\(\int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx\) [173]

Optimal result

Integrand size = 25, antiderivative size = 548 \[ \int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx=-\frac {2 \left (4 b^2 B-7 A b c-10 a B c\right ) \sqrt {x} \sqrt {a+b x+c x^2}}{105 c^2}+\frac {2 x^{3/2} (b B+7 A c+5 B c x) \sqrt {a+b x+c x^2}}{35 c}+\frac {\left (8 b^3 B-14 A b^2 c-29 a b B c+42 a A c^2\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{105 \sqrt {2} c^{7/2} \sqrt {a+b x+c x^2}}-\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \left (8 b^3 B-14 A b^2 c-29 a b B c+42 a A c^2-\frac {2 a c \left (4 b^2 B-7 A b c-10 a B c\right )}{b+\sqrt {b^2-4 a c}}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{105 \sqrt {2} c^{7/2} \sqrt {a+b x+c x^2}} \] Output:

-2/105*(-7*A*b*c-10*B*a*c+4*B*b^2)*x^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2+2/35*x^ 
(3/2)*(5*B*c*x+7*A*c+B*b)*(c*x^2+b*x+a)^(1/2)/c+1/210*(42*A*a*c^2-14*A*b^2 
*c-29*B*a*b*c+8*B*b^3)*(-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^(7/2)/(c*x^2+ 
b*x+a)^(1/2)-1/210*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a*c+b^2)^(1/2))*(8 
*B*b^3-14*A*b^2*c-29*B*a*b*c+42*A*a*c^2-2*a*c*(-7*A*b*c-10*B*a*c+4*B*b^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)*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^(7/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.30 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.16 \[ \int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {x} \sqrt {a+x (b+c x)} \left (-4 b^2 B+b c (7 A+3 B x)+c (10 a B+3 c x (7 A+5 B x))\right )}{105 c^2}-\frac {-4 \left (8 b^3 B-14 A b^2 c-29 a b B c+42 a A c^2\right ) \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))+i \left (8 b^3 B-14 A b^2 c-29 a b B c+42 a A c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (8 b^4 B+2 a c^2 \left (10 a B-21 A \sqrt {b^2-4 a c}\right )-2 b^3 \left (7 A c+4 B \sqrt {b^2-4 a c}\right )+a b c \left (56 A c+29 B \sqrt {b^2-4 a c}\right )+b^2 \left (-37 a B c+14 A c \sqrt {b^2-4 a c}\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{210 c^3 \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} \sqrt {x} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(2*Sqrt[x]*Sqrt[a + x*(b + c*x)]*(-4*b^2*B + b*c*(7*A + 3*B*x) + c*(10*a*B 
 + 3*c*x*(7*A + 5*B*x))))/(105*c^2) - (-4*(8*b^3*B - 14*A*b^2*c - 29*a*b*B 
*c + 42*a*A*c^2)*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(a + x*(b + c*x)) + I*(8* 
b^3*B - 14*A*b^2*c - 29*a*b*B*c + 42*a*A*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqr 
t[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*Sqr 
t[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[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])] + I*(8*b^4*B + 2*a*c^2*(10*a*B - 21*A*Sqrt[b^2 - 4*a*c 
]) - 2*b^3*(7*A*c + 4*B*Sqrt[b^2 - 4*a*c]) + a*b*c*(56*A*c + 29*B*Sqrt[b^2 
 - 4*a*c]) + b^2*(-37*a*B*c + 14*A*c*Sqrt[b^2 - 4*a*c]))*Sqrt[1 + (2*a)/(( 
b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*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])])/(210*c^3*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x]*Sqrt[a + x*(b + c* 
x)])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.80, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1236, 27, 1231, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 \int -\frac {(a B+(4 b B-7 A c) x) \sqrt {c x^2+b x+a}}{2 \sqrt {x}}dx}{7 c}+\frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\int \frac {(a B+(4 b B-7 A c) x) \sqrt {c x^2+b x+a}}{\sqrt {x}}dx}{7 c}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {2 \int \frac {a \left (4 B b^2-7 A c b-10 a B c\right )-\left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) x}{2 \sqrt {x} \sqrt {c x^2+b x+a}}dx}{15 c}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {\int \frac {a \left (4 B b^2-7 A c b-10 a B c\right )-\left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{15 c}}{7 c}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {2 \int \frac {a \left (4 B b^2-7 A c b-10 a B c\right )-\left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{15 c}}{7 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {2 \left (\frac {\sqrt {a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {a} \left (-\sqrt {a} \sqrt {c} \left (-10 a B c-7 A b c+4 b^2 B\right )-2 \left (b^2-3 a c\right ) (4 b B-7 A c)+5 a b B c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {2 \left (\frac {\left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {a} \left (-\sqrt {a} \sqrt {c} \left (-10 a B c-7 A b c+4 b^2 B\right )-2 \left (b^2-3 a c\right ) (4 b B-7 A c)+5 a b B c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c}}{7 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {2 \left (\frac {\left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (-\sqrt {a} \sqrt {c} \left (-10 a B c-7 A b c+4 b^2 B\right )-2 \left (b^2-3 a c\right ) (4 b B-7 A c)+5 a b B c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x+c x^2}}\right )}{15 c}}{7 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {\frac {2 \sqrt {x} \sqrt {a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{15 c}-\frac {2 \left (\frac {\left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (-\sqrt {a} \sqrt {c} \left (-10 a B c-7 A b c+4 b^2 B\right )-2 \left (b^2-3 a c\right ) (4 b B-7 A c)+5 a b B c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x+c x^2}}\right )}{15 c}}{7 c}\)

Input:

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

Output:

(2*B*Sqrt[x]*(a + b*x + c*x^2)^(3/2))/(7*c) - ((2*Sqrt[x]*(4*b^2*B - 7*A*b 
*c + 5*a*B*c + 3*c*(4*b*B - 7*A*c)*x)*Sqrt[a + b*x + c*x^2])/(15*c) - (2*( 
((5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b*B - 7*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] - (a^(1/4)*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b*B - 7*A*c) - Sqrt[a] 
*Sqrt[c]*(4*b^2*B - 7*A*b*c - 10*a*B*c))*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b 
*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/ 
a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x + c*x^2])) 
)/(15*c))/(7*c)
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1236

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

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 [A] (verified)

Time = 1.92 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.59

method	result	size
elliptic	\(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 B \,x^{2} \sqrt {c \,x^{3}+b \,x^{2}+a x}}{7}+\frac {2 \left (A c +\frac {B b}{7}\right ) x \sqrt {c \,x^{3}+b \,x^{2}+a x}}{5 c}+\frac {2 \left (A b +\frac {2 B a}{7}-\frac {4 b \left (A c +\frac {B b}{7}\right )}{5 c}\right ) \sqrt {c \,x^{3}+b \,x^{2}+a x}}{3 c}-\frac {a \left (A b +\frac {2 B a}{7}-\frac {4 b \left (A c +\frac {B b}{7}\right )}{5 c}\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 )}{3 c^{2} \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {\left (A a -\frac {3 a \left (A c +\frac {B b}{7}\right )}{5 c}-\frac {2 b \left (A b +\frac {2 B a}{7}-\frac {4 b \left (A c +\frac {B b}{7}\right )}{5 c}\right )}{3 c}\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}}\)	\(871\)
risch	\(\text {Expression too large to display}\)	\(1321\)
default	\(\text {Expression too large to display}\)	\(2884\)

Input:

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

Output:

(x*(c*x^2+b*x+a))^(1/2)/x^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/7*B*x^2*(c*x^3+b*x^ 
2+a*x)^(1/2)+2/5*(A*c+1/7*B*b)/c*x*(c*x^3+b*x^2+a*x)^(1/2)+2/3*(A*b+2/7*B* 
a-4/5*b/c*(A*c+1/7*B*b))/c*(c*x^3+b*x^2+a*x)^(1/2)-1/3*a/c^2*(A*b+2/7*B*a- 
4/5*b/c*(A*c+1/7*B*b))*(b+(-4*a*c+b^2)^(1/2))*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))+(A*a-3/5*a/c*(A*c+1/7*B*b)-2/3*b/c*(A*b+2/7*B 
*a-4/5*b/c*(A*c+1/7*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...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.48 \[ \int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx=-\frac {2 \, {\left ({\left (8 \, B b^{4} + 3 \, {\left (10 \, B a^{2} + 21 \, A a b\right )} c^{2} - {\left (41 \, B a b^{2} + 14 \, A b^{3}\right )} c\right )} \sqrt {c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (8 \, B b^{3} c + 42 \, A a c^{3} - {\left (29 \, B a b + 14 \, A b^{2}\right )} c^{2}\right )} \sqrt {c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (15 \, B c^{4} x^{2} - 4 \, B b^{2} c^{2} + {\left (10 \, B a + 7 \, A b\right )} c^{3} + 3 \, {\left (B b c^{3} + 7 \, A c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{315 \, c^{4}} \] Input:

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

Output:

-2/315*((8*B*b^4 + 3*(10*B*a^2 + 21*A*a*b)*c^2 - (41*B*a*b^2 + 14*A*b^3)*c 
)*sqrt(c)*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*(8*B*b^3*c + 42*A*a*c^3 - (29*B*a*b + 14*A* 
b^2)*c^2)*sqrt(c)*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*(15*B*c^4*x^2 - 4*B*b^2*c^2 + (10*B*a + 
7*A*b)*c^3 + 3*(B*b*c^3 + 7*A*c^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(x))/c^4
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {28 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a^{2} c -6 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}+28 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a b c x +4 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b^{3} x +20 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,x^{2}-42 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a^{2} c^{2}+43 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a \,b^{2} c -8 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) b^{4}-14 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{3} c +3 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{2} b^{2}}{70 b c} \] Input:

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

Output:

(28*sqrt(x)*sqrt(a + b*x + c*x**2)*a**2*c - 6*sqrt(x)*sqrt(a + b*x + c*x** 
2)*a*b**2 + 28*sqrt(x)*sqrt(a + b*x + c*x**2)*a*b*c*x + 4*sqrt(x)*sqrt(a + 
 b*x + c*x**2)*b**3*x + 20*sqrt(x)*sqrt(a + b*x + c*x**2)*b**2*c*x**2 - 42 
*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x)*a**2*c**2 + 
43*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x)*a*b**2*c - 
 8*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x)*b**4 - 14* 
int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)*a**3*c + 3 
*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)*a**2*b**2 
)/(70*b*c)