3.323 \(\int \frac{(d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 e \sqrt{b x+c x^2} (2 c d-b e)}{b^2 c}-\frac{2 (d+e x) (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.172165, antiderivative size = 101, 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 \[ \frac{2 e \sqrt{b x+c x^2} (2 c d-b e)}{b^2 c}-\frac{2 (d+e x) (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.4339, size = 94, normalized size = 0.93 \[ \frac{2 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{2 \left (d + e x\right ) \left (b d - x \left (b e - 2 c d\right )\right )}{b^{2} \sqrt{b x + c x^{2}}} - \frac{2 e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.125838, size = 102, normalized size = 1.01 \[ \frac{2 b^2 e^2 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )-2 \sqrt{c} \left (b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )}{b^2 c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*(2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x)) + 2*b^2*e^2*Sqrt[x]*Sq
rt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(b^2*c^(3/2)*Sqrt[x*(b + c*x
)])

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Maple [A]  time = 0.009, size = 97, normalized size = 1. \[ -2\,{\frac{{d}^{2} \left ( 2\,cx+b \right ) }{{b}^{2}\sqrt{c{x}^{2}+bx}}}-2\,{\frac{{e}^{2}x}{c\sqrt{c{x}^{2}+bx}}}+{{e}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+4\,{\frac{dex}{b\sqrt{c{x}^{2}+bx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d^2*(2*c*x+b)/b^2/(c*x^2+b*x)^(1/2)-2*e^2/c/(c*x^2+b*x)^(1/2)*x+e^2/c^(3/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+4*d*e/b/(c*x^2+b*x)^(1/2)*x

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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/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233149, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c x^{2} + b x} b^{2} e^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (b c d^{2} +{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x\right )} \sqrt{c}}{\sqrt{c x^{2} + b x} b^{2} c^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{c x^{2} + b x} b^{2} e^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (b c d^{2} +{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x\right )} \sqrt{-c}\right )}}{\sqrt{c x^{2} + b x} b^{2} \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(c*x^2 + b*x)*b^2*e^2*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) - 2
*(b*c*d^2 + (2*c^2*d^2 - 2*b*c*d*e + b^2*e^2)*x)*sqrt(c))/(sqrt(c*x^2 + b*x)*b^2
*c^(3/2)), 2*(sqrt(c*x^2 + b*x)*b^2*e^2*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))
- (b*c*d^2 + (2*c^2*d^2 - 2*b*c*d*e + b^2*e^2)*x)*sqrt(-c))/(sqrt(c*x^2 + b*x)*
b^2*sqrt(-c)*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.231579, size = 120, normalized size = 1.19 \[ -\frac{2 \,{\left (\frac{d^{2}}{b} + \frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x}{b^{2} c}\right )}}{\sqrt{c x^{2} + b x}} - \frac{e^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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