2)
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(5/2),x)
 

Output:

Integral(sqrt(c + d*x**2)/(a + b*x**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int((c + d*x^2)^(1/2)/(a + b*x^2)^(5/2),x)
 

Output:

int((c + d*x^2)^(1/2)/(a + b*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \] Input:

int((d*x^2+c)^(1/2)/(b*x^2+a)^(5/2),x)
 

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x 
**4 + b**3*x**6),x)