3.1255 \(\int (A+B x) \sqrt{d+e x} \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=433 \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (3 c e x (7 A c e-4 b B e+B c d)+7 A c e (b e+c d)-B \left (4 b^2 e^2-2 b c d e+4 c^2 d^2\right )\right )}{105 c^2 e^2}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (7 A c e (2 c d-b e)-B \left (-4 b^2 e^2-b c d e+8 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (7 A c e-4 b B e+B c d)+5 c d e (3 b B-7 A c) (2 c d-b e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 B \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

[Out]

(2*Sqrt[d + e*x]*(7*A*c*e*(c*d + b*e) - B*(4*c^2*d^2 - 2*b*c*d*e + 4*b^2*e^2) +
3*c*e*(B*c*d - 4*b*B*e + 7*A*c*e)*x)*Sqrt[b*x + c*x^2])/(105*c^2*e^2) + (2*B*Sqr
t[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) + (2*Sqrt[-b]*(5*c*(3*b*B - 7*A*c)*d*e*(2*
c*d - b*e) + (B*c*d - 4*b*B*e + 7*A*c*e)*(8*c^2*d^2 - 3*b*c*d*e - 2*b^2*e^2))*Sq
rt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b
]], (b*e)/(c*d)])/(105*c^(5/2)*e^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqr
t[-b]*d*(c*d - b*e)*(7*A*c*e*(2*c*d - b*e) - B*(8*c^2*d^2 - b*c*d*e - 4*b^2*e^2)
)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(5/2)*e^3*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.5086, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (3 c e x (7 A c e-4 b B e+B c d)+7 A c e (b e+c d)-B \left (4 b^2 e^2-2 b c d e+4 c^2 d^2\right )\right )}{105 c^2 e^2}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (7 A c e (2 c d-b e)-B \left (-4 b^2 e^2-b c d e+8 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (7 A c e-4 b B e+B c d)+5 c d e (3 b B-7 A c) (2 c d-b e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 B \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(7*A*c*e*(c*d + b*e) - B*(4*c^2*d^2 - 2*b*c*d*e + 4*b^2*e^2) +
3*c*e*(B*c*d - 4*b*B*e + 7*A*c*e)*x)*Sqrt[b*x + c*x^2])/(105*c^2*e^2) + (2*B*Sqr
t[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) + (2*Sqrt[-b]*(5*c*(3*b*B - 7*A*c)*d*e*(2*
c*d - b*e) + (B*c*d - 4*b*B*e + 7*A*c*e)*(8*c^2*d^2 - 3*b*c*d*e - 2*b^2*e^2))*Sq
rt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b
]], (b*e)/(c*d)])/(105*c^(5/2)*e^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqr
t[-b]*d*(c*d - b*e)*(7*A*c*e*(2*c*d - b*e) - B*(8*c^2*d^2 - b*c*d*e - 4*b^2*e^2)
)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(5/2)*e^3*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 154.508, size = 435, normalized size = 1. \[ \frac{2 B \sqrt{d + e x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 c} + \frac{8 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (\frac{5 c d e \left (7 A c - 3 B b\right )}{4} + \frac{3 c e x \left (7 A c e - 4 B b e + B c d\right )}{4} + \left (\frac{b e}{4} - c d\right ) \left (7 A c e - 4 B b e + B c d\right )\right )}{105 c^{2} e^{2}} + \frac{2 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (- 7 A b c e^{2} + 14 A c^{2} d e + 4 B b^{2} e^{2} + B b c d e - 8 B c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{105 c^{2} e^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (- 5 c d e \left (7 A c - 3 B b\right ) \left (b e - 2 c d\right ) + \left (2 b^{2} e^{2} + 3 b c d e - 8 c^{2} d^{2}\right ) \left (7 A c e - 4 B b e + B c d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{105 c^{\frac{5}{2}} e^{3} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(d + e*x)*(b*x + c*x**2)**(3/2)/(7*c) + 8*sqrt(d + e*x)*sqrt(b*x + c*x**
2)*(5*c*d*e*(7*A*c - 3*B*b)/4 + 3*c*e*x*(7*A*c*e - 4*B*b*e + B*c*d)/4 + (b*e/4 -
c*d)*(7*A*c*e - 4*B*b*e + B*c*d))/(105*c**2*e**2) + 2*sqrt(x)*(-d)**(3/2)*sqrt(
1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - c*d)*(-7*A*b*c*e**2 + 14*A*c**2*d*e + 4*B*b**2
*e**2 + B*b*c*d*e - 8*B*c**2*d**2)*elliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*
d/(b*e))/(105*c**2*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)*(-5*c*d*e*(7*A*c - 3*B*b)*(b*e - 2*c*d) + (2*b*
*2*e**2 + 3*b*c*d*e - 8*c**2*d**2)*(7*A*c*e - 4*B*b*e + B*c*d))*elliptic_e(asin(
sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(105*c**(5/2)*e**3*sqrt(1 + e*x/d)*sqrt(b*
x + c*x**2))

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Mathematica [C]  time = 5.90178, size = 461, normalized size = 1.06 \[ -\frac{2 \left (b e x (b+c x) (d+e x) \left (B \left (4 b^2 e^2-b c e (2 d+3 e x)+c^2 \left (4 d^2-3 d e x-15 e^2 x^2\right )\right )-7 A c e (b e+c (d+3 e x))\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (7 A c e (c d-2 b e)-B \left (-8 b^2 e^2+b c d e+4 c^2 d^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (14 A c e \left (b^2 e^2-b c d e+c^2 d^2\right )+B \left (-8 b^3 e^3+5 b^2 c d e^2+5 b c^2 d^2 e-8 c^3 d^3\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (14 A c e \left (b^2 e^2-b c d e+c^2 d^2\right )+B \left (-8 b^3 e^3+5 b^2 c d e^2+5 b c^2 d^2 e-8 c^3 d^3\right )\right )\right )\right )}{105 b c^2 e^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*e*x*(b + c*x)*(d + e*x)*(-7*A*c*e*(b*e + c*(d + 3*e*x)) + B*(4*b^2*e^2 -
b*c*e*(2*d + 3*e*x) + c^2*(4*d^2 - 3*d*e*x - 15*e^2*x^2))) + Sqrt[b/c]*(Sqrt[b/c
]*(14*A*c*e*(c^2*d^2 - b*c*d*e + b^2*e^2) + B*(-8*c^3*d^3 + 5*b*c^2*d^2*e + 5*b^
2*c*d*e^2 - 8*b^3*e^3))*(b + c*x)*(d + e*x) + I*b*e*(14*A*c*e*(c^2*d^2 - b*c*d*e
+ b^2*e^2) + B*(-8*c^3*d^3 + 5*b*c^2*d^2*e + 5*b^2*c*d*e^2 - 8*b^3*e^3))*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*(c*d - b*e)*(7*A*c*e*(c*d - 2*b*e) - B*(4*c^2*d^2 + b*c*d*e -
8*b^2*e^2))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqr
t[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(105*b*c^2*e^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.028, size = 1526, normalized size = 3.5 \[ \text{result too large to display} \]

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)^(1/2),x)

[Out]

2/105*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(-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^4
*c*d*e^3+3*B*((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^3*c^2*d^2*e^2+9*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^3*d^3*e+13*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*c*d*e
^3-13*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^3*d^3*e+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*c^4*d^4+14*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^4*c*e^4+23*B*x^3*b
*c^4*d*e^3+18*B*x^4*b*c^4*e^4-B*x^3*b^2*c^3*e^4-B*x^3*c^5*d^2*e^2+7*A*x^2*b^2*c^
3*e^4+7*A*x^2*c^5*d^2*e^2-4*B*x^2*b^3*c^2*e^4-4*B*x^2*c^5*d^3*e+35*A*x^2*b*c^4*d
*e^3+B*x^2*b^2*c^3*d*e^3+B*x^2*b*c^4*d^2*e^2+7*A*x*b^2*c^3*d*e^3+7*A*x*b*c^4*d^2
*e^2-4*B*x*b^3*c^2*d*e^3+2*B*x*b^2*c^3*d^2*e^2-4*B*x*b*c^4*d^3*e-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^5*e^4-8*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^4*d^4+18*
B*x^4*c^5*d*e^3+28*A*x^3*b*c^4*e^4+28*A*x^3*c^5*d*e^3+21*A*x^4*c^5*e^4+15*B*x^5*
c^5*e^4+7*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e^3-21*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^3*d^2*e^2+14*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*c^4*d^3
*e-28*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^2*d*e^3+28*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^3*d^2*e^2-14*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^4*d^3*e)/
x/(c*e*x^2+b*e*x+c*d*x+b*d)/c^4/e^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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