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

Optimal. Leaf size=162 \[ -\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{64 c^3}+\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac{e \left (b x+c x^2\right )^{3/2} (d+e x)}{4 c} \]

[Out]

((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^3) +
(5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*c^2) + (e*(d + e*x)*(b*x + c*x^2)^(
3/2))/(4*c) - (b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(64*c^(7/2))

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Rubi [A]  time = 0.314726, antiderivative size = 162, 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 \[ -\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{64 c^3}+\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac{e \left (b x+c x^2\right )^{3/2} (d+e x)}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^3) +
(5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*c^2) + (e*(d + e*x)*(b*x + c*x^2)^(
3/2))/(4*c) - (b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(64*c^(7/2))

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Rubi in Sympy [A]  time = 28.5175, size = 155, normalized size = 0.96 \[ - \frac{b^{2} \left (5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} + \frac{e \left (d + e x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 c} - \frac{5 e \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2}} + \frac{\left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{64 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2*(5*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x**
2))/(64*c**(7/2)) + e*(d + e*x)*(b*x + c*x**2)**(3/2)/(4*c) - 5*e*(b*e - 2*c*d)*
(b*x + c*x**2)**(3/2)/(24*c**2) + (b + 2*c*x)*sqrt(b*x + c*x**2)*(5*b**2*e**2 -
16*b*c*d*e + 16*c**2*d**2)/(64*c**3)

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Mathematica [A]  time = 0.276846, size = 166, normalized size = 1.02 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^3 e^2-2 b^2 c e (24 d+5 e x)+8 b c^2 \left (6 d^2+4 d e x+e^2 x^2\right )+16 c^3 x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )-\frac{3 b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3*e^2 - 2*b^2*c*e*(24*d + 5*e*x) + 8*b*c^2*(6*
d^2 + 4*d*e*x + e^2*x^2) + 16*c^3*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) - (3*b^2*(16*
c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[
x]*Sqrt[b + c*x])))/(192*c^(7/2))

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Maple [B]  time = 0.011, size = 287, normalized size = 1.8 \[{\frac{{d}^{2}x}{2}\sqrt{c{x}^{2}+bx}}+{\frac{{d}^{2}b}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}{d}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{e}^{2}x}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}b}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{e}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}{e}^{2}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{e}^{2}{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{2\,de}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bdex}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}de}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{de{b}^{3}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^2*(c*x^2+b*x)^(1/2)*x+1/4*d^2/c*(c*x^2+b*x)^(1/2)*b-1/8*d^2*b^2/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/4*e^2*x*(c*x^2+b*x)^(3/2)/c-5/24*e^2*b
/c^2*(c*x^2+b*x)^(3/2)+5/32*e^2*b^2/c^2*(c*x^2+b*x)^(1/2)*x+5/64*e^2*b^3/c^3*(c*
x^2+b*x)^(1/2)-5/128*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2
/3*d*e*(c*x^2+b*x)^(3/2)/c-1/2*d*e*b/c*(c*x^2+b*x)^(1/2)*x-1/4*d*e*b^2/c^2*(c*x^
2+b*x)^(1/2)+1/8*d*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23619, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, c^{3} e^{2} x^{3} + 48 \, b c^{2} d^{2} - 48 \, b^{2} c d e + 15 \, b^{3} e^{2} + 8 \,{\left (16 \, c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 3 \,{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{7}{2}}}, \frac{{\left (48 \, c^{3} e^{2} x^{3} + 48 \, b c^{2} d^{2} - 48 \, b^{2} c d e + 15 \, b^{3} e^{2} + 8 \,{\left (16 \, c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*c^3*e^2*x^3 + 48*b*c^2*d^2 - 48*b^2*c*d*e + 15*b^3*e^2 + 8*(16*c^3
*d*e + b*c^2*e^2)*x^2 + 2*(48*c^3*d^2 + 16*b*c^2*d*e - 5*b^2*c*e^2)*x)*sqrt(c*x^
2 + b*x)*sqrt(c) + 3*(16*b^2*c^2*d^2 - 16*b^3*c*d*e + 5*b^4*e^2)*log((2*c*x + b)
*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/c^(7/2), 1/192*((48*c^3*e^2*x^3 + 48*b*c^2*d^
2 - 48*b^2*c*d*e + 15*b^3*e^2 + 8*(16*c^3*d*e + b*c^2*e^2)*x^2 + 2*(48*c^3*d^2 +
16*b*c^2*d*e - 5*b^2*c*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(16*b^2*c^2*d^2 -
16*b^3*c*d*e + 5*b^4*e^2)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c
^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))*(d + e*x)**2, x)

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GIAC/XCAS [A]  time = 0.2213, size = 232, normalized size = 1.43 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x e^{2} + \frac{16 \, c^{3} d e + b c^{2} e^{2}}{c^{3}}\right )} x + \frac{48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2}}{c^{3}}\right )} x + \frac{3 \,{\left (16 \, b c^{2} d^{2} - 16 \, b^{2} c d e + 5 \, b^{3} e^{2}\right )}}{c^{3}}\right )} + \frac{{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*x*e^2 + (16*c^3*d*e + b*c^2*e^2)/c^3)*x + (48*c
^3*d^2 + 16*b*c^2*d*e - 5*b^2*c*e^2)/c^3)*x + 3*(16*b*c^2*d^2 - 16*b^2*c*d*e + 5
*b^3*e^2)/c^3) + 1/128*(16*b^2*c^2*d^2 - 16*b^3*c*d*e + 5*b^4*e^2)*ln(abs(-2*(sq
rt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2)