3.1257 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=283 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (-6 A c e-b B e+8 B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt{d+e x}} \]

[Out]

(2*(4*B*d - 3*A*e + B*e*x)*Sqrt[b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - (2*Sqrt[-b
]*(8*B*c*d - b*B*e - 6*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[
ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^3*Sqrt[1 + (e*x)/
d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*(B*d*(8*c*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*
Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/S
qrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^3*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.825028, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (-6 A c e-b B e+8 B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(4*B*d - 3*A*e + B*e*x)*Sqrt[b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - (2*Sqrt[-b
]*(8*B*c*d - b*B*e - 6*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[
ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^3*Sqrt[1 + (e*x)/
d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*(B*d*(8*c*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*
Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/S
qrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^3*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 89.2394, size = 274, normalized size = 0.97 \[ - \frac{4 \sqrt{b x + c x^{2}} \left (\frac{3 A e}{2} - 2 B d - \frac{B e x}{2}\right )}{3 e^{2} \sqrt{d + e x}} + \frac{2 \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (3 A b e^{2} - 6 A c d e - 5 B b d e + 8 B c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 e^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (6 A c e + B b e - 8 B c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} e^{3} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(b*x + c*x**2)*(3*A*e/2 - 2*B*d - B*e*x/2)/(3*e**2*sqrt(d + e*x)) + 2*sqr
t(x)*sqrt(-d)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(3*A*b*e**2 - 6*A*c*d*e - 5*B*b*d*
e + 8*B*c*d**2)*elliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(3*e**(7/2
)*sqrt(d + e*x)*sqrt(b*x + c*x**2)) + 2*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d
+ e*x)*(6*A*c*e + B*b*e - 8*B*c*d)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*
e/(c*d))/(3*sqrt(c)*e**3*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.9393, size = 269, normalized size = 0.95 \[ \frac{2 \left (b e x (b+c x) (-3 A e+4 B d+B e x)+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (-3 A c e-b B e+4 B c d) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (-6 A c e-b B e+8 B c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) (6 A c e+b B e-8 B c d)\right )\right )}{3 b e^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 804, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(3*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^2*c*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1
/2)-6*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d*e*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*A*EllipticE(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*b^2*c*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)+6*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d
*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-5*B*EllipticF((
(c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/
(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+8*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*b*c^2*d^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B
*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*e^2*((c*x+b)/b)^(1/2)*(-
(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+9*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*b^2*c*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/
b)^(1/2)-8*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^2*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*x^3*c^3*e^2-3*A*x^2*c^
3*e^2+B*x^2*b*c^2*e^2+4*B*x^2*c^3*d*e-3*A*x*b*c^2*e^2+4*B*x*b*c^2*d*e)/x/(c*e*x^
2+b*e*x+c*d*x+b*d)/e^3/c^2

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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