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

Optimal. Leaf size=383 \[ \frac {35 b^2 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{128 a^2}+\frac {\sqrt {a^2 x^2-b} \left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-9216 a^2 x^2-12288 a x+2450 b^2+2304 b\right )+\left (7680 a^2 x^2+6144 a x-3675 b^2-1920 b\right ) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )+\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-9216 a^3 x^3-12288 a^2 x^2+2450 a b^2 x+6912 a b x+1680 b^2+6144 b\right )+\left (7680 a^3 x^3+6144 a^2 x^2-3675 a b^2 x-5760 a b x-1960 b^2-3072 b\right ) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{13440 a^2 \left (2 a^2 x^2-b\right )+26880 a^3 x \sqrt {a^2 x^2-b}} \]

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Rubi [F]  time = 0.20, 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 {x}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

Defer[Int][x/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]], x]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {x}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 2.14, size = 444, normalized size = 1.16 \begin {gather*} -\frac {-\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{7/2}}{8 \left (\sqrt {a^2 x^2-b}+a x\right )^2}+\frac {25 b^2 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{5/2}}{48 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}}-\frac {163 b^2 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{3/2}}{192 \left (\sqrt {a^2 x^2-b}+a x\right )}+\frac {93 b^2 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{128 \sqrt {\sqrt {a^2 x^2-b}+a x}}+\frac {35}{256} b^2 \log \left (1-\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}\right )-\frac {35}{256} b^2 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}+1\right )-\frac {1}{7} \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{7/2}+\frac {3}{5} \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{5/2}-\left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{3/2}+\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

-((Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (93*b^2*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(128*Sqrt[a*x
+ Sqrt[-b + a^2*x^2]]) - (1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(3/2) - (163*b^2*(1 + Sqrt[a*x + Sqrt[-b + a^2*x
^2]])^(3/2))/(192*(a*x + Sqrt[-b + a^2*x^2])) + (3*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(5/2))/5 + (25*b^2*(1
+ Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(5/2))/(48*(a*x + Sqrt[-b + a^2*x^2])^(3/2)) - (1 + Sqrt[a*x + Sqrt[-b + a^2
*x^2]])^(7/2)/7 - (b^2*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(7/2))/(8*(a*x + Sqrt[-b + a^2*x^2])^2) + (35*b^2*
Log[1 - 1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/256 - (35*b^2*Log[1 + 1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2
*x^2]]]])/256)/a^2)

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IntegrateAlgebraic [A]  time = 0.85, size = 383, normalized size = 1.00 \begin {gather*} \frac {\left (6144 b+1680 b^2+6912 a b x+2450 a b^2 x-12288 a^2 x^2-9216 a^3 x^3\right ) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (-3072 b-1960 b^2-5760 a b x-3675 a b^2 x+6144 a^2 x^2+7680 a^3 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (2304 b+2450 b^2-12288 a x-9216 a^2 x^2\right ) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (-1920 b-3675 b^2+6144 a x+7680 a^2 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{26880 a^3 x \sqrt {-b+a^2 x^2}+13440 a^2 \left (-b+2 a^2 x^2\right )}+\frac {35 b^2 \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{128 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

((6144*b + 1680*b^2 + 6912*a*b*x + 2450*a*b^2*x - 12288*a^2*x^2 - 9216*a^3*x^3)*Sqrt[1 + Sqrt[a*x + Sqrt[-b +
a^2*x^2]]] + (-3072*b - 1960*b^2 - 5760*a*b*x - 3675*a*b^2*x + 6144*a^2*x^2 + 7680*a^3*x^3)*Sqrt[a*x + Sqrt[-b
+ a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*((2304*b + 2450*b^2 - 12288*a*x - 9
216*a^2*x^2)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-1920*b - 3675*b^2 + 6144*a*x + 7680*a^2*x^2)*Sqrt[a*
x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(26880*a^3*x*Sqrt[-b + a^2*x^2] + 13440*a^2
*(-b + 2*a^2*x^2)) + (35*b^2*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(128*a^2)

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fricas [A]  time = 0.53, size = 210, normalized size = 0.55 \begin {gather*} \frac {3675 \, b^{2} \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} + 1\right ) - 3675 \, b^{2} \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} - 1\right ) + 2 \, {\left (3360 \, a^{2} x^{2} + 2 \, {\left (1225 \, a b - 1152 \, a\right )} x - 2 \, \sqrt {a^{2} x^{2} - b} {\left (1680 \, a x + 1225 \, b + 1152\right )} - {\left (3920 \, a^{2} x^{2} + 15 \, {\left (245 \, a b - 128 \, a\right )} x - 5 \, \sqrt {a^{2} x^{2} - b} {\left (784 \, a x + 735 \, b + 384\right )} - 1960 \, b - 3072\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 1680 \, b - 6144\right )} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}{26880 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/26880*(3675*b^2*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) + 1) - 3675*b^2*log(sqrt(sqrt(a*x + sqrt(a^2*x^2
- b)) + 1) - 1) + 2*(3360*a^2*x^2 + 2*(1225*a*b - 1152*a)*x - 2*sqrt(a^2*x^2 - b)*(1680*a*x + 1225*b + 1152)
- (3920*a^2*x^2 + 15*(245*a*b - 128*a)*x - 5*sqrt(a^2*x^2 - b)*(784*a*x + 735*b + 384) - 1960*b - 3072)*sqrt(a
*x + sqrt(a^2*x^2 - b)) - 1680*b - 6144)*sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1))/a^2

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {1+\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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