3.1265 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=339 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (5 A c e-4 b B e+3 B c d) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (5 A c e-4 b B e+3 B c d)}{15 c^2}+\frac{2 B \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c} \]

[Out]

(2*(3*B*c*d - 4*b*B*e + 5*A*c*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*c^2) + (2*
B*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*c) + (2*Sqrt[-b]*(10*A*c*e*(2*c*d - b*e)
+ B*(3*c^2*d^2 - 13*b*c*d*e + 8*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*
x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*e*Sqr
t[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(3*B*c*d - 4*b*B*e
+ 5*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c
]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2
])

_______________________________________________________________________________________

Rubi [A]  time = 1.23992, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (5 A c e-4 b B e+3 B c d) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (5 A c e-4 b B e+3 B c d)}{15 c^2}+\frac{2 B \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*(3*B*c*d - 4*b*B*e + 5*A*c*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*c^2) + (2*
B*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*c) + (2*Sqrt[-b]*(10*A*c*e*(2*c*d - b*e)
+ B*(3*c^2*d^2 - 13*b*c*d*e + 8*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*
x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*e*Sqr
t[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(3*B*c*d - 4*b*B*e
+ 5*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c
]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2
])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 130.849, size = 330, normalized size = 0.97 \[ \frac{2 B \left (d + e x\right )^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{5 c} + \frac{2 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (5 A c e - 4 B b e + 3 B c d\right )}{15 c^{2}} - \frac{2 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (5 A c e - 4 B b e + 3 B c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{15 c^{2} e^{\frac{3}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (- 10 A b c e^{2} + 20 A c^{2} d e + 8 B b^{2} e^{2} - 13 B b c d e + 3 B c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 c^{\frac{5}{2}} e \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(d + e*x)**(3/2)*sqrt(b*x + c*x**2)/(5*c) + 2*sqrt(d + e*x)*sqrt(b*x + c*x**
2)*(5*A*c*e - 4*B*b*e + 3*B*c*d)/(15*c**2) - 2*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/
b)*sqrt(1 + e*x/d)*(b*e - c*d)*(5*A*c*e - 4*B*b*e + 3*B*c*d)*elliptic_f(asin(sqr
t(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(15*c**2*e**(3/2)*sqrt(d + e*x)*sqrt(b*x + c*
x**2)) + 2*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(-10*A*b*c*e**2 + 20*A
*c**2*d*e + 8*B*b**2*e**2 - 13*B*b*c*d*e + 3*B*c**2*d**2)*elliptic_e(asin(sqrt(c
)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(15*c**(5/2)*e*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2
))

_______________________________________________________________________________________

Mathematica [C]  time = 3.11666, size = 356, normalized size = 1.05 \[ \frac{2 \sqrt{x} \left (\frac{(b+c x) (d+e x) \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right )}{c e \sqrt{x}}+i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) \left (-b c (10 A e+9 B d)+15 A c^2 d+8 b^2 B e\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{b}+\sqrt{x} (b+c x) (d+e x) (5 A c e+B (-4 b e+6 c d+3 c e x))\right )}{15 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(((10*A*c*e*(2*c*d - b*e) + B*(3*c^2*d^2 - 13*b*c*d*e + 8*b^2*e^2))*(
b + c*x)*(d + e*x))/(c*e*Sqrt[x]) + Sqrt[x]*(b + c*x)*(d + e*x)*(5*A*c*e + B*(6*
c*d - 4*b*e + 3*c*e*x)) + I*Sqrt[b/c]*(10*A*c*e*(2*c*d - b*e) + B*(3*c^2*d^2 - 1
3*b*c*d*e + 8*b^2*e^2))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSin
h[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (I*Sqrt[b/c]*(-(c*d) + b*e)*(15*A*c^2*d + 8
*b^2*B*e - b*c*(9*B*d + 10*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF
[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(15*c^2*Sqrt[x*(b + c*x)]*Sqrt[
d + e*x])

_______________________________________________________________________________________

Maple [B]  time = 0.032, size = 1144, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(5*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c
^2*d*e^2-5*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^2*e+10*A*((c*x+b)/b)^(1/2)
*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b^3*c*e^3-30*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d*e^2+20
*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^2*e-4*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*b^3*c*d*e^2+7*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^2*e-3*B*((c*x+b)
/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*e^3+
21*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d*e^2-16*B*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*b^2*c^2*d^2*e+3*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3+3*B*x^4*c^4
*e^3+5*A*x^3*c^4*e^3-B*x^3*b*c^3*e^3+9*B*x^3*c^4*d*e^2+5*A*x^2*b*c^3*e^3+5*A*x^2
*c^4*d*e^2-4*B*x^2*b^2*c^2*e^3+5*B*x^2*b*c^3*d*e^2+6*B*x^2*c^4*d^2*e+5*A*x*b*c^3
*d*e^2-4*B*x*b^2*c^2*d*e^2+6*B*x*b*c^3*d^2*e)/e/x/(c*e*x^2+b*e*x+c*d*x+b*d)/c^4

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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