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

Optimal. Leaf size=86 \[ \frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.119989, antiderivative size = 86, 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 \[ \frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.6363, size = 75, normalized size = 0.87 \[ \frac{3 d e \sqrt{a + c x^{2}}}{2 c} + \frac{e \sqrt{a + c x^{2}} \left (d + e x\right )}{2 c} - \frac{\left (a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0647742, size = 71, normalized size = 0.83 \[ \frac{\left (2 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} e \sqrt{a+c x^2} (4 d+e x)}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*e*(4*d + e*x)*Sqrt[a + c*x^2] + (2*c*d^2 - a*e^2)*Log[c*x + Sqrt[c]*Sqr
t[a + c*x^2]])/(2*c^(3/2))

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Maple [A]  time = 0.01, size = 84, normalized size = 1. \[{{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{a{e}^{2}}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+a}}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*ln(c^(1/2)*x+(c*x^2+a)^(1/2))/c^(1/2)+1/2*e^2*x/c*(c*x^2+a)^(1/2)-1/2*e^2*a/
c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+2*d*e*(c*x^2+a)^(1/2)/c

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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 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241437, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (e^{2} x + 4 \, d e\right )} \sqrt{c x^{2} + a} \sqrt{c} -{\left (2 \, c d^{2} - a e^{2}\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{4 \, c^{\frac{3}{2}}}, \frac{{\left (e^{2} x + 4 \, d e\right )} \sqrt{c x^{2} + a} \sqrt{-c} +{\left (2 \, c d^{2} - a e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{2 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*(e^2*x + 4*d*e)*sqrt(c*x^2 + a)*sqrt(c) - (2*c*d^2 - a*e^2)*log(2*sqrt(c
*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/c^(3/2), 1/2*((e^2*x + 4*d*e)*sqrt(c*x^2
+ a)*sqrt(-c) + (2*c*d^2 - a*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)
*c)]

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Sympy [A]  time = 9.45749, size = 158, normalized size = 1.84 \[ \frac{\sqrt{a} e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{a e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + d^{2} \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + 2 d e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*e**2*x*sqrt(1 + c*x**2/a)/(2*c) - a*e**2*asinh(sqrt(c)*x/sqrt(a))/(2*c**
(3/2)) + d**2*Piecewise((sqrt(-a/c)*asin(x*sqrt(-c/a))/sqrt(a), (a > 0) & (c < 0
)), (sqrt(a/c)*asinh(x*sqrt(c/a))/sqrt(a), (a > 0) & (c > 0)), (sqrt(-a/c)*acosh
(x*sqrt(-c/a))/sqrt(-a), (c > 0) & (a < 0))) + 2*d*e*Piecewise((x**2/(2*sqrt(a))
, Eq(c, 0)), (sqrt(a + c*x**2)/c, True))

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GIAC/XCAS [A]  time = 0.218409, size = 85, normalized size = 0.99 \[ \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (\frac{x e^{2}}{c} + \frac{4 \, d e}{c}\right )} - \frac{{\left (2 \, c d^{2} - a e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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