3.1030 \(\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=373 \[ -\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} \left (-6 a B c-5 A b c+2 b^2 B\right )}{15 c^{3/2} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 \sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-6 a B c-5 A b c+2 b^2 B\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{7/4} \sqrt{a+b x+c x^2}}-\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-3 \sqrt{a} B \sqrt{c}-5 A c+2 b B\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{7/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c} \]

[Out]

(-2*(2*b^2*B - 5*A*b*c - 6*a*B*c)*Sqrt[x]*Sqrt[a + b*x + c*x^2])/(15*c^(3/2)*(Sq
rt[a] + Sqrt[c]*x)) + (2*Sqrt[x]*(b*B + 5*A*c + 3*B*c*x)*Sqrt[a + b*x + c*x^2])/
(15*c) + (2*a^(1/4)*(2*b^2*B - 5*A*b*c - 6*a*B*c)*(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])/(15*c^(7/4)*Sqrt[a + b*x + c*x^2]) - (a^(1/
4)*(b + 2*Sqrt[a]*Sqrt[c])*(2*b*B - 3*Sqrt[a]*B*Sqrt[c] - 5*A*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])/(15*c^(7/4)*Sqrt[a + b*x + c
*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.717183, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} \left (-6 a B c-5 A b c+2 b^2 B\right )}{15 c^{3/2} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 \sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-6 a B c-5 A b c+2 b^2 B\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{7/4} \sqrt{a+b x+c x^2}}-\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-3 \sqrt{a} B \sqrt{c}-5 A c+2 b B\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{7/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*(2*b^2*B - 5*A*b*c - 6*a*B*c)*Sqrt[x]*Sqrt[a + b*x + c*x^2])/(15*c^(3/2)*(Sq
rt[a] + Sqrt[c]*x)) + (2*Sqrt[x]*(b*B + 5*A*c + 3*B*c*x)*Sqrt[a + b*x + c*x^2])/
(15*c) + (2*a^(1/4)*(2*b^2*B - 5*A*b*c - 6*a*B*c)*(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])/(15*c^(7/4)*Sqrt[a + b*x + c*x^2]) - (a^(1/
4)*(b + 2*Sqrt[a]*Sqrt[c])*(2*b*B - 3*Sqrt[a]*B*Sqrt[c] - 5*A*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])/(15*c^(7/4)*Sqrt[a + b*x + c
*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 101.717, size = 364, normalized size = 0.98 \[ \frac{4 \sqrt [4]{a} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (- \frac{5 A b c}{2} - 3 B a c + B b^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{15 c^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt [4]{a} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (- 5 A b c - 6 B a c + 2 B b^{2} - \sqrt{a} \sqrt{c} \left (10 A c - B b\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{15 c^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} + \frac{4 \sqrt{x} \sqrt{a + b x + c x^{2}} \left (\frac{5 A c}{2} + \frac{B b}{2} + \frac{3 B c x}{2}\right )}{15 c} - \frac{4 \sqrt{x} \sqrt{a + b x + c x^{2}} \left (- \frac{5 A b c}{2} - 3 B a c + B b^{2}\right )}{15 c^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{c} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**(1/2),x)

[Out]

4*a**(1/4)*sqrt((a + b*x + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*
x)*(-5*A*b*c/2 - 3*B*a*c + B*b**2)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)),
1/2 - b/(4*sqrt(a)*sqrt(c)))/(15*c**(7/4)*sqrt(a + b*x + c*x**2)) - a**(1/4)*sq
rt((a + b*x + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(-5*A*b*c
- 6*B*a*c + 2*B*b**2 - sqrt(a)*sqrt(c)*(10*A*c - B*b))*elliptic_f(2*atan(c**(1/4
)*sqrt(x)/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(15*c**(7/4)*sqrt(a + b*x + c*
x**2)) + 4*sqrt(x)*sqrt(a + b*x + c*x**2)*(5*A*c/2 + B*b/2 + 3*B*c*x/2)/(15*c) -
4*sqrt(x)*sqrt(a + b*x + c*x**2)*(-5*A*b*c/2 - 3*B*a*c + B*b**2)/(15*c**(3/2)*(
sqrt(a) + sqrt(c)*x))

_______________________________________________________________________________________

Mathematica [C]  time = 7.43625, size = 550, normalized size = 1.47 \[ \frac{\frac{4 \sqrt{x} (a+x (b+c x)) (5 A c+b B+3 B c x)}{c}+\frac{x \left (-\frac{4 (a+x (b+c x)) \left (-6 a B c-5 A b c+2 b^2 B\right )}{x^{3/2}}+\frac{i \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \left (-6 a B c-5 A b c+2 b^2 B\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+\frac{i \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \left (-b^2 \left (2 B \sqrt{b^2-4 a c}+5 A c\right )+b \left (5 A c \sqrt{b^2-4 a c}-8 a B c\right )+2 a c \left (3 B \sqrt{b^2-4 a c}+10 A c\right )+2 b^3 B\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}\right )}{c^2}}{30 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((4*Sqrt[x]*(b*B + 5*A*c + 3*B*c*x)*(a + x*(b + c*x)))/c + (x*((-4*(2*b^2*B - 5*
A*b*c - 6*a*B*c)*(a + x*(b + c*x)))/x^(3/2) + (I*(2*b^2*B - 5*A*b*c - 6*a*B*c)*(
-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*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[(Sqr
t[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqr
t[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] + (I*(2*b^3*B - b^2*(5*A*c + 2
*B*Sqrt[b^2 - 4*a*c]) + 2*a*c*(10*A*c + 3*B*Sqrt[b^2 - 4*a*c]) + b*(-8*a*B*c + 5
*A*c*Sqrt[b^2 - 4*a*c]))*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*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[(Sq
rt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sq
rt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]))/c^2)/(30*Sqrt[a + x*(b + c*
x)])

_______________________________________________________________________________________

Maple [B]  time = 0.044, size = 2012, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(10*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*
x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1
/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^
(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*a*c^
2+20*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-
4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*
EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2
)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*b*c^2-5*A*((b+2*c*x+(-4*a
*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*
a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-
4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2))*b^3*c-5*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*
c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-
c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4
*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^
(1/2))*(-4*a*c+b^2)^(1/2)*b^2*c-12*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^
2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/
(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c
+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2
))*a^2*c^2+3*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-
2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2))
)^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/
2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*b^2*c-B*((b+2*c*x
+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2)
)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*
c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^
2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*a*b*c+24*B*((b+2*c*x+(-4
*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+
(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^2*c^2-14*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(
-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/
2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(
b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1
/2))^(1/2))*a*b^2*c-6*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1
/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^
2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))
^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^
2)^(1/2)*a*b*c+2*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(
(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1
/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2
),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*b^4+2*B*((b+2*c
*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+
2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+
b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*b^3+6*B*x^4*c^4+10*A*x
^3*c^4+8*B*x^3*b*c^3+10*A*x^2*b*c^3+6*B*x^2*a*c^3+2*B*x^2*b^2*c^2+10*A*x*a*c^3+2
*B*x*a*b*c^2)/(c*x^2+b*x+a)^(1/2)/x^(1/2)/c^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/sqrt(x),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{\sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/sqrt(x),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{a + b x + c x^{2}}}{\sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/sqrt(x),x, algorithm="giac")

[Out]

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