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

Optimal. Leaf size=67 \[ \frac{1}{2} d x \sqrt{a+c x^2}+\frac{a d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.0616367, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{2} d x \sqrt{a+c x^2}+\frac{a d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.84611, size = 58, normalized size = 0.87 \[ \frac{a d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c}} + \frac{d x \sqrt{a + c x^{2}}}{2} + \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.052258, size = 67, normalized size = 1. \[ \frac{\sqrt{a+c x^2} (2 a e+c x (3 d+2 e x))+3 a \sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(2*a*e + c*x*(3*d + 2*e*x)) + 3*a*Sqrt[c]*d*Log[c*x + Sqrt[c]*S
qrt[a + c*x^2]])/(6*c)

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Maple [A]  time = 0.007, size = 53, normalized size = 0.8 \[{\frac{dx}{2}\sqrt{c{x}^{2}+a}}+{\frac{ad}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a)^(1/2),x)

[Out]

1/2*d*x*(c*x^2+a)^(1/2)+1/2*d*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/3*e*(c*x
^2+a)^(3/2)/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(sqrt(c*x^2 + a)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2561, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a c d \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (2 \, c e x^{2} + 3 \, c d x + 2 \, a e\right )} \sqrt{c x^{2} + a} \sqrt{c}}{12 \, c^{\frac{3}{2}}}, \frac{3 \, a c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, c e x^{2} + 3 \, c d x + 2 \, a e\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{6 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*a*c*d*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(2*c*e*x^
2 + 3*c*d*x + 2*a*e)*sqrt(c*x^2 + a)*sqrt(c))/c^(3/2), 1/6*(3*a*c*d*arctan(sqrt(
-c)*x/sqrt(c*x^2 + a)) + (2*c*e*x^2 + 3*c*d*x + 2*a*e)*sqrt(c*x^2 + a)*sqrt(-c))
/(sqrt(-c)*c)]

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Sympy [A]  time = 7.70669, size = 70, normalized size = 1.04 \[ \frac{\sqrt{a} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*d*x*sqrt(1 + c*x**2/a)/2 + a*d*asinh(sqrt(c)*x/sqrt(a))/(2*sqrt(c)) + e*
Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True))

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GIAC/XCAS [A]  time = 0.214083, size = 77, normalized size = 1.15 \[ -\frac{a d{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} + \frac{1}{6} \, \sqrt{c x^{2} + a}{\left ({\left (2 \, x e + 3 \, d\right )} x + \frac{2 \, a e}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a*d*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/6*sqrt(c*x^2 + a)*((2
*x*e + 3*d)*x + 2*a*e/c)