3.1260 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=574 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+5 b c d e (2 c d-b e) (-9 A c e-3 b B e+8 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+9 A c e \left (b^2 e^2-11 b c d e+8 c^2 d^2\right )-2 B \left (2 b^3 e^3+3 b^2 c d e^2-42 b c^2 d^2 e+32 c^3 d^3\right )\right )}{315 c^2 e^4}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (9 A c e \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-4 b^3 e^3-9 b^2 c d e^2-120 b c^2 d^2 e+128 c^3 d^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-9 A c e-3 b B e+8 B c d-7 B c e x)}{63 c e^2} \]

[Out]

(2*Sqrt[d + e*x]*(9*A*c*e*(8*c^2*d^2 - 11*b*c*d*e + b^2*e^2) - 2*B*(32*c^3*d^3 -
42*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3) - 3*c*e*(9*A*c*e*(2*c*d - b*e) - B*
(16*c^2*d^2 - 7*b*c*d*e - 4*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^4) - (2*S
qrt[d + e*x]*(8*B*c*d - 3*b*B*e - 9*A*c*e - 7*B*c*e*x)*(b*x + c*x^2)^(3/2))/(63*
c*e^2) - (2*Sqrt[-b]*(5*b*c*d*e*(2*c*d - b*e)*(8*B*c*d - 3*b*B*e - 9*A*c*e) + (8
*c^2*d^2 - 3*b*c*d*e - 2*b^2*e^2)*(9*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 7*b*c
*d*e - 4*b^2*e^2)))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sq
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^5*Sqrt[1 + (e*x)/d]*Sqrt[
b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(9*A*c*e*(16*c^2*d^2 - 16*b*c*d*e - b^
2*e^2) - B*(128*c^3*d^3 - 120*b*c^2*d^2*e - 9*b^2*c*d*e^2 - 4*b^3*e^3))*Sqrt[x]*
Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(315*c^(5/2)*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.87611, antiderivative size = 574, 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 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+5 b c d e (2 c d-b e) (-9 A c e-3 b B e+8 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+9 A c e \left (b^2 e^2-11 b c d e+8 c^2 d^2\right )-2 B \left (2 b^3 e^3+3 b^2 c d e^2-42 b c^2 d^2 e+32 c^3 d^3\right )\right )}{315 c^2 e^4}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (9 A c e \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-4 b^3 e^3-9 b^2 c d e^2-120 b c^2 d^2 e+128 c^3 d^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-9 A c e-3 b B e+8 B c d-7 B c e x)}{63 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(9*A*c*e*(8*c^2*d^2 - 11*b*c*d*e + b^2*e^2) - 2*B*(32*c^3*d^3 -
42*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3) - 3*c*e*(9*A*c*e*(2*c*d - b*e) - B*
(16*c^2*d^2 - 7*b*c*d*e - 4*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^4) - (2*S
qrt[d + e*x]*(8*B*c*d - 3*b*B*e - 9*A*c*e - 7*B*c*e*x)*(b*x + c*x^2)^(3/2))/(63*
c*e^2) - (2*Sqrt[-b]*(5*b*c*d*e*(2*c*d - b*e)*(8*B*c*d - 3*b*B*e - 9*A*c*e) + (8
*c^2*d^2 - 3*b*c*d*e - 2*b^2*e^2)*(9*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 7*b*c
*d*e - 4*b^2*e^2)))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sq
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^5*Sqrt[1 + (e*x)/d]*Sqrt[
b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(9*A*c*e*(16*c^2*d^2 - 16*b*c*d*e - b^
2*e^2) - B*(128*c^3*d^3 - 120*b*c^2*d^2*e - 9*b^2*c*d*e^2 - 4*b^3*e^3))*Sqrt[x]*
Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(315*c^(5/2)*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 7.85782, size = 630, normalized size = 1.1 \[ -\frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (B \left (4 b^3 e^3-3 b^2 c e^2 (e x-2 d)+b c^2 e \left (-84 d^2+61 d e x-50 e^2 x^2\right )+c^3 \left (64 d^3-48 d^2 e x+40 d e^2 x^2-35 e^3 x^3\right )\right )-9 A c e \left (b^2 e^2+b c e (8 e x-11 d)+c^2 \left (8 d^2-6 d e x+5 e^2 x^2\right )\right )\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (9 A c e \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )+B \left (8 b^3 e^3+15 b^2 c d e^2+36 b c^2 d^2 e-64 c^3 d^3\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (18 A c e \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )-B \left (8 b^4 e^4+11 b^3 c d e^3+27 b^2 c^2 d^2 e^2-184 b c^3 d^3 e+128 c^4 d^4\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (18 A c e \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )-B \left (8 b^4 e^4+11 b^3 c d e^3+27 b^2 c^2 d^2 e^2-184 b c^3 d^3 e+128 c^4 d^4\right )\right )\right )\right )}{315 b c^2 e^5 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(-9*A*c*e*(b^2*e^2 + b*c*e*(-
11*d + 8*e*x) + c^2*(8*d^2 - 6*d*e*x + 5*e^2*x^2)) + B*(4*b^3*e^3 - 3*b^2*c*e^2*
(-2*d + e*x) + b*c^2*e*(-84*d^2 + 61*d*e*x - 50*e^2*x^2) + c^3*(64*d^3 - 48*d^2*
e*x + 40*d*e^2*x^2 - 35*e^3*x^3))) + Sqrt[b/c]*(Sqrt[b/c]*(18*A*c*e*(8*c^3*d^3 -
12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 184*b*c^3*d^3*e +
27*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 + 8*b^4*e^4))*(b + c*x)*(d + e*x) + I*b*e*(1
8*A*c*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4
- 184*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 + 8*b^4*e^4))*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)*(9*A*c*e*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + B*(-64
*c^3*d^3 + 36*b*c^2*d^2*e + 15*b^2*c*d*e^2 + 8*b^3*e^3))*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)])))/(31
5*b*c^2*e^5*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.035, size = 2112, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(144*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^5*d^4*e+18*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^2*d*e^4-252*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^3*d^2*e^3+360*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^4*d^3*e^2-144*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^5*d^4*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^5*c*d*e^4-5*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^2*d^2
*e^3-111*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^3*d^3*e^2+248*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^4*d^4*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^5*c*d*e^
4-16*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^4*c^2*d^2*e^3+211*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^3*d^3*e^2+9*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^4*c^2*d*e^
4+135*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^3*c^3*d^2*e^3-288*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^4*d^3*e^2+85*B*x^5*b*c^5*e^5-312*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^4*d^4*e+35*B*x^6*c^6*e^5+45*A*x^5*c^6*e^5-5*B*x^5*c^6*d*e^4+117*
A*x^4*b*c^5*e^5-9*A*x^4*c^6*d*e^4+53*B*x^4*b^2*c^4*e^5+8*B*x^4*c^6*d^2*e^3+81*A*
x^3*b^2*c^4*e^5+18*A*x^3*c^6*d^2*e^3-B*x^3*b^3*c^3*e^5-16*B*x^3*c^6*d^3*e^2+9*A*
x^2*b^3*c^3*e^5+72*A*x^2*c^6*d^3*e^2+68*B*x^2*b*c^5*d^3*e^2+9*A*x*b^3*c^3*d*e^4-
99*A*x*b^2*c^4*d^2*e^3+72*A*x*b*c^5*d^3*e^2-4*B*x*b^4*c^2*d*e^4-6*B*x*b^3*c^3*d^
2*e^3+84*B*x*b^2*c^4*d^3*e^2-64*B*x*b*c^5*d^4*e-16*B*x^4*b*c^5*d*e^4-36*A*x^3*b*
c^5*d*e^4-14*B*x^3*b^2*c^4*d*e^4+31*B*x^3*b*c^5*d^2*e^3-18*A*x^2*b^2*c^4*d*e^4-8
1*A*x^2*b*c^5*d^2*e^3-7*B*x^2*b^3*c^3*d*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)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*b^6*e^5-128*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^5*d^5+128*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^5*d^5+17*B*x^2*b^2*c^4*d^2*e^3-4*B*x^2*b^4*c^2*e^5-64*B*x^2
*c^6*d^4*e+18*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c*e^5)/c^4/e^5/x/(c*e*x^2+b
*e*x+c*d*x+b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*c*x^3 + A*b*x + (B*b + A*c)*x^2)*sqrt(c*x^2 + b*x)/sqrt(e*x + d), 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)*(c*x**2+b*x)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.22202, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done