\(\int \frac {(c+d x^2)^{5/2}}{(a+b x^2)^{3/2}} \, dx\) [151]

Optimal result

Integrand size = 23, antiderivative size = 279 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {2 d (3 b c-2 a d) x \sqrt {c+d x^2}}{3 b^2 \sqrt {a+b x^2}}+\frac {d x \left (c+d x^2\right )^{3/2}}{3 b \sqrt {a+b x^2}}+\frac {\left (3 b^2 c^2-13 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {2 \sqrt {a} d (3 b c-2 a d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

2/3*d*(-2*a*d+3*b*c)*x*(d*x^2+c)^(1/2)/b^2/(b*x^2+a)^(1/2)+1/3*d*x*(d*x^2+ 
c)^(3/2)/b/(b*x^2+a)^(1/2)+1/3*(8*a^2*d^2-13*a*b*c*d+3*b^2*c^2)*(d*x^2+c)^ 
(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^( 
1/2)/b^(5/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+2/3*a^(1/2)*d 
*(-2*a*d+3*b*c)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)), 
(1-a*d/b/c)^(1/2))/b^(5/2)/(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 = 4.31 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (3 b^2 c^2+4 a^2 d^2+a b d \left (-6 c+d x^2\right )\right )+i c \left (3 b^2 c^2-13 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (3 b^2 c^2-7 a b c d+4 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(c + d*x^2)^(5/2)/(a + b*x^2)^(3/2),x]
 

Output:

(Sqrt[b/a]*(Sqrt[b/a]*x*(c + d*x^2)*(3*b^2*c^2 + 4*a^2*d^2 + a*b*d*(-6*c + 
 d*x^2)) + I*c*(3*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sq 
rt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(3* 
b^2*c^2 - 7*a*b*c*d + 4*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E 
llipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b^3*Sqrt[a + b*x^2]*Sqr 
t[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {315, 27, 403, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 315

\(\displaystyle \frac {\int \frac {d \sqrt {d x^2+c} \left (a c-(3 b c-4 a d) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {\sqrt {d x^2+c} \left (a c-(3 b c-4 a d) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {d \left (\frac {\int \frac {2 a c (3 b c-2 a d)-\left (3 b^2 c^2-13 a b d c+8 a^2 d^2\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 b}\right )}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {d \left (\frac {2 a c (3 b c-2 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (8 a^2 d^2-13 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 b}\right )}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {d \left (\frac {\frac {2 c^{3/2} \sqrt {a+b x^2} (3 b c-2 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (8 a^2 d^2-13 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 b}\right )}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {d \left (\frac {\frac {2 c^{3/2} \sqrt {a+b x^2} (3 b c-2 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (8 a^2 d^2-13 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 b}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 b}\right )}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {d \left (\frac {\frac {2 c^{3/2} \sqrt {a+b x^2} (3 b c-2 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (8 a^2 d^2-13 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 b}\right )}{a b}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{a b \sqrt {a+b x^2}}\)

Input:

Int[(c + d*x^2)^(5/2)/(a + b*x^2)^(3/2),x]
 

Output:

((b*c - a*d)*x*(c + d*x^2)^(3/2))/(a*b*Sqrt[a + b*x^2]) + (d*(-1/3*((3*b*c 
 - 4*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/b + (-((3*b^2*c^2 - 13*a*b*c* 
d + 8*a^2*d^2)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a 
+ b*x^2]*EllipticE[ArcTan[(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]))) + (2*c^(3/2)*( 
3*b*c - 2*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - 
(b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^ 
2]))/(3*b)))/(a*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[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 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 403

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(250)=500\).

Time = 17.00 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.83

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{a \,b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {d^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2}}+\frac {\left (\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right )}{b^{3}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right )}{b^{3} a}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{2} a}-\frac {d^{2} a c}{3 b^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {d^{2} \left (a d -3 b c \right )}{b^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{b^{2} a}-\frac {d^{2} \left (2 a d +2 b c \right )}{3 b^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(510\)
default	\(\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+3 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+13 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -6 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x +3 \sqrt {-\frac {b}{a}}\, b^{2} c^{3} x \right )}{3 b^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}\, a}\)	\(525\)
risch	\(\frac {d^{2} x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b^{2}}-\frac {\left (\frac {d \left (-\frac {b \left (5 a d -7 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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 a^{2} d^{2} \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 {9 b^{2} c^{2} \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 {10 a b c d \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}}\right )}{b}+\frac {3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \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 {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\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 {b 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 )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(813\)

Input:

int((d*x^2+c)^(5/2)/(b*x^2+a)^(3/2),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)*((b*d*x^2+b*c) 
*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a/b^3*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3*d 
^2/b^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(d*(a^2*d^2-3*a*b*c*d+3*b^2*c 
^2)/b^3-(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3*(a*d-b*c)/a-1/b^2*c*(a^2*d^2-2*a*b 
*c*d+b^2*c^2)/a-1/3*d^2/b^2*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))-(-1/b^2*d^2*(a*d-3*b*c)-(a^2*d^2-2*a*b*c*d+b^2*c^2)/b 
^2*d/a-1/3*d^2/b^2*(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))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.24 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, b^{3} c^{3} - 13 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{3} - 13 \, a^{2} b c^{2} d + 8 \, a^{3} c d^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, b^{3} c^{3} - 13 \, a b^{2} c^{2} d + 4 \, a^{2} b d^{3} + 2 \, {\left (4 \, a^{2} b - 3 \, a b^{2}\right )} c d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{3} - 13 \, a^{2} b c^{2} d + 4 \, a^{3} d^{3} + 2 \, {\left (4 \, a^{3} - 3 \, a^{2} b\right )} c d^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (a b^{2} d^{3} x^{4} - 3 \, a b^{2} c^{2} d + 13 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} + {\left (7 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a b^{4} d x^{3} + a^{2} b^{3} d x\right )}} \] Input:

integrate((d*x^2+c)^(5/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

1/3*(((3*b^3*c^3 - 13*a*b^2*c^2*d + 8*a^2*b*c*d^2)*x^3 + (3*a*b^2*c^3 - 13 
*a^2*b*c^2*d + 8*a^3*c*d^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt 
(-c/d)/x), a*d/(b*c)) - ((3*b^3*c^3 - 13*a*b^2*c^2*d + 4*a^2*b*d^3 + 2*(4* 
a^2*b - 3*a*b^2)*c*d^2)*x^3 + (3*a*b^2*c^3 - 13*a^2*b*c^2*d + 4*a^3*d^3 + 
2*(4*a^3 - 3*a^2*b)*c*d^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt( 
-c/d)/x), a*d/(b*c)) + (a*b^2*d^3*x^4 - 3*a*b^2*c^2*d + 13*a^2*b*c*d^2 - 8 
*a^3*d^3 + (7*a*b^2*c*d^2 - 4*a^2*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + 
 c))/(a*b^4*d*x^3 + a^2*b^3*d*x)
 

Sympy [F]

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

integrate((d*x**2+c)**(5/2)/(b*x**2+a)**(3/2),x)
 

Output:

Integral((c + d*x**2)**(5/2)/(a + b*x**2)**(3/2), x)
 

Maxima [F]

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

integrate((d*x^2+c)^(5/2)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^(3/2), x)
 

Giac [F]

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

integrate((d*x^2+c)^(5/2)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((c + d*x^2)^(5/2)/(a + b*x^2)^(3/2),x)
 

Output:

int((c + d*x^2)^(5/2)/(a + b*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^(5/2)/(b*x^2+a)^(3/2),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*x + 2*sqrt(c + d*x**2)*sqrt( 
a + b*x**2)*a*d**2*x**3 + 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2*x - 8 
*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a* 
b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**3*d**3 + 17*int 
((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c* 
x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c*d**2 - 8*int( 
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x 
**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**3*x**2 - 9*in 
t((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c 
*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c**2*d + 17*in 
t((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c 
*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d**2*x**2 - 
9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a 
*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*b**3*c**2*d*x**2 
+ 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b* 
c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**3*c**2*d - 3*int( 
(sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 
 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c**3 + 3*int((sqrt(c 
+ d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d 
*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c**2*d*x**2 - 3*int((sqrt(...