3.4 \(\int (A+B x) \sqrt{a+b x^2} \, dx\)

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

[Out]

(A*x*Sqrt[a + b*x^2])/2 + (B*(a + b*x^2)^(3/2))/(3*b) + (a*A*ArcTanh[(Sqrt[b]*x)
/Sqrt[a + b*x^2]])/(2*Sqrt[b])

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Rubi [A]  time = 0.0639736, 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 \[ \frac{1}{2} A x \sqrt{a+b x^2}+\frac{a A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b}}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*Sqrt[a + b*x^2],x]

[Out]

(A*x*Sqrt[a + b*x^2])/2 + (B*(a + b*x^2)^(3/2))/(3*b) + (a*A*ArcTanh[(Sqrt[b]*x)
/Sqrt[a + b*x^2]])/(2*Sqrt[b])

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Rubi in Sympy [A]  time = 7.01506, size = 58, normalized size = 0.87 \[ \frac{A a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 \sqrt{b}} + \frac{A x \sqrt{a + b x^{2}}}{2} + \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2*sqrt(b)) + A*x*sqrt(a + b*x**2)/2 + B*(
a + b*x**2)**(3/2)/(3*b)

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Mathematica [A]  time = 0.0584011, size = 67, normalized size = 1. \[ \frac{\sqrt{a+b x^2} (2 a B+b x (3 A+2 B x))+3 a A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{6 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(2*a*B + b*x*(3*A + 2*B*x)) + 3*a*A*Sqrt[b]*Log[b*x + Sqrt[b]*S
qrt[a + b*x^2]])/(6*b)

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Maple [A]  time = 0.007, size = 53, normalized size = 0.8 \[{\frac{Ax}{2}\sqrt{b{x}^{2}+a}}+{\frac{Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{B}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x^2+a)^(1/2),x)

[Out]

1/2*A*x*(b*x^2+a)^(1/2)+1/2*A*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/3*B*(b*x
^2+a)^(3/2)/b

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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(b*x^2 + a)*(B*x + A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262778, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, A a b \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, B b x^{2} + 3 \, A b x + 2 \, B a\right )} \sqrt{b x^{2} + a} \sqrt{b}}{12 \, b^{\frac{3}{2}}}, \frac{3 \, A a b \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, B b x^{2} + 3 \, A b x + 2 \, B a\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{6 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*A*a*b*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(2*B*b*x^
2 + 3*A*b*x + 2*B*a)*sqrt(b*x^2 + a)*sqrt(b))/b^(3/2), 1/6*(3*A*a*b*arctan(sqrt(
-b)*x/sqrt(b*x^2 + a)) + (2*B*b*x^2 + 3*A*b*x + 2*B*a)*sqrt(b*x^2 + a)*sqrt(-b))
/(sqrt(-b)*b)]

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Sympy [A]  time = 3.74226, size = 70, normalized size = 1.04 \[ \frac{A \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + B \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*sqrt(a)*x*sqrt(1 + b*x**2/a)/2 + A*a*asinh(sqrt(b)*x/sqrt(a))/(2*sqrt(b)) + B*
Piecewise((sqrt(a)*x**2/2, Eq(b, 0)), ((a + b*x**2)**(3/2)/(3*b), True))

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GIAC/XCAS [A]  time = 0.230956, size = 74, normalized size = 1.1 \[ -\frac{A a{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, \sqrt{b}} + \frac{1}{6} \, \sqrt{b x^{2} + a}{\left ({\left (2 \, B x + 3 \, A\right )} x + \frac{2 \, B a}{b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A*a*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/6*sqrt(b*x^2 + a)*((2
*B*x + 3*A)*x + 2*B*a/b)