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

Optimal. Leaf size=99 \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c} \]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c^2) + (e*(b*x + c*x^2)^(3/2))/
(3*c) - (b^2*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(5/2))

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Rubi [A]  time = 0.106552, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c^2) + (e*(b*x + c*x^2)^(3/2))/
(3*c) - (b^2*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(5/2))

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Rubi in Sympy [A]  time = 11.3516, size = 87, normalized size = 0.88 \[ \frac{b^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{8 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(b*e - 2*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*c**(5/2)) + e*(b*x + c
*x**2)**(3/2)/(3*c) - (b + 2*c*x)*(b*e - 2*c*d)*sqrt(b*x + c*x**2)/(8*c**2)

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Mathematica [A]  time = 0.167974, size = 110, normalized size = 1.11 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-3 b^2 e+2 b c (3 d+e x)+4 c^2 x (3 d+2 e x)\right )+\frac{3 b^2 (b e-2 c d) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{24 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3*b^2*e + 2*b*c*(3*d + e*x) + 4*c^2*x*(3*d + 2*e*x
)) + (3*b^2*(-2*c*d + b*e)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt
[b + c*x])))/(24*c^(5/2))

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Maple [A]  time = 0.009, size = 157, normalized size = 1.6 \[{\frac{dx}{2}\sqrt{c{x}^{2}+bx}}+{\frac{bd}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}d}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bex}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{e{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*(c*x^2+b*x)^(1/2)*x+1/4*d/c*(c*x^2+b*x)^(1/2)*b-1/8*d*b^2/c^(3/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/3*e*(c*x^2+b*x)^(3/2)/c-1/4*e*b/c*(c*x^2+b*x
)^(1/2)*x-1/8*e*b^2/c^2*(c*x^2+b*x)^(1/2)+1/16*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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229839, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, c^{2} e x^{2} + 6 \, b c d - 3 \, b^{2} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{48 \, c^{\frac{5}{2}}}, \frac{{\left (8 \, c^{2} e x^{2} + 6 \, b c d - 3 \, b^{2} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{24 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(8*c^2*e*x^2 + 6*b*c*d - 3*b^2*e + 2*(6*c^2*d + b*c*e)*x)*sqrt(c*x^2 +
b*x)*sqrt(c) - 3*(2*b^2*c*d - b^3*e)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*
x)*c))/c^(5/2), 1/24*((8*c^2*e*x^2 + 6*b*c*d - 3*b^2*e + 2*(6*c^2*d + b*c*e)*x)*
sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(2*b^2*c*d - b^3*e)*arctan(sqrt(c*x^2 + b*x)*sqrt
(-c)/(c*x)))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221164, size = 146, normalized size = 1.47 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, x e + \frac{6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac{3 \,{\left (2 \, b c d - b^{2} e\right )}}{c^{2}}\right )} + \frac{{\left (2 \, b^{2} c d - b^{3} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*(4*x*e + (6*c^2*d + b*c*e)/c^2)*x + 3*(2*b*c*d - b^2*e
)/c^2) + 1/16*(2*b^2*c*d - b^3*e)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt
(c) - b))/c^(5/2)