3.285 \(\int (d+e x)^3 \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=210 \[ -\frac{b^2 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{128 c^4}+\frac{e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2+42 c e x (2 c d-b e)-150 b c d e+192 c^2 d^2\right )}{240 c^3}+\frac{e \left (b x+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^
2])/(128*c^4) + (e*(d + e*x)^2*(b*x + c*x^2)^(3/2))/(5*c) + (e*(192*c^2*d^2 - 15
0*b*c*d*e + 35*b^2*e^2 + 42*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(240*c^3)
- (b^2*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/S
qrt[b*x + c*x^2]])/(128*c^(9/2))

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Rubi [A]  time = 0.602971, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{b^2 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{128 c^4}+\frac{e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2+42 c e x (2 c d-b e)-150 b c d e+192 c^2 d^2\right )}{240 c^3}+\frac{e \left (b x+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^
2])/(128*c^4) + (e*(d + e*x)^2*(b*x + c*x^2)^(3/2))/(5*c) + (e*(192*c^2*d^2 - 15
0*b*c*d*e + 35*b^2*e^2 + 42*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(240*c^3)
- (b^2*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/S
qrt[b*x + c*x^2]])/(128*c^(9/2))

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Rubi in Sympy [A]  time = 39.4333, size = 209, normalized size = 1. \[ \frac{b^{2} \left (b e - 2 c d\right ) \left (7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{128 c^{\frac{9}{2}}} + \frac{e \left (d + e x\right )^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 c} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{35 b^{2} e^{2}}{4} - \frac{75 b c d e}{2} + 48 c^{2} d^{2} - \frac{21 c e x \left (b e - 2 c d\right )}{2}\right )}{60 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}} \left (7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{128 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(b*e - 2*c*d)*(7*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*atanh(sqrt(c)*x/sqr
t(b*x + c*x**2))/(128*c**(9/2)) + e*(d + e*x)**2*(b*x + c*x**2)**(3/2)/(5*c) + e
*(b*x + c*x**2)**(3/2)*(35*b**2*e**2/4 - 75*b*c*d*e/2 + 48*c**2*d**2 - 21*c*e*x*
(b*e - 2*c*d)/2)/(60*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*sqrt(b*x + c*x**2)*(7*b**
2*e**2 - 16*b*c*d*e + 16*c**2*d**2)/(128*c**4)

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Mathematica [A]  time = 0.397606, size = 232, normalized size = 1.1 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^2 \left (7 b^3 e^3-30 b^2 c d e^2+48 b c^2 d^2 e-32 c^3 d^3\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-105 b^4 e^3+10 b^3 c e^2 (45 d+7 e x)-4 b^2 c^2 e \left (180 d^2+75 d e x+14 e^2 x^2\right )+48 b c^3 \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )+96 c^4 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )\right )}{1920 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^4*e^3 + 10*b^3*c*e^2*(45*d + 7*e*x) - 4*b^2*
c^2*e*(180*d^2 + 75*d*e*x + 14*e^2*x^2) + 48*b*c^3*(10*d^3 + 10*d^2*e*x + 5*d*e^
2*x^2 + e^3*x^3) + 96*c^4*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)) +
(15*b^2*(-32*c^3*d^3 + 48*b*c^2*d^2*e - 30*b^2*c*d*e^2 + 7*b^3*e^3)*Log[c*Sqrt[x
] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(1920*c^(9/2))

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Maple [B]  time = 0.014, size = 444, normalized size = 2.1 \[{\frac{{d}^{3}x}{2}\sqrt{c{x}^{2}+bx}}+{\frac{{d}^{3}b}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{d}^{3}{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{e}^{3}{x}^{2}}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,b{e}^{3}x}{40\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{e}^{3}{b}^{2}}{48\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{3}{e}^{3}x}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{e}^{3}{b}^{4}}{128\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{e}^{3}{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{3\,d{e}^{2}x}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bd{e}^{2}}{8\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{b}^{2}d{e}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{15\,{b}^{3}d{e}^{2}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{15\,d{e}^{2}{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{d}^{2}e}{c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,b{d}^{2}ex}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{d}^{2}e{b}^{2}}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{d}^{2}e{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^3*(c*x^2+b*x)^(1/2)*x+1/4*d^3/c*(c*x^2+b*x)^(1/2)*b-1/8*d^3*b^2/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/5*e^3*x^2*(c*x^2+b*x)^(3/2)/c-7/40*e^3
*b/c^2*x*(c*x^2+b*x)^(3/2)+7/48*e^3*b^2/c^3*(c*x^2+b*x)^(3/2)-7/64*e^3*b^3/c^3*(
c*x^2+b*x)^(1/2)*x-7/128*e^3*b^4/c^4*(c*x^2+b*x)^(1/2)+7/256*e^3*b^5/c^(9/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/4*d*e^2*x*(c*x^2+b*x)^(3/2)/c-5/8*d*e^2
*b/c^2*(c*x^2+b*x)^(3/2)+15/32*d*e^2*b^2/c^2*(c*x^2+b*x)^(1/2)*x+15/64*d*e^2*b^3
/c^3*(c*x^2+b*x)^(1/2)-15/128*d*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x)^(1/2))+d^2*e*(c*x^2+b*x)^(3/2)/c-3/4*d^2*e*b/c*(c*x^2+b*x)^(1/2)*x-3/8*d^2*e*
b^2/c^2*(c*x^2+b*x)^(1/2)+3/16*d^2*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x)^(1/2))

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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(sqrt(c*x^2 + b*x)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236028, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, c^{4} e^{3} x^{4} + 480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 450 \, b^{3} c d e^{2} - 105 \, b^{4} e^{3} + 48 \,{\left (30 \, c^{4} d e^{2} + b c^{3} e^{3}\right )} x^{3} + 8 \,{\left (240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3}\right )} x^{2} + 10 \,{\left (96 \, c^{4} d^{3} + 48 \, b c^{3} d^{2} e - 30 \, b^{2} c^{2} d e^{2} + 7 \, b^{3} c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 15 \,{\left (32 \, b^{2} c^{3} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 30 \, b^{4} c d e^{2} - 7 \, b^{5} e^{3}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{3840 \, c^{\frac{9}{2}}}, \frac{{\left (384 \, c^{4} e^{3} x^{4} + 480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 450 \, b^{3} c d e^{2} - 105 \, b^{4} e^{3} + 48 \,{\left (30 \, c^{4} d e^{2} + b c^{3} e^{3}\right )} x^{3} + 8 \,{\left (240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3}\right )} x^{2} + 10 \,{\left (96 \, c^{4} d^{3} + 48 \, b c^{3} d^{2} e - 30 \, b^{2} c^{2} d e^{2} + 7 \, b^{3} c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 15 \,{\left (32 \, b^{2} c^{3} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 30 \, b^{4} c d e^{2} - 7 \, b^{5} e^{3}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{1920 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(384*c^4*e^3*x^4 + 480*b*c^3*d^3 - 720*b^2*c^2*d^2*e + 450*b^3*c*d*e^
2 - 105*b^4*e^3 + 48*(30*c^4*d*e^2 + b*c^3*e^3)*x^3 + 8*(240*c^4*d^2*e + 30*b*c^
3*d*e^2 - 7*b^2*c^2*e^3)*x^2 + 10*(96*c^4*d^3 + 48*b*c^3*d^2*e - 30*b^2*c^2*d*e^
2 + 7*b^3*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15*(32*b^2*c^3*d^3 - 48*b^3*c^2*
d^2*e + 30*b^4*c*d*e^2 - 7*b^5*e^3)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x
)*c))/c^(9/2), 1/1920*((384*c^4*e^3*x^4 + 480*b*c^3*d^3 - 720*b^2*c^2*d^2*e + 45
0*b^3*c*d*e^2 - 105*b^4*e^3 + 48*(30*c^4*d*e^2 + b*c^3*e^3)*x^3 + 8*(240*c^4*d^2
*e + 30*b*c^3*d*e^2 - 7*b^2*c^2*e^3)*x^2 + 10*(96*c^4*d^3 + 48*b*c^3*d^2*e - 30*
b^2*c^2*d*e^2 + 7*b^3*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 15*(32*b^2*c^3*d^3
- 48*b^3*c^2*d^2*e + 30*b^4*c*d*e^2 - 7*b^5*e^3)*arctan(sqrt(c*x^2 + b*x)*sqrt(-
c)/(c*x)))/(sqrt(-c)*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221498, size = 338, normalized size = 1.61 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x e^{3} + \frac{30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac{240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3}}{c^{4}}\right )} x + \frac{5 \,{\left (96 \, c^{4} d^{3} + 48 \, b c^{3} d^{2} e - 30 \, b^{2} c^{2} d e^{2} + 7 \, b^{3} c e^{3}\right )}}{c^{4}}\right )} x + \frac{15 \,{\left (32 \, b c^{3} d^{3} - 48 \, b^{2} c^{2} d^{2} e + 30 \, b^{3} c d e^{2} - 7 \, b^{4} e^{3}\right )}}{c^{4}}\right )} + \frac{{\left (32 \, b^{2} c^{3} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 30 \, b^{4} c d e^{2} - 7 \, b^{5} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*x^2 + b*x)*(2*(4*(6*(8*x*e^3 + (30*c^4*d*e^2 + b*c^3*e^3)/c^4)*x +
(240*c^4*d^2*e + 30*b*c^3*d*e^2 - 7*b^2*c^2*e^3)/c^4)*x + 5*(96*c^4*d^3 + 48*b*
c^3*d^2*e - 30*b^2*c^2*d*e^2 + 7*b^3*c*e^3)/c^4)*x + 15*(32*b*c^3*d^3 - 48*b^2*c
^2*d^2*e + 30*b^3*c*d*e^2 - 7*b^4*e^3)/c^4) + 1/256*(32*b^2*c^3*d^3 - 48*b^3*c^2
*d^2*e + 30*b^4*c*d*e^2 - 7*b^5*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*s
qrt(c) - b))/c^(9/2)