\(\int \frac {(e x)^{3/2}}{(c+d x) \sqrt {a-b x^2}} \, dx\) [1393]

Optimal result

Integrand size = 27, antiderivative size = 252 \[ \int \frac {(e x)^{3/2}}{(c+d x) \sqrt {a-b x^2}} \, dx=\frac {2 a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} d \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{3/4} d^2 \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} c e^{3/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^2 \sqrt {a-b x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 11.43 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.94 \[ \int \frac {(e x)^{3/2}}{(c+d x) \sqrt {a-b x^2}} \, dx=\frac {2 e \sqrt {e x} \sqrt {a-b x^2} \left (\left (-\sqrt {a} \sqrt {b} c d+a d^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt {1+\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )\right |2\right )+\left (-b c^2+\sqrt {a} \sqrt {b} c d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),2\right )+b c^2 \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} d}{-\sqrt {b} c+\sqrt {a} d},\arcsin \left (\frac {\sqrt {1+\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),2\right )\right )}{\sqrt {b} d^2 \left (-a \sqrt {b} c+a^{3/2} d\right ) \sqrt {-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {1-\frac {b x^2}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {616, 27, 1665, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}

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.

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

\(\Big \downarrow \) 616

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1665

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

2*(-((-((a^(3/4)*d*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*S 
qrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(3/4)*Sqrt[a - b*x^2])) + (a^(1/4)*( 
c + (Sqrt[a]*d)/Sqrt[b])*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)*Sqrt[a - b*x^2]))/d^2) + 
 (a^(1/4)*c*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]*d)/(Sqrt[b]* 
c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*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 1665

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.17

method	result	size
default	\(\frac {\left (2 \sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) c d -2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) c d -\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}-\operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}+2 \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}+\operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}\right ) \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {a b}\, \sqrt {2}\, e \sqrt {e x}}{b \sqrt {-b \,x^{2}+a}\, x \,d^{2} \left (b c -d \sqrt {a b}\right )}\)	\(296\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {c \,e^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} b \sqrt {-b e \,x^{3}+a e x}}+\frac {e^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{d b \sqrt {-b e \,x^{3}+a e x}}+\frac {c^{2} e^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{3} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-b \,x^{2}+a}}\)	\(458\)

Input:

int((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(2*(a*b)^(1/2)*EllipticF(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2) 
)*c*d-2*(a*b)^(1/2)*EllipticE(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^ 
(1/2))*c*d-EllipticF(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a* 
d^2-EllipticF(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*b*c^2+2*E 
llipticE(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2+Elliptic 
Pi(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),(a*b)^(1/2)*d/(d*(a*b)^(1/2)-b*c) 
,1/2*2^(1/2))*b*c^2)*(-b*x/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a*b)^(1 
/2))^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*(a*b)^(1/2)*2^(1/2)*e*(e* 
x)^(1/2)/b/(-b*x^2+a)^(1/2)/x/d^2/(b*c-d*(a*b)^(1/2))
 

Fricas [F(-1)]

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

integrate((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(3/2)/(d*x+c)/(-b*x**2+a)**(1/2),x)
                                                                                    
                                                                                    
 

Output:

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

Maxima [F]

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

integrate((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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