\(\int \frac {A+B x}{\sqrt {d+e x} (a-c x^2)^{3/2}} \, dx\) [281]

Optimal result

Integrand size = 27, antiderivative size = 351 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{a \left (c d^2-a e^2\right ) \sqrt {a-c x^2}}+\frac {(A c d-a B e) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {c} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {A \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {c} \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.63 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.46 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{3/2}} \, dx=\frac {-e^2 (A c d-a B e) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )-c e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} (d+e x) (-A c d x+a (-B d+A e+B e x))+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) (A c d-a B e) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} \left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {c} e \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{a c e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}} \] Input:

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

Output:

(-(e^2*(A*c*d - a*B*e)*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) - c*e* 
Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(d + e*x)*(-(A*c*d*x) + a*(-(B*d) + A*e + B 
*e*x)) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(A*c*d - a*B*e)*Sqrt[(e*(Sqrt[a 
]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]* 
(d + e*x)^(3/2)*EllipticE[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)] + I*Sqrt[a]*(Sqr 
t[a]*B + A*Sqrt[c])*Sqrt[c]*e*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[(e*(Sqrt[a]/Sqr 
t[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + 
e*x)^(3/2)*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)])/(a*c*e*Sqrt[-d + (Sq 
rt[a]*e)/Sqrt[c]]*(c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[a - c*x^2])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {686, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(a*(c*d^2 - a*e^2)*Sqr 
t[a - c*x^2]) + (e*((2*Sqrt[a]*(A*c*d - a*B*e)*Sqrt[d + e*x]*Sqrt[1 - (c*x 
^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sq 
rt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + S 
qrt[a]*e)]*Sqrt[a - c*x^2]) - (2*Sqrt[a]*A*(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[Sq 
rt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(S 
qrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2])))/(2*a*(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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -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. \(719\) vs. \(2(293)=586\).

Time = 6.30 (sec) , antiderivative size = 720, normalized size of antiderivative = 2.05

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (-c e x -c d \right ) \left (-\frac {\left (A c d -B a e \right ) x}{2 \left (a \,e^{2}-c \,d^{2}\right ) a c}+\frac {A e -B d}{2 c \left (a \,e^{2}-c \,d^{2}\right )}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (\frac {A}{a}-\frac {e \left (A e -B d \right )}{2 \left (a \,e^{2}-c \,d^{2}\right )}+\frac {d \left (A c d -B a e \right )}{\left (a \,e^{2}-c \,d^{2}\right ) a}\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 {e \left (A c d -B a e \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}}}\, \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 ) a \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(720\)
default	\(\text {Expression too large to display}\)	\(1257\)

Input:

int((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^(3/2),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-c*d) 
*(-1/2*(A*c*d-B*a*e)/(a*e^2-c*d^2)/a/c*x+1/2*(A*e-B*d)/c/(a*e^2-c*d^2))/(( 
x^2-a/c)*(-c*e*x-c*d))^(1/2)+2*(A/a-1/2*e*(A*e-B*d)/(a*e^2-c*d^2)+d*(A*c*d 
-B*a*e)/(a*e^2-c*d^2)/a)*(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)* 
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))+e*(A*c*d-B*a*e)/(a*e^2-c*d^2)/a*(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 = 0.11 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^{3/2}} \, dx=-\frac {{\left (A a c d^{2} + 2 \, B a^{2} d e - 3 \, A a^{2} e^{2} - {\left (A c^{2} d^{2} + 2 \, B a c d e - 3 \, A a c e^{2}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (A a c d e - B a^{2} e^{2} - {\left (A c^{2} d e - B a c e^{2}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (B a c d e - A a c e^{2} + {\left (A c^{2} d e - B a c e^{2}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a^{2} c^{2} d^{2} e - a^{3} c e^{3} - {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} x^{2}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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