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

Optimal. Leaf size=168 \[ \sqrt {a^2 x^2+b^2} \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}-\frac {b}{2 a}\right )+\frac {\left (a x+3 b^2\right ) \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}+\frac {b \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 a}+\frac {\left (-b^3-b\right ) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+b\right )}{a}-\frac {b x}{2} \]

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Rubi [F]  time = 13.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x/(4*b) - (b*x)/4 - ((1 + b^2)*x)/(4*b) + Sqrt[b^2 + a^2*x^2]/(4*a*b) + ((1 - b^2)*Sqrt[b^2 + a^2*x^2])/(4*a*b
) - ((1 + b^2)*Sqrt[b^2 + a^2*x^2])/(4*a*b) - (x*Sqrt[b^2 + a^2*x^2])/(4*b) - ((1 - a*x)*Sqrt[b^2 + a^2*x^2])/
(4*a*b) + 1/(4*a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) - b^2/(4*a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) - (1 + b^2)/(4*a
*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + ((1 + b^2)^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(4*a*b^2) - ((1 - b^4)*Sqrt[
a*x + Sqrt[b^2 + a^2*x^2]])/(4*a*b^2) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(12*a) + ((1 + b^(-2))*(a*x + Sqrt[b
^2 + a^2*x^2])^(3/2))/(12*a) - (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(12*a*b^2) - ((1 + b^2)^2*ArcTan[Sqrt[a*x + S
qrt[b^2 + a^2*x^2]]])/(4*a*b^2) + ((1 - b^4)*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]])/(4*a*b^2) - ArcTanh[(a*x
)/Sqrt[b^2 + a^2*x^2]]/(8*a*b) - (b*ArcTanh[(a*x)/Sqrt[b^2 + a^2*x^2]])/(4*a) - ((1 - b^2)^2*ArcTanh[(a*x)/Sqr
t[b^2 + a^2*x^2]])/(8*a*b) + ((1 + 2*b^2)*ArcTanh[(a*x)/Sqrt[b^2 + a^2*x^2]])/(8*a*b) + ((1 - b^4)*ArcTanh[(a*
x)/Sqrt[b^2 + a^2*x^2]])/(8*a*b) + ((1 + b^2)^2*ArcTanh[(2*b^2 - a*(1 - b^2)*x)/((1 + b^2)*Sqrt[b^2 + a^2*x^2]
)])/(8*a*b) - ((1 - b^4)*ArcTanh[(2*b^2 - a*(1 - b^2)*x)/((1 + b^2)*Sqrt[b^2 + a^2*x^2])])/(8*a*b) - ((1 + b^2
)^2*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/b])/(4*a*b) + ((1 - b^4)*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/b
])/(4*a*b) - ((1 + b^2)^2*Log[1 - b^2 + 2*a*x])/(8*a*b) + ((1 - b^4)*Log[1 - b^2 + 2*a*x])/(8*a*b) - Defer[Int
][(Sqrt[b^2 + a^2*x^2]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(-1 + b^2 - 2*a*x), x]/b^2 + (1 - b^(-2))*Defer[Int][(
Sqrt[b^2 + a^2*x^2]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(1 - b^2 + 2*a*x), x]

Rubi steps

\begin {align*} \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {a x}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=a \int \frac {x}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+\int \frac {\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=a \int \left (\frac {-1-b^2}{4 a b}+\frac {x}{2 b}+\frac {-1+b^4}{4 a b \left (-1+b^2-2 a x\right )}-\frac {\sqrt {b^2+a^2 x^2}}{2 a b}+\frac {\left (1-b^2\right ) \sqrt {b^2+a^2 x^2}}{2 a b \left (1-b^2+2 a x\right )}+\frac {\left (1+\frac {1}{b^2}\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a}-\frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 b^2}-\frac {\left (1-b^4\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a b^2 \left (1-b^2+2 a x\right )}+\frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 a b^2}+\frac {\left (1-\frac {1}{b^2}\right ) \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 a \left (1-b^2+2 a x\right )}\right ) \, dx+\int \left (\frac {a x \sqrt {b^2+a^2 x^2}}{b (1+2 a x)}+\frac {b (1+a x) \sqrt {b^2+a^2 x^2}}{\left (-1+b^2-2 a x\right ) (1+2 a x)}-\frac {b^2+a^2 x^2}{b (1+2 a x)}+\frac {b \left (b^2+a^2 x^2\right )}{\left (-1+b^2-2 a x\right ) (1+2 a x)}-\frac {a x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 (1+2 a x)}-\frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\left (-1+b^2-2 a x\right ) (1+2 a x)}+\frac {a x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{(1+2 a x) \left (1-b^2+2 a x\right )}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 (1+2 a x)}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{(1+2 a x) \left (1-b^2+2 a x\right )}\right ) \, dx\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}

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Mathematica [B]  time = 3.79, size = 423, normalized size = 2.52 \begin {gather*} \frac {b \left (-2 \sqrt {a^2 x^2+b^2}+b^2 \log \left (b^2 \left (\sqrt {a^2 x^2+b^2}+a x+2\right )+\sqrt {a^2 x^2+b^2}-a x\right )-\left (b^2-1\right ) \log \left (\sqrt {a^2 x^2+b^2}+a x\right )+\log \left (b^2 \left (\sqrt {a^2 x^2+b^2}+a x+2\right )+\sqrt {a^2 x^2+b^2}-a x\right )-2 \left (b^2+1\right ) \log \left (2 a x-b^2+1\right )-2 a x\right )}{4 a}+\frac {a b^2 x \left (a x \left (6 \sqrt {a^2 x^2+b^2}+5\right )+3 \sqrt {a^2 x^2+b^2}+6 a^2 x^2\right )-3 \left (b^2+1\right ) b \sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (a x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{b}\right )+b^4 \left (3 \sqrt {a^2 x^2+b^2}+6 a x+1\right )+4 a^3 x^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )}{3 a \sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (a x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^3*x^3*(a*x + Sqrt[b^2 + a^2*x^2]) + b^4*(1 + 6*a*x + 3*Sqrt[b^2 + a^2*x^2]) + a*b^2*x*(6*a^2*x^2 + 3*Sqrt
[b^2 + a^2*x^2] + a*x*(5 + 6*Sqrt[b^2 + a^2*x^2])) - 3*b*(1 + b^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]*(b^2 + a*x*
(a*x + Sqrt[b^2 + a^2*x^2]))*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/b])/(3*a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]*
(b^2 + a*x*(a*x + Sqrt[b^2 + a^2*x^2]))) + (b*(-2*a*x - 2*Sqrt[b^2 + a^2*x^2] - 2*(1 + b^2)*Log[1 - b^2 + 2*a*
x] - (-1 + b^2)*Log[a*x + Sqrt[b^2 + a^2*x^2]] + Log[-(a*x) + Sqrt[b^2 + a^2*x^2] + b^2*(2 + a*x + Sqrt[b^2 +
a^2*x^2])] + b^2*Log[-(a*x) + Sqrt[b^2 + a^2*x^2] + b^2*(2 + a*x + Sqrt[b^2 + a^2*x^2])]))/(4*a)

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IntegrateAlgebraic [A]  time = 0.32, size = 168, normalized size = 1.00 \begin {gather*} \sqrt {a^2 x^2+b^2} \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}-\frac {b}{2 a}\right )+\frac {\left (a x+3 b^2\right ) \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{3 a}+\frac {b \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 a}+\frac {\left (-b^3-b\right ) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+b\right )}{a}-\frac {b x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*x + Sqrt[b^2 + a^2*x^2])/(b + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

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

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fricas [A]  time = 0.49, size = 122, normalized size = 0.73 \begin {gather*} -\frac {3 \, a b x + 6 \, {\left (b^{3} + b\right )} \log \left (b + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - 6 \, b \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - 2 \, {\left (3 \, b^{2} + a x + \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} + 3 \, \sqrt {a^{2} x^{2} + b^{2}} b}{6 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*b*x + 6*(b^3 + b)*log(b + sqrt(a*x + sqrt(a^2*x^2 + b^2))) - 6*b*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))
) - 2*(3*b^2 + a*x + sqrt(a^2*x^2 + b^2))*sqrt(a*x + sqrt(a^2*x^2 + b^2)) + 3*sqrt(a^2*x^2 + b^2)*b)/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + \sqrt {a^{2} x^{2} + b^{2}}}{b + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x + sqrt(a^2*x^2 + b^2))/(b + sqrt(a*x + sqrt(a^2*x^2 + b^2))), x)

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x +\sqrt {a^{2} x^{2}+b^{2}}}{b +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+(a^2*x^2+b^2)^(1/2))/(b+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x)

[Out]

int((a*x+(a^2*x^2+b^2)^(1/2))/(b+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a x^{2}}{4 \, b} + \frac {\frac {b^{2} \operatorname {arsinh}\left (\frac {a x}{b}\right )}{2 \, a} + \frac {1}{2} \, \sqrt {a^{2} x^{2} + b^{2}} x}{2 \, b} - \int -\frac {a b^{2} x - 2 \, a^{2} x^{2} - b^{2} + \sqrt {a^{2} x^{2} + b^{2}} {\left (b^{2} - 2 \, a x\right )}}{2 \, {\left (b^{3} + a b x + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} b^{2} + \sqrt {a^{2} x^{2} + b^{2}} b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^2/b + 1/2*integrate(sqrt(a^2*x^2 + b^2), x)/b - integrate(-1/2*(a*b^2*x - 2*a^2*x^2 - b^2 + sqrt(a^2*x
^2 + b^2)*(b^2 - 2*a*x))/(b^3 + a*b*x + 2*sqrt(a*x + sqrt(a^2*x^2 + b^2))*b^2 + sqrt(a^2*x^2 + b^2)*b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x+\sqrt {a^2\,x^2+b^2}}{b+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + (b^2 + a^2*x^2)^(1/2))/(b + (a*x + (b^2 + a^2*x^2)^(1/2))^(1/2)),x)

[Out]

int((a*x + (b^2 + a^2*x^2)^(1/2))/(b + (a*x + (b^2 + a^2*x^2)^(1/2))^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + \sqrt {a^{2} x^{2} + b^{2}}}{b + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*x + sqrt(a**2*x**2 + b**2))/(b + sqrt(a*x + sqrt(a**2*x**2 + b**2))), x)

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