3.1191 \(\int \frac{(A+B x) (d+e x)^3}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=305 \[ \frac{\sqrt{b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (-105 b^3 e^3+360 b^2 c d e^2-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac{\sqrt{b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac{B \sqrt{b x+c x^2} (d+e x)^3}{4 c} \]

[Out]

((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[b*x + c*x^2])/(24*c^2) + (B*(d +
e*x)^3*Sqrt[b*x + c*x^2])/(4*c) + ((8*A*c*e*(64*c^2*d^2 - 54*b*c*d*e + 15*b^2*e
^2) + B*(96*c^3*d^3 - 376*b*c^2*d^2*e + 360*b^2*c*d*e^2 - 105*b^3*e^3) + 2*c*e*(
40*A*c*e*(2*c*d - b*e) + B*(24*c^2*d^2 - 64*b*c*d*e + 35*b^2*e^2))*x)*Sqrt[b*x +
c*x^2])/(192*c^4) + ((128*A*c^4*d^3 + 35*b^4*B*e^3 + 144*b^2*c^2*d*e*(B*d + A*e
) - 40*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*ArcTanh[(Sqrt[c]*x)
/Sqrt[b*x + c*x^2]])/(64*c^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.01188, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (-105 b^3 e^3+360 b^2 c d e^2-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac{\sqrt{b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac{B \sqrt{b x+c x^2} (d+e x)^3}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[b*x + c*x^2])/(24*c^2) + (B*(d +
e*x)^3*Sqrt[b*x + c*x^2])/(4*c) + ((8*A*c*e*(64*c^2*d^2 - 54*b*c*d*e + 15*b^2*e
^2) + B*(96*c^3*d^3 - 376*b*c^2*d^2*e + 360*b^2*c*d*e^2 - 105*b^3*e^3) + 2*c*e*(
40*A*c*e*(2*c*d - b*e) + B*(24*c^2*d^2 - 64*b*c*d*e + 35*b^2*e^2))*x)*Sqrt[b*x +
c*x^2])/(192*c^4) + ((128*A*c^4*d^3 + 35*b^4*B*e^3 + 144*b^2*c^2*d*e*(B*d + A*e
) - 40*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*ArcTanh[(Sqrt[c]*x)
/Sqrt[b*x + c*x^2]])/(64*c^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 128.959, size = 357, normalized size = 1.17 \[ \frac{B \left (d + e x\right )^{3} \sqrt{b x + c x^{2}}}{4 c} + \frac{\left (d + e x\right )^{2} \sqrt{b x + c x^{2}} \left (8 A c e - 7 B b e + 6 B c d\right )}{24 c^{2}} - \frac{\sqrt{b x + c x^{2}} \left (- 15 A b^{2} c e^{3} + 54 A b c^{2} d e^{2} - 64 A c^{3} d^{2} e + \frac{105 B b^{3} e^{3}}{8} - 45 B b^{2} c d e^{2} + 47 B b c^{2} d^{2} e - 12 B c^{3} d^{3} - \frac{c e x \left (- 40 A b c e^{2} + 80 A c^{2} d e + 35 B b^{2} e^{2} - 64 B b c d e + 24 B c^{2} d^{2}\right )}{4}\right )}{24 c^{4}} + \frac{\left (- 40 A b^{3} c e^{3} + 144 A b^{2} c^{2} d e^{2} - 192 A b c^{3} d^{2} e + 128 A c^{4} d^{3} + 35 B b^{4} e^{3} - 120 B b^{3} c d e^{2} + 144 B b^{2} c^{2} d^{2} e - 64 B b c^{3} d^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(d + e*x)**3*sqrt(b*x + c*x**2)/(4*c) + (d + e*x)**2*sqrt(b*x + c*x**2)*(8*A*c
*e - 7*B*b*e + 6*B*c*d)/(24*c**2) - sqrt(b*x + c*x**2)*(-15*A*b**2*c*e**3 + 54*A
*b*c**2*d*e**2 - 64*A*c**3*d**2*e + 105*B*b**3*e**3/8 - 45*B*b**2*c*d*e**2 + 47*
B*b*c**2*d**2*e - 12*B*c**3*d**3 - c*e*x*(-40*A*b*c*e**2 + 80*A*c**2*d*e + 35*B*
b**2*e**2 - 64*B*b*c*d*e + 24*B*c**2*d**2)/4)/(24*c**4) + (-40*A*b**3*c*e**3 + 1
44*A*b**2*c**2*d*e**2 - 192*A*b*c**3*d**2*e + 128*A*c**4*d**3 + 35*B*b**4*e**3 -
120*B*b**3*c*d*e**2 + 144*B*b**2*c**2*d**2*e - 64*B*b*c**3*d**3)*atanh(sqrt(c)*
x/sqrt(b*x + c*x**2))/(64*c**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.496579, size = 287, normalized size = 0.94 \[ \frac{\sqrt{x} \left (\frac{\sqrt{x} (b+c x) \left (8 A c e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+10 b^2 c e^2 (36 d+7 e x)-8 b c^2 e \left (54 d^2+30 d e x+7 e^2 x^2\right )+48 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )\right )}{3 c^4}+\frac{\sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{c^{9/2}}\right )}{64 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*((Sqrt[x]*(b + c*x)*(8*A*c*e*(15*b^2*e^2 - 2*b*c*e*(27*d + 5*e*x) + 4*c
^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + B*(-105*b^3*e^3 + 10*b^2*c*e^2*(36*d + 7*e*
x) - 8*b*c^2*e*(54*d^2 + 30*d*e*x + 7*e^2*x^2) + 48*c^3*(4*d^3 + 6*d^2*e*x + 4*d
*e^2*x^2 + e^3*x^3))))/(3*c^4) + ((128*A*c^4*d^3 + 35*b^4*B*e^3 + 144*b^2*c^2*d*
e*(B*d + A*e) - 40*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*Sqrt[b
+ c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/c^(9/2)))/(64*Sqrt[x*(b + c*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.015, size = 646, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*d^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))/c^(1/2)+1/3*x^2/c*(c*x^2+b*x)^(1
/2)*A*e^3+x^2/c*(c*x^2+b*x)^(1/2)*B*d*e^2-5/12*b/c^2*x*(c*x^2+b*x)^(1/2)*A*e^3-5
/4*b/c^2*x*(c*x^2+b*x)^(1/2)*B*d*e^2+5/8*b^2/c^3*(c*x^2+b*x)^(1/2)*A*e^3+15/8*b^
2/c^3*(c*x^2+b*x)^(1/2)*B*d*e^2-5/16*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x)^(1/2))*A*e^3-15/16*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d
*e^2+3/2*x/c*(c*x^2+b*x)^(1/2)*A*d*e^2+3/2*x/c*(c*x^2+b*x)^(1/2)*B*d^2*e-9/4*b/c
^2*(c*x^2+b*x)^(1/2)*A*d*e^2-9/4*b/c^2*(c*x^2+b*x)^(1/2)*B*d^2*e+9/8*b^2/c^(5/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d*e^2+9/8*b^2/c^(5/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d^2*e+3/c*(c*x^2+b*x)^(1/2)*A*d^2*e+1/c*(c*x^2+b*
x)^(1/2)*B*d^3-3/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d^2*e-1
/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d^3+1/4*B*e^3*x^3/c*(c*
x^2+b*x)^(1/2)-7/24*B*e^3*b/c^2*x^2*(c*x^2+b*x)^(1/2)+35/96*B*e^3*b^2/c^3*x*(c*x
^2+b*x)^(1/2)-35/64*B*e^3*b^3/c^4*(c*x^2+b*x)^(1/2)+35/128*B*e^3*b^4/c^(9/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.350967, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (48 \, B c^{3} e^{3} x^{3} + 192 \, B c^{3} d^{3} - 144 \,{\left (3 \, B b c^{2} - 4 \, A c^{3}\right )} d^{2} e + 72 \,{\left (5 \, B b^{2} c - 6 \, A b c^{2}\right )} d e^{2} - 15 \,{\left (7 \, B b^{3} - 8 \, A b^{2} c\right )} e^{3} + 8 \,{\left (24 \, B c^{3} d e^{2} -{\left (7 \, B b c^{2} - 8 \, A c^{3}\right )} e^{3}\right )} x^{2} + 2 \,{\left (144 \, B c^{3} d^{2} e - 24 \,{\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} d e^{2} + 5 \,{\left (7 \, B b^{2} c - 8 \, A b c^{2}\right )} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 3 \,{\left (64 \,{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \,{\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \,{\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \,{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{9}{2}}}, \frac{{\left (48 \, B c^{3} e^{3} x^{3} + 192 \, B c^{3} d^{3} - 144 \,{\left (3 \, B b c^{2} - 4 \, A c^{3}\right )} d^{2} e + 72 \,{\left (5 \, B b^{2} c - 6 \, A b c^{2}\right )} d e^{2} - 15 \,{\left (7 \, B b^{3} - 8 \, A b^{2} c\right )} e^{3} + 8 \,{\left (24 \, B c^{3} d e^{2} -{\left (7 \, B b c^{2} - 8 \, A c^{3}\right )} e^{3}\right )} x^{2} + 2 \,{\left (144 \, B c^{3} d^{2} e - 24 \,{\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} d e^{2} + 5 \,{\left (7 \, B b^{2} c - 8 \, A b c^{2}\right )} e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (64 \,{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \,{\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \,{\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \,{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*B*c^3*e^3*x^3 + 192*B*c^3*d^3 - 144*(3*B*b*c^2 - 4*A*c^3)*d^2*e +
72*(5*B*b^2*c - 6*A*b*c^2)*d*e^2 - 15*(7*B*b^3 - 8*A*b^2*c)*e^3 + 8*(24*B*c^3*d*
e^2 - (7*B*b*c^2 - 8*A*c^3)*e^3)*x^2 + 2*(144*B*c^3*d^2*e - 24*(5*B*b*c^2 - 6*A*
c^3)*d*e^2 + 5*(7*B*b^2*c - 8*A*b*c^2)*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 3*(64
*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*A*b*c^3)*d^2*e + 24*(5*B*b^3*c -
6*A*b^2*c^2)*d*e^2 - 5*(7*B*b^4 - 8*A*b^3*c)*e^3)*log((2*c*x + b)*sqrt(c) - 2*sq
rt(c*x^2 + b*x)*c))/c^(9/2), 1/192*((48*B*c^3*e^3*x^3 + 192*B*c^3*d^3 - 144*(3*B
*b*c^2 - 4*A*c^3)*d^2*e + 72*(5*B*b^2*c - 6*A*b*c^2)*d*e^2 - 15*(7*B*b^3 - 8*A*b
^2*c)*e^3 + 8*(24*B*c^3*d*e^2 - (7*B*b*c^2 - 8*A*c^3)*e^3)*x^2 + 2*(144*B*c^3*d^
2*e - 24*(5*B*b*c^2 - 6*A*c^3)*d*e^2 + 5*(7*B*b^2*c - 8*A*b*c^2)*e^3)*x)*sqrt(c*
x^2 + b*x)*sqrt(-c) - 3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*A*b*c^
3)*d^2*e + 24*(5*B*b^3*c - 6*A*b^2*c^2)*d*e^2 - 5*(7*B*b^4 - 8*A*b^3*c)*e^3)*arc
tan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^4)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.332475, size = 420, normalized size = 1.38 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (\frac{6 \, B x e^{3}}{c} + \frac{24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac{144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac{3 \,{\left (64 \, B c^{3} d^{3} - 144 \, B b c^{2} d^{2} e + 192 \, A c^{3} d^{2} e + 120 \, B b^{2} c d e^{2} - 144 \, A b c^{2} d e^{2} - 35 \, B b^{3} e^{3} + 40 \, A b^{2} c e^{3}\right )}}{c^{4}}\right )} + \frac{{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 144 \, A b^{2} c^{2} d e^{2} - 35 \, B b^{4} e^{3} + 40 \, A b^{3} c e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*x*e^3/c + (24*B*c^3*d*e^2 - 7*B*b*c^2*e^3 + 8
*A*c^3*e^3)/c^4)*x + (144*B*c^3*d^2*e - 120*B*b*c^2*d*e^2 + 144*A*c^3*d*e^2 + 35
*B*b^2*c*e^3 - 40*A*b*c^2*e^3)/c^4)*x + 3*(64*B*c^3*d^3 - 144*B*b*c^2*d^2*e + 19
2*A*c^3*d^2*e + 120*B*b^2*c*d*e^2 - 144*A*b*c^2*d*e^2 - 35*B*b^3*e^3 + 40*A*b^2*
c*e^3)/c^4) + 1/128*(64*B*b*c^3*d^3 - 128*A*c^4*d^3 - 144*B*b^2*c^2*d^2*e + 192*
A*b*c^3*d^2*e + 120*B*b^3*c*d*e^2 - 144*A*b^2*c^2*d*e^2 - 35*B*b^4*e^3 + 40*A*b^
3*c*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)