\(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{3/2}} \, dx\) [175]

Optimal result

Integrand size = 25, antiderivative size = 432 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{3/2}} \, dx=-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}+\frac {(b B+6 A c) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt {2} c^{3/2} \sqrt {a+b x+c x^2}}-\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b^2 B-4 a B c+\sqrt {b^2-4 a c} (b B+6 A c)\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt {2} c^{3/2} \sqrt {a+b x+c x^2}} \] Output:

-2/3*(-B*x+3*A)*(c*x^2+b*x+a)^(1/2)/x^(1/2)+1/6*(6*A*c+B*b)*(-b+(-4*a*c+b^ 
2)^(1/2))^(1/2)*(b+(-4*a*c+b^2)^(1/2))*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1 
/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1/2)*x^(1/ 
2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^( 
1/2)))^(1/2))*2^(1/2)/c^(3/2)/(c*x^2+b*x+a)^(1/2)-1/6*(-b+(-4*a*c+b^2)^(1/ 
2))^(1/2)*(B*b^2-4*B*a*c+(-4*a*c+b^2)^(1/2)*(6*A*c+B*b))*(1+2*c*x/(b-(-4*a 
*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(2^( 
1/2)*c^(1/2)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2),((b-(-4*a*c+b^2)^(1/2)) 
/(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)/c^(3/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.50 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{3/2}} \, dx=\frac {\frac {4 (b B+6 A c) (a+x (b+c x))}{c \sqrt {x}}+\frac {4 (-3 A+B x) (a+x (b+c x))}{\sqrt {x}}-\frac {i (b B+6 A c) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (-b^2 B+4 a B c+b B \sqrt {b^2-4 a c}+6 A c \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}}{6 \sqrt {a+x (b+c x)}} \] Input:

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

Output:

((4*(b*B + 6*A*c)*(a + x*(b + c*x)))/(c*Sqrt[x]) + (4*(-3*A + B*x)*(a + x* 
(b + c*x)))/Sqrt[x] - (I*(b*B + 6*A*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + ( 
4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x) 
/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqr 
t[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] 
)])/(c*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]) + (I*(-(b^2*B) + 4*a*B*c + b*B*Sqr 
t[b^2 - 4*a*c] + 6*A*c*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 
4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a 
*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt 
[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(c*Sqrt[a/(b + Sqr 
t[b^2 - 4*a*c])]))/(6*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1230, 27, 1240, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {2}{3} \int -\frac {3 A b+2 a B+(b B+6 A c) x}{2 \sqrt {x} \sqrt {c x^2+b x+a}}dx-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 A b+2 a B+(b B+6 A c) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2}{3} \int \frac {3 A b+2 a B+(b B+6 A c) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {2}{3} \left (\frac {\left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a} B+3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {a} (6 A c+b B) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {\left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a} B+3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {(6 A c+b B) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {2}{3} \left (\frac {\left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a+b x+c x^2}}-\frac {(6 A c+b B) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2}{3} \left (\frac {\left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a+b x+c x^2}}-\frac {(6 A c+b B) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )-\frac {2 (3 A-B x) \sqrt {a+b x+c x^2}}{3 \sqrt {x}}\)

Input:

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

Output:

(-2*(3*A - B*x)*Sqrt[a + b*x + c*x^2])/(3*Sqrt[x]) + (2*(-(((b*B + 6*A*c)* 
(-((Sqrt[x]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt 
[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE 
[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/ 
4)*Sqrt[a + b*x + c*x^2])))/Sqrt[c]) + ((b + 2*Sqrt[a]*Sqrt[c])*(Sqrt[a]*B 
 + 3*A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sq 
rt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a] 
*Sqrt[c]))/4])/(2*a^(1/4)*c^(3/4)*Sqrt[a + b*x + c*x^2])))/3
 

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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (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 + b*x + 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 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, 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])
 

rule 1240

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), 
x_Symbol] :> Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, 
 Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(768\) vs. \(2(356)=712\).

Time = 1.70 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.78

method	result	size
elliptic	\(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (c \,x^{2}+b x +a \right ) A}{\sqrt {x \left (c \,x^{2}+b x +a \right )}}+\frac {2 B \sqrt {c \,x^{3}+b \,x^{2}+a x}}{3}+\frac {\left (A b +\frac {2 B a}{3}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {\left (2 A c +\frac {B b}{3}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}\right )}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\)	\(769\)
risch	\(\text {Expression too large to display}\)	\(1003\)
default	\(\text {Expression too large to display}\)	\(1652\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.44 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{3/2}} \, dx=-\frac {2 \, {\left ({\left (B b^{2} - 3 \, {\left (2 \, B a + A b\right )} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (B b c + 6 \, A c^{2}\right )} \sqrt {c} x {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (B c^{2} x - 3 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{9 \, c^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{3/2}} \, dx=\frac {-6 \sqrt {c \,x^{2}+b x +a}\, a +2 \sqrt {c \,x^{2}+b x +a}\, b x +5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a b +6 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{2}+b x +a}d x \right ) a c +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{2}+b x +a}d x \right ) b^{2}}{3 \sqrt {x}} \] Input:

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

Output:

( - 6*sqrt(a + b*x + c*x**2)*a + 2*sqrt(a + b*x + c*x**2)*b*x + 5*sqrt(x)* 
int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)*a*b + 6*sq 
rt(x)*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a + b*x + c*x**2),x)*a*c + sqr 
t(x)*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a + b*x + c*x**2),x)*b**2)/(3*s 
qrt(x))