\(\int \frac {(e x)^{5/2}}{(c+d x) (a-b x^2)^{3/2}} \, dx\) [1408]

Optimal result

Integrand size = 27, antiderivative size = 348 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=-\frac {e^2 \sqrt {e x} (a d-b c x)}{b \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {a^{3/4} c e^{5/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {\sqrt [4]{a} \left (2 \sqrt {b} c+\sqrt {a} d\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{5/4} d \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \left (b c^2-a d^2\right ) \sqrt {a-b x^2}} \] Output:

-e^2*(e*x)^(1/2)*(-b*c*x+a*d)/b/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)-a^(3/4)*c* 
e^(5/2)*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I) 
/b^(3/4)/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)-a^(1/4)*(2*b^(1/2)*c+a^(1/2)*d)*e 
^(5/2)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/ 
b^(5/4)/d/(b^(1/2)*c+a^(1/2)*d)/(-b*x^2+a)^(1/2)+2*a^(1/4)*c^2*e^(5/2)*(1- 
b*x^2/a)^(1/2)*EllipticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2)*d/b 
^(1/2)/c,I)/b^(1/4)/d/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.66 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.91 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {e^3 \left (-a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c d+a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} d^2 x-i \sqrt {a} \sqrt {b} c d \sqrt {1-\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+2 i b c^2 \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b d \left (-b c^2+a d^2\right ) \sqrt {e x} \sqrt {a-b x^2}} \] Input:

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

Output:

(e^3*(-(a*Sqrt[-(Sqrt[a]/Sqrt[b])]*c*d) + a*Sqrt[-(Sqrt[a]/Sqrt[b])]*d^2*x 
 - I*Sqrt[a]*Sqrt[b]*c*d*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[S 
qrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] + I*Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]*d 
)*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])] 
/Sqrt[x]], -1] + (2*I)*b*c^2*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticPi[-((Sqr 
t[b]*c)/(Sqrt[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1]))/( 
Sqrt[-(Sqrt[a]/Sqrt[b])]*b*d*(-(b*c^2) + a*d^2)*Sqrt[e*x]*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {616, 27, 1641, 1543, 1542, 2397, 27, 1513, 27, 765, 762, 1390, 1389, 327}

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.

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

\(\Big \downarrow \) 616

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1641

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {-\frac {\left (b c^2-a d^2\right ) x^2 e^2}{d}-a c x e^2+\frac {a c^2 e^2}{d}}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 2397

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^2 \int \frac {a \left (\left (\frac {2 b c^2}{d}-a d\right ) e+b c x e\right )}{e \sqrt {a-b x^2}}d\sqrt {e x}}{2 a b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \int \frac {\left (\frac {2 b c^2}{d}-a d\right ) e+b c x e}{\sqrt {a-b x^2}}d\sqrt {e x}}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 1513

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\frac {e \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}}{d}+\sqrt {a} \sqrt {b} c e \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\frac {e \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}}{d}+\sqrt {b} c \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\frac {e \sqrt {1-\frac {b x^2}{a}} \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{d \sqrt {a-b x^2}}+\sqrt {b} c \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\sqrt {b} c \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 1390

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\frac {\sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 1389

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\frac {\sqrt {a} \sqrt {b} c e \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

\(\Big \downarrow \) 327

\(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e \left (\frac {a^{3/4} \sqrt [4]{b} c e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-\sqrt {a} \sqrt {b} c d-a d^2+2 b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}\right )}{2 b}+\frac {e \sqrt {e x} (a d e-b c e x)}{2 b \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\)

Input:

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

Output:

2*(-(((e*Sqrt[e*x]*(a*d*e - b*c*e*x))/(2*b*Sqrt[a - b*x^2]) + (e*((a^(3/4) 
*b^(1/4)*c*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[e*x] 
)/(a^(1/4)*Sqrt[e])], -1])/Sqrt[a - b*x^2] + (a^(1/4)*(2*b*c^2 - Sqrt[a]*S 
qrt[b]*c*d - a*d^2)*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)* 
Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*d*Sqrt[a - b*x^2])))/(2*b))/( 
b*c^2 - a*d^2)) + (a^(1/4)*c^2*e^(5/2)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((S 
qrt[a]*d)/(Sqrt[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1] 
)/(b^(1/4)*d*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]))
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 616

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) - 
 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; 
FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1513

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[-c/a, 2]}, Simp[(d*q - e)/q   Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q 
  Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg 
Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

rule 1641

Int[((x_)^(m_)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] 
 :> Simp[(-d/e)^(m/2)*((c*d^2 + a*e^2)^(p + 1/2)/e^(2*p + 1))   Int[1/((d + 
 e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[(c*d^2 + a*e^2)^(p + 1/2)   Int[(a 
+ c*x^4)^p*ExpandToSum[((c*d^2 + a*e^2)^(-p - 1/2)*x^m - e^(-2*p - 1)*(-d/e 
)^(m/2)*(a + c*x^4)^(-p - 1/2))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, 
e}, x] && ILtQ[p + 1/2, 0] && IGtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ 
[c/a]
 

rule 2397

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, 
x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, 
 x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S 
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] 
 + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))   Int[(a + b*x^n)^(p + 1)* 
ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, 
n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(584\) vs. \(2(278)=556\).

Time = 1.33 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.68

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {2 b e x \left (-\frac {e^{2} c x}{2 b \left (a \,d^{2}-b \,c^{2}\right )}+\frac {a d \,e^{2}}{2 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\sqrt {-\left (x^{2}-\frac {a}{b}\right ) b e x}}+\frac {\left (-\frac {e^{3}}{b d}+\frac {a d \,e^{3}}{2 b \left (a \,d^{2}-b \,c^{2}\right )}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {e^{3} c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right ) b \sqrt {-b e \,x^{3}+a e x}}-\frac {c^{3} e^{3} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) d^{2} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-b \,x^{2}+a}}\)	\(585\)
default	\(\frac {\left (\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} d^{3} \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b \,c^{2} d \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-2 \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,d^{2} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+2 \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, b \,c^{3} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-2 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b \,c^{2} d \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+2 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,d^{2} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-2 \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, b \,c^{3} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-2 b^{2} c^{2} d \,x^{2}+2 b c \,d^{2} x^{2} \sqrt {a b}+2 a b c \,d^{2} x -2 a \,d^{3} x \sqrt {a b}\right ) e^{2} \sqrt {e x}}{2 b x \sqrt {-b \,x^{2}+a}\, d \left (b c -d \sqrt {a b}\right ) \left (a \,d^{2}-b \,c^{2}\right )}\)	\(701\)

Input:

int((e*x)^(5/2)/(d*x+c)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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