3.578 \(\int \frac{(a+b x)^2}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)+\frac{3}{2} a b \sqrt{x^2+1}+\frac{1}{2} b \sqrt{x^2+1} (a+b x) \]

[Out]

(3*a*b*Sqrt[1 + x^2])/2 + (b*(a + b*x)*Sqrt[1 + x^2])/2 + ((2*a^2 - b^2)*ArcSinh
[x])/2

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Rubi [A]  time = 0.0655453, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)+\frac{3}{2} a b \sqrt{x^2+1}+\frac{1}{2} b \sqrt{x^2+1} (a+b x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/Sqrt[1 + x^2],x]

[Out]

(3*a*b*Sqrt[1 + x^2])/2 + (b*(a + b*x)*Sqrt[1 + x^2])/2 + ((2*a^2 - b^2)*ArcSinh
[x])/2

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Rubi in Sympy [A]  time = 9.71251, size = 42, normalized size = 0.81 \[ \frac{3 a b \sqrt{x^{2} + 1}}{2} + \frac{b \left (a + b x\right ) \sqrt{x^{2} + 1}}{2} + \left (a^{2} - \frac{b^{2}}{2}\right ) \operatorname{asinh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/(x**2+1)**(1/2),x)

[Out]

3*a*b*sqrt(x**2 + 1)/2 + b*(a + b*x)*sqrt(x**2 + 1)/2 + (a**2 - b**2/2)*asinh(x)

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Mathematica [A]  time = 0.039404, size = 36, normalized size = 0.69 \[ \left (a^2-\frac{b^2}{2}\right ) \sinh ^{-1}(x)+\frac{1}{2} b \sqrt{x^2+1} (4 a+b x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2/Sqrt[1 + x^2],x]

[Out]

(b*(4*a + b*x)*Sqrt[1 + x^2])/2 + (a^2 - b^2/2)*ArcSinh[x]

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Maple [A]  time = 0.01, size = 38, normalized size = 0.7 \[{a}^{2}{\it Arcsinh} \left ( x \right ) +{b}^{2} \left ({\frac{x}{2}\sqrt{{x}^{2}+1}}-{\frac{{\it Arcsinh} \left ( x \right ) }{2}} \right ) +2\,ab\sqrt{{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(x^2+1)^(1/2),x)

[Out]

a^2*arcsinh(x)+b^2*(1/2*x*(x^2+1)^(1/2)-1/2*arcsinh(x))+2*a*b*(x^2+1)^(1/2)

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Maxima [A]  time = 0.791892, size = 51, normalized size = 0.98 \[ \frac{1}{2} \, \sqrt{x^{2} + 1} b^{2} x + a^{2} \operatorname{arsinh}\left (x\right ) - \frac{1}{2} \, b^{2} \operatorname{arsinh}\left (x\right ) + 2 \, \sqrt{x^{2} + 1} a b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 + 1)*b^2*x + a^2*arcsinh(x) - 1/2*b^2*arcsinh(x) + 2*sqrt(x^2 + 1)*
a*b

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Fricas [A]  time = 0.220081, size = 196, normalized size = 3.77 \[ -\frac{2 \, b^{2} x^{4} + 8 \, a b x^{3} + 2 \, b^{2} x^{2} + 8 \, a b x +{\left (2 \,{\left (2 \, a^{2} - b^{2}\right )} x^{2} - 2 \,{\left (2 \, a^{2} - b^{2}\right )} \sqrt{x^{2} + 1} x + 2 \, a^{2} - b^{2}\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) -{\left (2 \, b^{2} x^{3} + 8 \, a b x^{2} + b^{2} x + 4 \, a b\right )} \sqrt{x^{2} + 1}}{2 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

-1/2*(2*b^2*x^4 + 8*a*b*x^3 + 2*b^2*x^2 + 8*a*b*x + (2*(2*a^2 - b^2)*x^2 - 2*(2*
a^2 - b^2)*sqrt(x^2 + 1)*x + 2*a^2 - b^2)*log(-x + sqrt(x^2 + 1)) - (2*b^2*x^3 +
8*a*b*x^2 + b^2*x + 4*a*b)*sqrt(x^2 + 1))/(2*x^2 - 2*sqrt(x^2 + 1)*x + 1)

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Sympy [A]  time = 0.605186, size = 42, normalized size = 0.81 \[ a^{2} \operatorname{asinh}{\left (x \right )} + 2 a b \sqrt{x^{2} + 1} + \frac{b^{2} x \sqrt{x^{2} + 1}}{2} - \frac{b^{2} \operatorname{asinh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/(x**2+1)**(1/2),x)

[Out]

a**2*asinh(x) + 2*a*b*sqrt(x**2 + 1) + b**2*x*sqrt(x**2 + 1)/2 - b**2*asinh(x)/2

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GIAC/XCAS [A]  time = 0.214587, size = 61, normalized size = 1.17 \[ -\frac{1}{2} \,{\left (2 \, a^{2} - b^{2}\right )}{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) + \frac{1}{2} \,{\left (b^{2} x + 4 \, a b\right )} \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

-1/2*(2*a^2 - b^2)*ln(-x + sqrt(x^2 + 1)) + 1/2*(b^2*x + 4*a*b)*sqrt(x^2 + 1)