\(\int \frac {A+B x}{\sqrt {d+e x} (a+b x+c x^2)^{3/2}} \, dx\) [1095]

Optimal result

Integrand size = 29, antiderivative size = 508 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x} \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} (b B d-2 A c d+A b e-2 a B e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} (b B-2 A c) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2*(e*x+d)^(1/2)*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d)+c*(A*b*e-2*A*c*d 
-2*B*a*e+B*b*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(1/2)-2^ 
(1/2)*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a* 
c+b^2))^(1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2) 
,(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/(-4*a*c 
+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e 
))^(1/2)/(c*x^2+b*x+a)^(1/2)+2*2^(1/2)*(-2*A*c+B*b)*(c*(e*x+d)/(2*c*d-(b+( 
-4*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*Ellipti 
cF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/ 
2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c/(-4*a*c+b^2)^(1/2)/(e*x+d) 
^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.85 (sec) , antiderivative size = 893, normalized size of antiderivative = 1.76 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x} \left (A b c d-2 a B c d-A b^2 e+a b B e+2 a A c e-b B c d x+2 A c^2 d x-A b c e x+2 a B c e x\right ) \left (a+b x+c x^2\right )}{\left (-b^2+4 a c\right ) \left (c d^2-b d e+a e^2\right ) (a+x (b+c x))^{3/2}}-\frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \left (-\left ((b B d-2 A c d+A b e-2 a B e) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )\right )+\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left ((b B d-2 A c d+A b e-2 a B e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\left (-2 A c d \sqrt {\left (b^2-4 a c\right ) e^2}+b^2 e (B d-A e)+b \sqrt {\left (b^2-4 a c\right ) e^2} (B d+A e)-2 a e \left (2 B c d-2 A c e+B \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )}{\left (-b^2+4 a c\right ) e \left (c d^2-b d e+a e^2\right ) (a+x (b+c x))^{3/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \] Input:

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

Output:

(2*Sqrt[d + e*x]*(A*b*c*d - 2*a*B*c*d - A*b^2*e + a*b*B*e + 2*a*A*c*e - b* 
B*c*d*x + 2*A*c^2*d*x - A*b*c*e*x + 2*a*B*c*e*x)*(a + b*x + c*x^2))/((-b^2 
 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + x*(b + c*x))^(3/2)) - (2*(d + e*x)^ 
(3/2)*(a + b*x + c*x^2)^(3/2)*(-((b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(c*(- 
1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x) 
)) + ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[( 
b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((- 
2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((b*B*d - 2*A*c*d + A*b 
*e - 2*a*B*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*EllipticE[I*ArcSinh[ 
(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e 
^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - 
 b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (-2*A*c*d*Sqrt[(b^2 - 4*a*c)*e^2] + b^ 
2*e*(B*d - A*e) + b*Sqrt[(b^2 - 4*a*c)*e^2]*(B*d + A*e) - 2*a*e*(2*B*c*d - 
 2*A*c*e + B*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[( 
c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + 
 e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^ 
2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b* 
e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/((-b^2 + 4*a*c)*e*(c*d^2 - 
b*d*e + a*e^2)*(a + x*(b + c*x))^(3/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e 
*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1235, 27, 1269, 1172, 321, 327}

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.

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(2*Sqrt[d + e*x]*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B 
*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2) 
*Sqrt[a + b*x + c*x^2]) - (e*((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(b*B*d - 2*A*c*d 
+ A*b*e - 2*a*B*e)*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c 
))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c 
]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)] 
)/(e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x 
+ c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(b*B - 2*A*c)*(c*d^2 - b*d*e + a* 
e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a 
+ b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a* 
c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - 
 (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])))/ 
((b^2 - 4*a*c)*(c*d^2 - b*d*e + 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 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1276\) vs. \(2(460)=920\).

Time = 7.89 (sec) , antiderivative size = 1277, normalized size of antiderivative = 2.51

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1277\)
default	\(\text {Expression too large to display}\)	\(5059\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 956 vs. \(2 (468) = 936\).

Time = 0.17 (sec) , antiderivative size = 956, normalized size of antiderivative = 1.88 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/3*(((B*a*b*c - 2*A*a*c^2)*d^2 - 2*(B*a*b^2 - (2*B*a^2 + A*a*b)*c)*d*e + 
(B*a^2*b + A*a*b^2 - 6*A*a^2*c)*e^2 + ((B*b*c^2 - 2*A*c^3)*d^2 - 2*(B*b^2* 
c - (2*B*a + A*b)*c^2)*d*e - (6*A*a*c^2 - (B*a*b + A*b^2)*c)*e^2)*x^2 + (( 
B*b^2*c - 2*A*b*c^2)*d^2 - 2*(B*b^3 - (2*B*a*b + A*b^2)*c)*d*e + (B*a*b^2 
+ A*b^3 - 6*A*a*b*c)*e^2)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - 
b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 
 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e* 
x + c*d + b*e)/(c*e)) - 3*((2*B*a^2 - A*a*b)*c*e^2 - (B*a*b*c - 2*A*a*c^2) 
*d*e + ((2*B*a - A*b)*c^2*e^2 - (B*b*c^2 - 2*A*c^3)*d*e)*x^2 + ((2*B*a*b - 
 A*b^2)*c*e^2 - (B*b^2*c - 2*A*b*c^2)*d*e)*x)*sqrt(c*e)*weierstrassZeta(4/ 
3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3* 
b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3) 
, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2 
), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 
 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*((2*B*a - 
A*b)*c^2*d*e - (2*A*a*c^2 + (B*a*b - A*b^2)*c)*e^2 - ((2*B*a - A*b)*c^2*e^ 
2 - (B*b*c^2 - 2*A*c^3)*d*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/((a*b 
^2*c^2 - 4*a^2*c^3)*d^2*e - (a*b^3*c - 4*a^2*b*c^2)*d*e^2 + (a^2*b^2*c - 4 
*a^3*c^2)*e^3 + ((b^2*c^3 - 4*a*c^4)*d^2*e - (b^3*c^2 - 4*a*b*c^3)*d*e^2 + 
 (a*b^2*c^2 - 4*a^2*c^3)*e^3)*x^2 + ((b^3*c^2 - 4*a*b*c^3)*d^2*e - (b^4...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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