\(\int \frac {A+B x}{(d+e x)^{3/2} \sqrt {a-c x^2}} \, dx\) [276]

Optimal result

Integrand size = 27, antiderivative size = 339 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {a-c x^2}} \, dx=-\frac {2 (B d-A e) \sqrt {a-c x^2}}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {c} (B d-A e) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{e \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {2 \sqrt {a} B \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {c} e \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

-2*(-A*e+B*d)*(-c*x^2+a)^(1/2)/(-a*e^2+c*d^2)/(e*x+d)^(1/2)+2*a^(1/2)*c^(1 
/2)*(-A*e+B*d)*(e*x+d)^(1/2)*(1-c*x^2/a)^(1/2)*EllipticE(1/2*(1-c^(1/2)*x/ 
a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/e/ 
(-a*e^2+c*d^2)/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1 
/2)-2*a^(1/2)*B*(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c*x^2/a)^ 
(1/2)*EllipticF(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e 
/(c^(1/2)*d+a^(1/2)*e))^(1/2))/c^(1/2)/e/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.01 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {a-c x^2}} \, dx=\frac {2 i \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x) \left (\sqrt {c} (B d-A e) E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+\left (\sqrt {a} B+A \sqrt {c}\right ) e \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{e^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \sqrt {a-c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {688, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {B \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (B d-A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {B \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (B d-A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (B d-A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a} B \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (B d-A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a} B \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\)

Input:

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

Output:

(-2*(B*d - A*e)*Sqrt[a - c*x^2])/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((2*Sqr 
t[a]*Sqrt[c]*(B*d - A*e)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSi 
n[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)] 
)/(e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) - 
(2*Sqrt[a]*B*(c*d^2 - a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a] 
*e)]*Sqrt[1 - (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sq 
rt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a 
- c*x^2]))/(c*d^2 - a*e^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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

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 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

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

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

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

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(653\) vs. \(2(281)=562\).

Time = 5.99 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.93

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (-c e \,x^{2}+a e \right ) \left (A e -B d \right )}{e \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {B}{e}-\frac {c d \left (A e -B d \right )}{e \left (a \,e^{2}-c \,d^{2}\right )}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}-\frac {2 \left (A e -B d \right ) c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(654\)
default	\(\text {Expression too large to display}\)	\(1236\)

Input:

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

Output:

((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2*(-c*e*x^2+a* 
e)/e/(a*e^2-c*d^2)*(A*e-B*d)/((x+d/e)*(-c*e*x^2+a*e))^(1/2)+2*(B/e-c*d/e*( 
A*e-B*d)/(a*e^2-c*d^2))*(d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2 
)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c) 
^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*E 
llipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d 
/e-1/c*(a*c)^(1/2)))^(1/2))-2*(A*e-B*d)*c/(a*e^2-c*d^2)*(d/e-1/c*(a*c)^(1/ 
2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*( 
a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c* 
e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))*EllipticE(((x+d/e)/ 
(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2) 
))^(1/2))+1/c*(a*c)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2), 
((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {a-c x^2}} \, dx=-\frac {2 \, {\left ({\left (B c d^{3} + 2 \, A c d^{2} e - 3 \, B a d e^{2} + {\left (B c d^{2} e + 2 \, A c d e^{2} - 3 \, B a e^{3}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (B c d^{2} e - A c d e^{2} + {\left (B c d e^{2} - A c e^{3}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (B c d e^{2} - A c e^{3}\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{3 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4} + {\left (c^{2} d^{2} e^{3} - a c e^{5}\right )} x\right )}} \] Input:

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

Output:

-2/3*((B*c*d^3 + 2*A*c*d^2*e - 3*B*a*d*e^2 + (B*c*d^2*e + 2*A*c*d*e^2 - 3* 
B*a*e^3)*x)*sqrt(-c*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), 
-8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 3*(B*c*d^2*e - A*c 
*d*e^2 + (B*c*d*e^2 - A*c*e^3)*x)*sqrt(-c*e)*weierstrassZeta(4/3*(c*d^2 + 
3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), weierstrassPInverse(4 
/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e* 
x + d)/e)) + 3*(B*c*d*e^2 - A*c*e^3)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c^2* 
d^3*e^2 - a*c*d*e^4 + (c^2*d^2*e^3 - a*c*e^5)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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