3.2636 \(\int \frac{(A+B x) \sqrt{d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=460 \[ -\frac{2 \sqrt{d+e x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b B-2 A c) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

[Out]

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

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Rubi [A]  time = 1.18515, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 \sqrt{d+e x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b B-2 A c) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 171.547, size = 440, normalized size = 0.96 \[ - \frac{2 \sqrt{d + e x} \left (A b - 2 B a + x \left (2 A c - B b\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{2 \sqrt{2} \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (A b e - 2 A c d - 2 B a e + B b d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{c \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \left (2 A c - B b\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{c \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

-2*sqrt(d + e*x)*(A*b - 2*B*a + x*(2*A*c - B*b))/((-4*a*c + b**2)*sqrt(a + b*x +
c*x**2)) + 2*sqrt(2)*sqrt(c*(-d - e*x)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))*s
qrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(A*b*e - 2*A*c*d - 2*B*a*e + B*b*d)*ell
iptic_f(asin(sqrt(2)*sqrt((b + 2*c*x + sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2))
/2), 2*e*sqrt(-4*a*c + b**2)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))/(c*sqrt(d +
e*x)*sqrt(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)) + sqrt(2)*sqrt(c*(a + b*x + c*x
**2)/(4*a*c - b**2))*sqrt(d + e*x)*(2*A*c - B*b)*elliptic_e(asin(sqrt(2)*sqrt((b
+ 2*c*x + sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2))/2), 2*e*sqrt(-4*a*c + b**2)
/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))/(c*sqrt(c*(-d - e*x)/(b*e - 2*c*d + e*sq
rt(-4*a*c + b**2)))*sqrt(-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 13.1687, size = 5246, normalized size = 11.4 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.081, size = 3171, normalized size = 6.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-4*A*2^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1
/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*
a*c*e^2*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)
*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+
d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)+A*2^(1/2)*EllipticF(2^(1/2)*(-(e*x+
d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/
(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*e^2*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*
e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4
*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^
(1/2)+A*2^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^
(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)
)*b*e^2*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)
*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-4*a*c
+b^2)^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)-2*A*2^(1/2)*Elli
pticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b
^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*c*d*e*((-2*c*x+(-4
*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2
)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-4*a*c+b^2)^(1/2)*(-(e*x+d
)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)+4*A*2^(1/2)*EllipticE(2^(1/2)*(-(e*x
+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)
/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c*e^2*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)
*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-
4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))
^(1/2)-4*A*2^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d
))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1
/2))*b*c*d*e*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^
(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-
(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)+4*A*2^(1/2)*EllipticE(2^(1/2)*
(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-
2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*c^2*d^2*((-2*c*x+(-4*a*c+b^2)^(1
/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e
/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2
*c*d))^(1/2)+4*B*2^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e
-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2
)))^(1/2))*a*c*d*e*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1
/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1
/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)-2*B*2^(1/2)*EllipticF(2^
(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2
)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*e^2*((-2*c*x+(-4*a*c+b^2
)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+
b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-4*a*c+b^2)^(1/2)*(-(e*x+d)*c/(e*(
-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)-B*2^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(
-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*
e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*d*e*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-
b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)
^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)+B*2
^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(
e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b*d*e*(
(-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+
(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-4*a*c+b^2)^(1/
2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)-2*B*2^(1/2)*EllipticE(2^(
1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)
+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*e^2*((-2*c*x+(-4*a*c+b^
2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)
+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+
b*e-2*c*d))^(1/2)+2*B*2^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2
)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)
^(1/2)))^(1/2))*b^2*d*e*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^
2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d
))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)-2*B*2^(1/2)*Ellipti
cE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)
^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b*c*d^2*((-2*c*x+(-4*
a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)
^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^
(1/2)+b*e-2*c*d))^(1/2)+4*A*x^2*c^2*e^2-2*B*x^2*b*c*e^2+2*A*b*c*e^2*x+4*A*c^2*d*
e*x-4*B*a*c*e^2*x-2*B*b*c*d*e*x+2*A*b*c*d*e-4*B*a*c*d*e)*(e*x+d)^(1/2)*(c*x^2+b*
x+a)^(1/2)/c/e/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)/(4*a*c-b^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \sqrt{e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out