\(\int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{3/2}} \, dx\) [264]

Optimal result

Integrand size = 27, antiderivative size = 346 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 (4 B d-3 A e+B e x) \sqrt {a-c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {4 \sqrt {a} \sqrt {c} (4 B d-3 A e) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{3 e^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {4 \sqrt {a} \left (4 B c d^2-3 A c d e-a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{3 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

2/3*(B*e*x-3*A*e+4*B*d)*(-c*x^2+a)^(1/2)/e^2/(e*x+d)^(1/2)-4/3*a^(1/2)*c^( 
1/2)*(-3*A*e+4*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^3/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)+4/3* 
a^(1/2)*(-3*A*c*d*e-B*a*e^2+4*B*c*d^2)*(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^3/(e*x+d)^ 
(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.64 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {a-c x^2} \left (3 B d-3 A e+B (d+e x)-\frac {2 \left (e^2 (4 B d-3 A e) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) (4 B d-3 A e) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )-i \sqrt {a} e \left (-4 B \sqrt {c} d+\sqrt {a} B e+3 A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )}\right )}{3 e^2 \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[a - c*x^2]*(3*B*d - 3*A*e + B*(d + e*x) - (2*(e^2*(4*B*d - 3*A*e)* 
Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a 
]*e)*(4*B*d - 3*A*e)*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sq 
rt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sq 
rt[-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]*e*(-4*B*Sqrt[c]*d + Sqrt[a]*B*e + 3*A*Sqrt 
[c]*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)*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[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2))))/(3*e^2*Sqrt[d + 
 e*x])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {681, 25, 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.

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 \left (\frac {\left (a B e^2-c d (4 B d-3 A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {c (4 B d-3 A e) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 \left (\frac {\left (a B e^2-c d (4 B d-3 A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {c \sqrt {1-\frac {c x^2}{a}} (4 B d-3 A e) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 \left (\frac {\left (a B e^2-c d (4 B d-3 A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 B d-3 A e) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \left (\frac {\left (a B e^2-c d (4 B d-3 A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 B d-3 A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {2 \left (\frac {\sqrt {1-\frac {c x^2}{a}} \left (a B e^2-c d (4 B d-3 A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 B d-3 A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (a B e^2-c d (4 B d-3 A e)\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 B d-3 A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (a B e^2-c d (4 B d-3 A e)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 B d-3 A e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}\)

Input:

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

Output:

(2*(4*B*d - 3*A*e + B*e*x)*Sqrt[a - c*x^2])/(3*e^2*Sqrt[d + e*x]) + (2*((- 
2*Sqrt[a]*Sqrt[c]*(4*B*d - 3*A*e)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*Ellipt 
icE[ArcSin[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]*(a*B*e^2 - c*d*(4*B*d - 3*A*e))*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]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]* 
e*Sqrt[d + e*x]*Sqrt[a - c*x^2])))/(3*e^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (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. \(680\) vs. \(2(282)=564\).

Time = 6.83 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.97

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (-c e \,x^{2}+a e \right ) \left (A e -B d \right )}{e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+a e \right )}}+\frac {2 B \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 e^{2}}+\frac {2 \left (\frac {A c d e +B a \,e^{2}-B c \,d^{2}}{e^{3}}-\frac {\left (A e -B d \right ) c d}{e^{3}}-\frac {B a}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (-\frac {2 \left (A e -B d \right ) c}{e^{2}}+\frac {2 d B c}{3 e^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(681\)
risch	\(\text {Expression too large to display}\)	\(1012\)
default	\(\text {Expression too large to display}\)	\(1392\)

Input:

int((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(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^2+a* 
e)*(A*e-B*d)/e^3/((x+d/e)*(-c*e*x^2+a*e))^(1/2)+2/3*B/e^2*(-c*e*x^3-c*d*x^ 
2+a*e*x+a*d)^(1/2)+2*((A*c*d*e+B*a*e^2-B*c*d^2)/e^3-(A*e-B*d)*c/e^3*d-1/3* 
B*a/e)*(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))+2*(-2*(A*e-B*d)*c/e^2+2/3*d/e^2*B*c)*(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 = 0.10 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (2 \, {\left (4 \, B c d^{3} - 3 \, A c d^{2} e - 3 \, B a d e^{2} + {\left (4 \, B c d^{2} e - 3 \, A c d e^{2} - 3 \, B a e^{3}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (4 \, B c d^{2} e - 3 \, A c d e^{2} + {\left (4 \, B c d e^{2} - 3 \, A c e^{3}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (B c e^{3} x + 4 \, B c d e^{2} - 3 \, A c e^{3}\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (c e^{5} x + c d e^{4}\right )}} \] Input:

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

Output:

2/9*(2*(4*B*c*d^3 - 3*A*c*d^2*e - 3*B*a*d*e^2 + (4*B*c*d^2*e - 3*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) + 6*(4*B*c*d^2* 
e - 3*A*c*d*e^2 + (4*B*c*d*e^2 - 3*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), weierstr 
assPInverse(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*e^3*x + 4*B*c*d*e^2 - 3*A*c*e^3)*sqrt(-c* 
x^2 + a)*sqrt(e*x + d))/(c*e^5*x + c*d*e^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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