\(\int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [1594]

Optimal result

Integrand size = 29, antiderivative size = 415 \[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {a-b x^2}}{5 a c e (e x)^{5/2}}+\frac {8 d \sqrt {c+d x} \sqrt {a-b x^2}}{15 a c^2 e^2 (e x)^{3/2}}+\frac {2 \sqrt {b} \left (9 b c^2+8 a d^2\right ) \sqrt {\frac {b-\frac {a}{x^2}}{b}} \sqrt {e x} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b} c}{\sqrt {b} c+\sqrt {a} d}\right )}{15 a^{3/2} c^3 e^4 \sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {b} d \left (7 b c^2+8 a d^2\right ) \sqrt {\frac {b-\frac {a}{x^2}}{b}} \sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{\sqrt {b} c+\sqrt {a} d}} (e x)^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b} c}{\sqrt {b} c+\sqrt {a} d}\right )}{15 a^{3/2} c^3 e^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/5*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/c/e/(e*x)^(5/2)+8/15*d*(d*x+c)^(1/2) 
*(-b*x^2+a)^(1/2)/a/c^2/e^2/(e*x)^(3/2)+2/15*b^(1/2)*(8*a*d^2+9*b*c^2)*((b 
-a/x^2)/b)^(1/2)*(e*x)^(1/2)*(d*x+c)^(1/2)*EllipticE(1/2*(1-a^(1/2)/b^(1/2 
)/x)^(1/2)*2^(1/2),2^(1/2)*(b^(1/2)*c/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2 
)/c^3/e^4/(a^(1/2)*(d+c/x)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-2 
/15*b^(1/2)*d*(8*a*d^2+7*b*c^2)*((b-a/x^2)/b)^(1/2)*(a^(1/2)*(d+c/x)/(b^(1 
/2)*c+a^(1/2)*d))^(1/2)*(e*x)^(3/2)*EllipticF(1/2*(1-a^(1/2)/b^(1/2)/x)^(1 
/2)*2^(1/2),2^(1/2)*(b^(1/2)*c/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/c^3/e 
^5/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(879\) vs. \(2(415)=830\).

Time = 19.97 (sec) , antiderivative size = 879, normalized size of antiderivative = 2.12 \[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 x \sqrt {a-b x^2} \left (-\left ((c+d x) \left (9 b c^2 x^2+a \left (3 c^2-4 c d x+8 d^2 x^2\right )\right )\right )-x^3 \left (-\left (\left (9 b c^2+8 a d^2\right ) \left (d+\frac {c}{x}\right )\right )+\frac {2 \sqrt {a} b c^2 d \sqrt {1+\frac {\sqrt {a}}{\sqrt {b} x}} \sqrt {\frac {\sqrt {a} (c+d x)}{\left (\sqrt {b} c+\sqrt {a} d\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b} c}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {1-\frac {\sqrt {a}}{\sqrt {b} x}} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {9 b c^2 \sqrt {\frac {c \left (-\sqrt {a}+\sqrt {b} x\right )}{\left (\sqrt {b} c+\sqrt {a} d\right ) x}} \sqrt {-\frac {\sqrt {a} (c+d x)}{\left (\sqrt {b} c-\sqrt {a} d\right ) x}} \left (\left (\sqrt {b} c+\sqrt {a} d\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-\sqrt {b} c+\sqrt {a} d}}\right )|\frac {-\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )-\sqrt {b} c \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-\sqrt {b} c+\sqrt {a} d}}\right ),\frac {-\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )\right )}{\left (\sqrt {a}-\sqrt {b} x\right ) \sqrt {\frac {\sqrt {a} c+\sqrt {b} c x}{\sqrt {b} c x-\sqrt {a} d x}}}-\frac {8 a d^2 \sqrt {\frac {c \left (-\sqrt {a}+\sqrt {b} x\right )}{\left (\sqrt {b} c+\sqrt {a} d\right ) x}} \sqrt {\frac {\sqrt {a} (c+d x)}{\left (-\sqrt {b} c+\sqrt {a} d\right ) x}} \left (\left (\sqrt {b} c+\sqrt {a} d\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-\sqrt {b} c+\sqrt {a} d}}\right )|\frac {-\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )-\sqrt {b} c \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {a} \left (d+\frac {c}{x}\right )}{-\sqrt {b} c+\sqrt {a} d}}\right ),\frac {-\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )\right )}{\left (-\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {\sqrt {a} c+\sqrt {b} c x}{\sqrt {b} c x-\sqrt {a} d x}}}\right )\right )}{15 a^2 c^3 (e x)^{7/2} \sqrt {c+d x}} \] Input:

Integrate[1/((e*x)^(7/2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
 

Output:

(2*x*Sqrt[a - b*x^2]*(-((c + d*x)*(9*b*c^2*x^2 + a*(3*c^2 - 4*c*d*x + 8*d^ 
2*x^2))) - x^3*(-((9*b*c^2 + 8*a*d^2)*(d + c/x)) + (2*Sqrt[a]*b*c^2*d*Sqrt 
[1 + Sqrt[a]/(Sqrt[b]*x)]*Sqrt[(Sqrt[a]*(c + d*x))/((Sqrt[b]*c + Sqrt[a]*d 
)*x)]*EllipticF[ArcSin[Sqrt[1 - Sqrt[a]/(Sqrt[b]*x)]/Sqrt[2]], (2*Sqrt[b]* 
c)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[1 - Sqrt[a]/(Sqrt[b]*x)]*(Sqrt[a] + Sqr 
t[b]*x)) + (9*b*c^2*Sqrt[(c*(-Sqrt[a] + Sqrt[b]*x))/((Sqrt[b]*c + Sqrt[a]* 
d)*x)]*Sqrt[-((Sqrt[a]*(c + d*x))/((Sqrt[b]*c - Sqrt[a]*d)*x))]*((Sqrt[b]* 
c + Sqrt[a]*d)*EllipticE[ArcSin[Sqrt[(Sqrt[a]*(d + c/x))/(-(Sqrt[b]*c) + S 
qrt[a]*d)]], (-(Sqrt[b]*c) + Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)] - Sqrt[b] 
*c*EllipticF[ArcSin[Sqrt[(Sqrt[a]*(d + c/x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]], 
 (-(Sqrt[b]*c) + Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)]))/((Sqrt[a] - Sqrt[b] 
*x)*Sqrt[(Sqrt[a]*c + Sqrt[b]*c*x)/(Sqrt[b]*c*x - Sqrt[a]*d*x)]) - (8*a*d^ 
2*Sqrt[(c*(-Sqrt[a] + Sqrt[b]*x))/((Sqrt[b]*c + Sqrt[a]*d)*x)]*Sqrt[(Sqrt[ 
a]*(c + d*x))/((-(Sqrt[b]*c) + Sqrt[a]*d)*x)]*((Sqrt[b]*c + Sqrt[a]*d)*Ell 
ipticE[ArcSin[Sqrt[(Sqrt[a]*(d + c/x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]], (-(Sq 
rt[b]*c) + Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)] - Sqrt[b]*c*EllipticF[ArcSi 
n[Sqrt[(Sqrt[a]*(d + c/x))/(-(Sqrt[b]*c) + Sqrt[a]*d)]], (-(Sqrt[b]*c) + S 
qrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)]))/((-Sqrt[a] + Sqrt[b]*x)*Sqrt[(Sqrt[a] 
*c + Sqrt[b]*c*x)/(Sqrt[b]*c*x - Sqrt[a]*d*x)]))))/(15*a^2*c^3*(e*x)^(7/2) 
*Sqrt[c + d*x])
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{(e x)^{7/2} \sqrt {a-b x^2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 636

\(\displaystyle \frac {\int -\frac {-\frac {2 b d x^2}{c}-3 b x+\frac {4 a d}{c}}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {-\frac {2 b d x^2}{c}-3 b x+\frac {4 a d}{c}}{(e x)^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 2352

\(\displaystyle -\frac {-\frac {\int \frac {a \left (\frac {8 a d^2}{c}+2 b x d+9 b c\right )}{(e x)^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 a c e}-\frac {8 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 e (e x)^{3/2}}}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\frac {8 a d^2}{c}+2 b x d+9 b c}{(e x)^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 c e}-\frac {8 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 e (e x)^{3/2}}}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 2352

\(\displaystyle -\frac {-\frac {-\frac {\int -\frac {-2 b d \left (\frac {8 a d^2}{c}+9 b c\right ) x^2-b \left (9 b c^2+8 a d^2\right ) x+2 a b c d}{\sqrt {e x} \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a c e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (8 a d^2+9 b c^2\right )}{a c^2 e \sqrt {e x}}}{3 c e}-\frac {8 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 e (e x)^{3/2}}}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\frac {\int \frac {-2 b d \left (\frac {8 a d^2}{c}+9 b c\right ) x^2-b \left (9 b c^2+8 a d^2\right ) x+2 a b c d}{\sqrt {e x} \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a c e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (8 a d^2+9 b c^2\right )}{a c^2 e \sqrt {e x}}}{3 c e}-\frac {8 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 e (e x)^{3/2}}}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 2354

\(\displaystyle -\frac {-\frac {\frac {2 \int \frac {-2 b d \left (\frac {8 a d^2}{c}+9 b c\right ) x^2-b \left (9 b c^2+8 a d^2\right ) x+2 a b c d}{\sqrt {c+d x} \sqrt {a-b x^2}}d\sqrt {e x}}{a c e^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (8 a d^2+9 b c^2\right )}{a c^2 e \sqrt {e x}}}{3 c e}-\frac {8 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 e (e x)^{3/2}}}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

\(\Big \downarrow \) 2261

\(\displaystyle -\frac {-\frac {\frac {2 \int \frac {-2 b d \left (\frac {8 a d^2}{c}+9 b c\right ) x^2-b \left (9 b c^2+8 a d^2\right ) x+2 a b c d}{\sqrt {c+d x} \sqrt {a-b x^2}}d\sqrt {e x}}{a c e^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (8 a d^2+9 b c^2\right )}{a c^2 e \sqrt {e x}}}{3 c e}-\frac {8 d \sqrt {a-b x^2} \sqrt {c+d x}}{3 c^2 e (e x)^{3/2}}}{5 a e}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x}}{5 a c e (e x)^{5/2}}\)

Input:

Int[1/((e*x)^(7/2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

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

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 636

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/Sqrt[(a_) + (b_.)*(x_)^2], 
 x_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2] 
/(a*e*(m + 1))), x] - Simp[1/(2*a*e*(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + 
 d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2 
)*d*(2*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x 
^3 - 2*a*(m + 1)*(c + d*x)^(n + 1/2))/x, x], x], x] /; FreeQ[{a, b, c, d, e 
}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

rule 2261

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, 
p, q}, x] && PolyQ[Px, x]
 

rule 2352

Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x 
_)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m 
+ 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* 
(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 
2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 
2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, 
x] && LtQ[m, -1]
 

rule 2354

Int[(Px_)*((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[(Px /. 
 x -> x^k/e)*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, n, p}, x] && PolyQ[Px, x] 
&& FractionQ[m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1417\) vs. \(2(337)=674\).

Time = 15.57 (sec) , antiderivative size = 1418, normalized size of antiderivative = 3.42

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1418\)
default	\(\text {Expression too large to display}\)	\(4847\)

Input:

int(1/(e*x)^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(e*x*(d*x+c)*(-b*x^2+a))^(1/2)/(e*x)^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)* 
(-2/5/e^4/a/c*(-b*d*e*x^4-b*c*e*x^3+a*d*e*x^2+a*c*e*x)^(1/2)/x^3+8/15/a/c^ 
2/e^4*d*(-b*d*e*x^4-b*c*e*x^3+a*d*e*x^2+a*c*e*x)^(1/2)/x^2-2/15*(-b*d*e*x^ 
3-b*c*e*x^2+a*d*e*x+a*c*e)/e^4/a^2/c^3*(8*a*d^2+9*b*c^2)/(x*(-b*d*e*x^3-b* 
c*e*x^2+a*d*e*x+a*c*e))^(1/2)-4/15/a*d^2/c^2/e^3*(a*b)^(1/2)*(-(c/d-1/b*(a 
*b)^(1/2))*x*b/(a*b)^(1/2)/(x+c/d))^(1/2)*(x+c/d)^2*(-c/d*(x-1/b*(a*b)^(1/ 
2))*b/(a*b)^(1/2)/(x+c/d))^(1/2)*(c/d*(x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2)/(x 
+c/d))^(1/2)/(c/d-1/b*(a*b)^(1/2))/(-b*d*e*x*(x+c/d)*(x-1/b*(a*b)^(1/2))*( 
x+1/b*(a*b)^(1/2)))^(1/2)*EllipticF((-(c/d-1/b*(a*b)^(1/2))*x*b/(a*b)^(1/2 
)/(x+c/d))^(1/2),(-(-c/d-1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2))+2 
/15/c^3*(8*a*d^2+9*b*c^2)/a^2/e^3*(a*b)^(1/2)*(-(c/d-1/b*(a*b)^(1/2))*x*b/ 
(a*b)^(1/2)/(x+c/d))^(1/2)*(x+c/d)^2*(-c/d*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/ 
2)/(x+c/d))^(1/2)*(c/d*(x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2)/(x+c/d))^(1/2)/(c 
/d-1/b*(a*b)^(1/2))*d/(-b*d*e*x*(x+c/d)*(x-1/b*(a*b)^(1/2))*(x+1/b*(a*b)^( 
1/2)))^(1/2)*(-c/d*EllipticF((-(c/d-1/b*(a*b)^(1/2))*x*b/(a*b)^(1/2)/(x+c/ 
d))^(1/2),(-(-c/d-1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2))+c/d*Elli 
pticPi((-(c/d-1/b*(a*b)^(1/2))*x*b/(a*b)^(1/2)/(x+c/d))^(1/2),-1/b*(a*b)^( 
1/2)/(c/d-1/b*(a*b)^(1/2)),(-(-c/d-1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)) 
)^(1/2)))-2/15*b*d*(8*a*d^2+9*b*c^2)/a^2/c^3/e^3*(x*(x-1/b*(a*b)^(1/2))*(x 
+1/b*(a*b)^(1/2))+1/b*(a*b)^(1/2)*(-(c/d-1/b*(a*b)^(1/2))*x*b/(a*b)^(1/...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (3 \, b c^{2} d + 8 \, a d^{3}\right )} \sqrt {a c e} x^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, b c^{2} + a d^{2}\right )}}{3 \, a c^{2}}, \frac {8 \, {\left (9 \, b c^{2} d - a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right ) + 3 \, {\left (9 \, b c^{3} + 8 \, a c d^{2}\right )} \sqrt {a c e} x^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (3 \, b c^{2} + a d^{2}\right )}}{3 \, a c^{2}}, \frac {8 \, {\left (9 \, b c^{2} d - a d^{3}\right )}}{27 \, a c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, b c^{2} + a d^{2}\right )}}{3 \, a c^{2}}, \frac {8 \, {\left (9 \, b c^{2} d - a d^{3}\right )}}{27 \, a c^{3}}, \frac {d x + 3 \, c}{3 \, c x}\right )\right ) + 3 \, {\left (4 \, a c^{2} d x - 3 \, a c^{3}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c} \sqrt {e x}\right )}}{45 \, a^{2} c^{4} e^{4} x^{3}} \] Input:

integrate(1/(e*x)^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="frica 
s")
 

Output:

2/45*((3*b*c^2*d + 8*a*d^3)*sqrt(a*c*e)*x^3*weierstrassPInverse(4/3*(3*b*c 
^2 + a*d^2)/(a*c^2), 8/27*(9*b*c^2*d - a*d^3)/(a*c^3), 1/3*(d*x + 3*c)/(c* 
x)) + 3*(9*b*c^3 + 8*a*c*d^2)*sqrt(a*c*e)*x^3*weierstrassZeta(4/3*(3*b*c^2 
 + a*d^2)/(a*c^2), 8/27*(9*b*c^2*d - a*d^3)/(a*c^3), weierstrassPInverse(4 
/3*(3*b*c^2 + a*d^2)/(a*c^2), 8/27*(9*b*c^2*d - a*d^3)/(a*c^3), 1/3*(d*x + 
 3*c)/(c*x))) + 3*(4*a*c^2*d*x - 3*a*c^3)*sqrt(-b*x^2 + a)*sqrt(d*x + c)*s 
qrt(e*x))/(a^2*c^4*e^4*x^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(7/2)/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x + c} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="maxim 
a")
 

Output:

integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x + c)*(e*x)^(7/2)), x)
 

Giac [F]

\[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {1}{\sqrt {-b x^{2} + a} \sqrt {d x + c} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x + c)*(e*x)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{7/2}\,\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/((e*x)^(7/2)*(a - b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
 

Output:

int(1/((e*x)^(7/2)*(a - b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1}{(e x)^{7/2} \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {1}{\left (e x \right )^{\frac {7}{2}} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}d x \] Input:

int(1/(e*x)^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
 

Output:

int(1/(e*x)^(7/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)