3.1258 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=346 \[ -\frac{2 \sqrt{b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (-2 A c e-3 b B e+8 B c d) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]

[Out]

(-2*(d^2*(4*B*c*d - 3*b*B*e - A*c*e) + e*(B*d*(5*c*d - 4*b*e) - A*e*(2*c*d - b*e
))*x)*Sqrt[b*x + c*x^2])/(3*d*e^2*(c*d - b*e)*(d + e*x)^(3/2)) + (2*Sqrt[-b]*Sqr
t[c]*(B*d*(8*c*d - 7*b*e) - A*e*(2*c*d - b*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*d*e^3*(c*d
- b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(8*B*c*d - 3*b*B*e -
2*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*S
qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^3*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.06537, antiderivative size = 346, 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 \[ -\frac{2 \sqrt{b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (-2 A c e-3 b B e+8 B c d) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^(5/2),x]

[Out]

(-2*(d^2*(4*B*c*d - 3*b*B*e - A*c*e) + e*(B*d*(5*c*d - 4*b*e) - A*e*(2*c*d - b*e
))*x)*Sqrt[b*x + c*x^2])/(3*d*e^2*(c*d - b*e)*(d + e*x)^(3/2)) + (2*Sqrt[-b]*Sqr
t[c]*(B*d*(8*c*d - 7*b*e) - A*e*(2*c*d - b*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*d*e^3*(c*d
- b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(8*B*c*d - 3*b*B*e -
2*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*S
qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^3*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 110.859, size = 326, normalized size = 0.94 \[ - \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e \left (A e - 7 B d\right ) - 2 c d \left (A e - 4 B d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d e^{3} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (2 A c e + 3 B b e - 8 B c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 e^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{4 \sqrt{b x + c x^{2}} \left (\frac{d^{2} \left (A c e + 3 B b e - 4 B c d\right )}{2} - \frac{e x \left (A b e^{2} - 2 A c d e - 4 B b d e + 5 B c d^{2}\right )}{2}\right )}{3 d e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} \]

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)**(5/2),x)

[Out]

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

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Mathematica [C]  time = 2.58296, size = 346, normalized size = 1. \[ \frac{2 \left (e x \sqrt{\frac{b}{c}} (b+c x) \left (A e \left (c d (d+2 e x)-b e^2 x\right )+B d (b e (3 d+4 e x)-c d (4 d+5 e x))\right )+(d+e x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (4 B d-A e) (c d-b e) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (A e (2 c d-b e)+B d (7 b e-8 c d)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) (A e (b e-2 c d)+B d (8 c d-7 b e))\right )\right )}{3 d e^3 \sqrt{\frac{b}{c}} \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^(5/2),x]

[Out]

(2*(Sqrt[b/c]*e*x*(b + c*x)*(A*e*(-(b*e^2*x) + c*d*(d + 2*e*x)) + B*d*(b*e*(3*d
+ 4*e*x) - c*d*(4*d + 5*e*x))) + (d + e*x)*(Sqrt[b/c]*(B*d*(8*c*d - 7*b*e) + A*e
*(-2*c*d + b*e))*(b + c*x)*(d + e*x) - I*b*e*(A*e*(2*c*d - b*e) + B*d*(-8*c*d +
7*b*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*d - A*e)*(c*d - b*e)*Sqrt[1 + b/(c*x)]*Sqr
t[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(
3*Sqrt[b/c]*d*e^3*(c*d - b*e)*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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Maple [B]  time = 0.048, size = 1959, normalized size = 5.7 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

2/3*(2*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^3*e*((c*x+b)
/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-11*B*EllipticF(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)+15*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b^2*c*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-4*B*x^
3*b*c^2*d*e^3+2*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^2*e
^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-2*A*EllipticF((
(c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d)
)^(1/2))*b^2*c*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)-2*A*x^3*c^3*d*e^3-2*A*x^2*b*c^2*d*e^3-4*B*x^2*b^2*c*d*e^3+2*B*x^2*b*c^2*d^2
*e^2-A*b*c^2*d^2*e^2*x-3*B*b^2*c*d^2*e^2*x+4*B*b*c^2*d^3*e*x-7*B*EllipticE(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*d^2*e^2*((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^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+A*E
llipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*e^4*((c*x+b)/b)^(1/2)*(-
(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*b^3*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)
^(1/2)+3*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*d^2*e^2*((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^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d)
)^(1/2)*(-c*x/b)^(1/2)+3*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*
b^3*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-7*B*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*d*e^3*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+5*B*x^3*c^3*d^2*e^2+A*x^2*b^2*c*e^4-A*x
^2*c^3*d^2*e^2+4*B*x^2*c^3*d^3*e+A*x^3*b*c^2*e^4-2*A*EllipticF(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*x*b*c^2*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)-3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b
^2*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*A*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^2*d^2*e^2*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-11*B*EllipticF(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*x*b^2*c*d^2*e^2*((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))*x*b*c
^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*B*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c*d^2*e^2*((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))*x*b*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)
*(-c*x/b)^(1/2)+2*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c*d
*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2))*(x*(c*x+b))^
(1/2)/c/(b*e-c*d)/d/(c*x+b)/x/e^3/(e*x+d)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2), x)