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

Optimal. Leaf size=110 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c} \]

[Out]

(3*e*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sqrt[b*x + c*x^2])/
(2*c) + ((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(4*c^(5/2))

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Rubi [A]  time = 0.206391, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sqrt[b*x + c*x^2])/
(2*c) + ((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(4*c^(5/2))

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Rubi in Sympy [A]  time = 22.9823, size = 102, normalized size = 0.93 \[ \frac{e \left (d + e x\right ) \sqrt{b x + c x^{2}}}{2 c} - \frac{3 e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{4 c^{2}} + \frac{\left (3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{5}{2}}} \]

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]

e*(d + e*x)*sqrt(b*x + c*x**2)/(2*c) - 3*e*(b*e - 2*c*d)*sqrt(b*x + c*x**2)/(4*c
**2) + (3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2
))/(4*c**(5/2))

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Mathematica [A]  time = 0.147584, size = 110, normalized size = 1. \[ \frac{\sqrt{x} \sqrt{b+c x} \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+\sqrt{c} e x (b+c x) (-3 b e+8 c d+2 c e x)}{4 c^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*e*x*(b + c*x)*(8*c*d - 3*b*e + 2*c*e*x) + (8*c^2*d^2 - 8*b*c*d*e + 3*b^
2*e^2)*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(4*c^(5/2)*
Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.011, size = 158, normalized size = 1.4 \[{{d}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{e}^{2}b}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+bx}}{c}}-{bde\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

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]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249755, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, c e^{2} x + 8 \, c d e - 3 \, b e^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c} +{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{8 \, c^{\frac{5}{2}}}, \frac{{\left (2 \, c e^{2} x + 8 \, c d e - 3 \, b e^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c} +{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{4 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*(2*c*e^2*x + 8*c*d*e - 3*b*e^2)*sqrt(c*x^2 + b*x)*sqrt(c) + (8*c^2*d^2 -
8*b*c*d*e + 3*b^2*e^2)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(5/2
), 1/4*((2*c*e^2*x + 8*c*d*e - 3*b*e^2)*sqrt(c*x^2 + b*x)*sqrt(-c) + (8*c^2*d^2
- 8*b*c*d*e + 3*b^2*e^2)*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. \[ \int \frac{\left (d + e x\right )^{2}}{\sqrt{x \left (b + c x\right )}}\, dx \]

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((d + e*x)**2/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.234101, size = 131, normalized size = 1.19 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (\frac{2 \, x e^{2}}{c} + \frac{8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x)*(2*x*e^2/c + (8*c*d*e - 3*b*e^2)/c^2) - 1/8*(8*c^2*d^2 - 8
*b*c*d*e + 3*b^2*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^
(5/2)