3.549 \(\int (2+3 x) \sqrt{-5+7 x^2} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{7} \left (7 x^2-5\right )^{3/2}+x \sqrt{7 x^2-5}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{7 x^2-5}}\right )}{\sqrt{7}} \]

[Out]

x*Sqrt[-5 + 7*x^2] + (-5 + 7*x^2)^(3/2)/7 - (5*ArcTanh[(Sqrt[7]*x)/Sqrt[-5 + 7*x
^2]])/Sqrt[7]

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Rubi [A]  time = 0.0510203, antiderivative size = 55, 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 \[ \frac{1}{7} \left (7 x^2-5\right )^{3/2}+x \sqrt{7 x^2-5}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{7 x^2-5}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)*Sqrt[-5 + 7*x^2],x]

[Out]

x*Sqrt[-5 + 7*x^2] + (-5 + 7*x^2)^(3/2)/7 - (5*ArcTanh[(Sqrt[7]*x)/Sqrt[-5 + 7*x
^2]])/Sqrt[7]

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Rubi in Sympy [A]  time = 3.82091, size = 49, normalized size = 0.89 \[ x \sqrt{7 x^{2} - 5} + \frac{\left (7 x^{2} - 5\right )^{\frac{3}{2}}}{7} - \frac{5 \sqrt{7} \operatorname{atanh}{\left (\frac{\sqrt{7} x}{\sqrt{7 x^{2} - 5}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)*(7*x**2-5)**(1/2),x)

[Out]

x*sqrt(7*x**2 - 5) + (7*x**2 - 5)**(3/2)/7 - 5*sqrt(7)*atanh(sqrt(7)*x/sqrt(7*x*
*2 - 5))/7

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Mathematica [A]  time = 0.034554, size = 50, normalized size = 0.91 \[ \left (x^2+x-\frac{5}{7}\right ) \sqrt{7 x^2-5}-\frac{5 \log \left (\sqrt{7} \sqrt{7 x^2-5}+7 x\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)*Sqrt[-5 + 7*x^2],x]

[Out]

(-5/7 + x + x^2)*Sqrt[-5 + 7*x^2] - (5*Log[7*x + Sqrt[7]*Sqrt[-5 + 7*x^2]])/Sqrt
[7]

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \[ x\sqrt{7\,{x}^{2}-5}-{\frac{5\,\sqrt{7}}{7}\ln \left ( x\sqrt{7}+\sqrt{7\,{x}^{2}-5} \right ) }+{\frac{1}{7} \left ( 7\,{x}^{2}-5 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)*(7*x^2-5)^(1/2),x)

[Out]

x*(7*x^2-5)^(1/2)-5/7*ln(x*7^(1/2)+(7*x^2-5)^(1/2))*7^(1/2)+1/7*(7*x^2-5)^(3/2)

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Maxima [A]  time = 0.797274, size = 63, normalized size = 1.15 \[ \frac{1}{7} \,{\left (7 \, x^{2} - 5\right )}^{\frac{3}{2}} + \sqrt{7 \, x^{2} - 5} x - \frac{5}{7} \, \sqrt{7} \log \left (2 \, \sqrt{7} \sqrt{7 \, x^{2} - 5} + 14 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(7*x^2 - 5)*(3*x + 2),x, algorithm="maxima")

[Out]

1/7*(7*x^2 - 5)^(3/2) + sqrt(7*x^2 - 5)*x - 5/7*sqrt(7)*log(2*sqrt(7)*sqrt(7*x^2
- 5) + 14*x)

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Fricas [A]  time = 0.222607, size = 77, normalized size = 1.4 \[ \frac{1}{98} \, \sqrt{7}{\left (2 \, \sqrt{7}{\left (7 \, x^{2} + 7 \, x - 5\right )} \sqrt{7 \, x^{2} - 5} + 35 \, \log \left (\sqrt{7}{\left (14 \, x^{2} - 5\right )} - 14 \, \sqrt{7 \, x^{2} - 5} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(7*x^2 - 5)*(3*x + 2),x, algorithm="fricas")

[Out]

1/98*sqrt(7)*(2*sqrt(7)*(7*x^2 + 7*x - 5)*sqrt(7*x^2 - 5) + 35*log(sqrt(7)*(14*x
^2 - 5) - 14*sqrt(7*x^2 - 5)*x))

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Sympy [A]  time = 0.685909, size = 56, normalized size = 1.02 \[ x^{2} \sqrt{7 x^{2} - 5} + x \sqrt{7 x^{2} - 5} - \frac{5 \sqrt{7 x^{2} - 5}}{7} - \frac{5 \sqrt{7} \operatorname{acosh}{\left (\frac{\sqrt{35} x}{5} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)*(7*x**2-5)**(1/2),x)

[Out]

x**2*sqrt(7*x**2 - 5) + x*sqrt(7*x**2 - 5) - 5*sqrt(7*x**2 - 5)/7 - 5*sqrt(7)*ac
osh(sqrt(35)*x/5)/7

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GIAC/XCAS [A]  time = 0.217511, size = 58, normalized size = 1.05 \[ \frac{1}{7} \,{\left (7 \,{\left (x + 1\right )} x - 5\right )} \sqrt{7 \, x^{2} - 5} + \frac{5}{7} \, \sqrt{7}{\rm ln}\left ({\left | -\sqrt{7} x + \sqrt{7 \, x^{2} - 5} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(7*x^2 - 5)*(3*x + 2),x, algorithm="giac")

[Out]

1/7*(7*(x + 1)*x - 5)*sqrt(7*x^2 - 5) + 5/7*sqrt(7)*ln(abs(-sqrt(7)*x + sqrt(7*x
^2 - 5)))