3.2.74 \(\int \frac {(c+d x^2)^{3/2}}{\sqrt {a+b x^2}} \, dx\) [174]

3.2.74.1 Optimal result
3.2.74.2 Mathematica [C] (verified)
3.2.74.3 Rubi [A] (verified)
3.2.74.4 Maple [A] (verified)
3.2.74.5 Fricas [A] (verification not implemented)
3.2.74.6 Sympy [F]
3.2.74.7 Maxima [F]
3.2.74.8 Giac [F]
3.2.74.9 Mupad [F(-1)]

3.2.74.1 Optimal result

Integrand size = 23, antiderivative size = 273 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 b^2 \sqrt {c+d x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}-\frac {2 \sqrt {c} \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} (3 b c-a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

output
2/3*d*(-a*d+2*b*c)*x*(b*x^2+a)^(1/2)/b^2/(d*x^2+c)^(1/2)+1/3*c^(3/2)*(-a*d 
+3*b*c)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2 
)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*(b*x^2+a)^(1/2)/a/b/d^(1/2)/(c*(b*x 
^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-2/3*(-a*d+2*b*c)*(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-b 
*c/a/d)^(1/2))*c^(1/2)*d^(1/2)*(b*x^2+a)^(1/2)/b^2/(c*(b*x^2+a)/a/(d*x^2+c 
))^(1/2)/(d*x^2+c)^(1/2)+1/3*d*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b
 
3.2.74.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.93 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.73 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )+2 i c (-2 b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

input
Integrate[(c + d*x^2)^(3/2)/Sqrt[a + b*x^2],x]
 
output
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) + (2*I)*c*(-2*b*c + a*d)*Sqrt[1 + ( 
b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c 
)] - I*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ 
I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*b*Sqrt[b/a]*Sqrt[a + b*x^2]*Sqrt[ 
c + d*x^2])
 
3.2.74.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {318, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

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

3.2.74.3.1 Defintions of rubi rules used

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 318
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 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 388
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] - Simp[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 406
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 
3.2.74.4 Maple [A] (verified)

Time = 4.42 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.14

method result size
elliptic \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {d x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b}+\frac {\left (c^{2}-\frac {d a c}{3 b}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 c d -\frac {d \left (2 a d +2 b c \right )}{3 b}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) \(310\)
default \(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, b \,d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b c d \,x^{3}+a c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d +4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+\sqrt {-\frac {b}{a}}\, a c d x \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) b \sqrt {-\frac {b}{a}}}\) \(330\)
risch \(\frac {d x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{3 b}-\frac {\left (\frac {a c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {3 b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a \,d^{2}-4 b c d \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 b \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) \(383\)

input
int((d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),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*d/b*x*(b* 
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(c^2-1/3*d/b*a*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))-(2*c*d-1/3*d/b*(2*a*d+2*b*c))*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)/d*(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))))
 
3.2.74.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.61 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left (2 \, b c^{2} - a c d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (4 \, b c^{2} - {\left (2 \, a - 3 \, b\right )} c d - a d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b d^{2} x^{2} + 4 \, b c d - 2 \, a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b^{2} d x} \]

input
integrate((d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 
output
-1/3*(2*(2*b*c^2 - a*c*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c 
/d)/x), a*d/(b*c)) - (4*b*c^2 - (2*a - 3*b)*c*d - a*d^2)*sqrt(b*d)*x*sqrt( 
-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (b*d^2*x^2 + 4*b*c*d - 
 2*a*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d*x)
 
3.2.74.6 Sympy [F]

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

input
integrate((d*x**2+c)**(3/2)/(b*x**2+a)**(1/2),x)
 
output
Integral((c + d*x**2)**(3/2)/sqrt(a + b*x**2), x)
 
3.2.74.7 Maxima [F]

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

input
integrate((d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 
output
integrate((d*x^2 + c)^(3/2)/sqrt(b*x^2 + a), x)
 
3.2.74.8 Giac [F]

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

input
integrate((d*x^2+c)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="giac")
 
output
integrate((d*x^2 + c)^(3/2)/sqrt(b*x^2 + a), x)
 
3.2.74.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{\sqrt {b\,x^2+a}} \,d x \]

input
int((c + d*x^2)^(3/2)/(a + b*x^2)^(1/2),x)
 
output
int((c + d*x^2)^(3/2)/(a + b*x^2)^(1/2), x)