3.31.83 \(\int \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\)

Optimal. Leaf size=515 \[ \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{128 a c^{7/2}}-\frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{a}+\frac {\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (2560 a^3 c^4 x^3+2048 a^2 c^6 x^2-1575 a b^2 x+1920 a b c^4 x-840 b^2 c^2+1024 b c^6\right )+\sqrt {a^2 x^2+b} \left (\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (2560 a^2 c^4 x^2+2048 a c^6 x-1575 b^2+640 b c^4\right )+\left (35840 a^3 c^3 x^3-3072 a^2 c^5 x^2+179200 a b c^3 x-4096 a c^7 x+1050 b^2 c-768 b c^5\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )+\left (35840 a^4 c^3 x^4-3072 a^3 c^5 x^3+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2+1050 a b^2 c x-2304 a b c^5 x+80080 b^2 c^3-2048 b c^7\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{40320 a c^3 \left (2 a^2 x^2+b\right )+80640 a^2 c^3 x \sqrt {a^2 x^2+b}} \]

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Rubi [F]  time = 0.31, 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 \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[b + a^2*x^2]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]

[Out]

Defer[Int][Sqrt[b + a^2*x^2]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]], x]

Rubi steps

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

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Mathematica [A]  time = 1.26, size = 392, normalized size = 0.76 \begin {gather*} \frac {\left (\left (\sqrt {a^2 x^2+b}+a x\right )^2+b\right ) \left (\sqrt {c} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (-105 b^2 \left (8 c^2 \sqrt {\sqrt {a^2 x^2+b}+a x}-10 c \left (\sqrt {a^2 x^2+b}+a x\right )+15 \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}+48 c^3\right )+80640 b c^3 \left (\sqrt {a^2 x^2+b}+a x\right )^2-128 c^3 \left (\sqrt {a^2 x^2+b}+a x\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b}+a x}+c\right ) \left (-24 c^2 \sqrt {\sqrt {a^2 x^2+b}+a x}+30 c \left (\sqrt {a^2 x^2+b}+a x\right )-35 \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}+16 c^3\right )\right )+315 b \left (5 b-256 c^4\right ) \left (\sqrt {a^2 x^2+b}+a x\right )^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )\right )}{80640 a c^{7/2} \sqrt {a^2 x^2+b} \left (\sqrt {a^2 x^2+b}+a x\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b + a^2*x^2]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]

[Out]

((b + (a*x + Sqrt[b + a^2*x^2])^2)*(Sqrt[c]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]*(80640*b*c^3*(a*x + Sqrt[b
+ a^2*x^2])^2 - 128*c^3*(a*x + Sqrt[b + a^2*x^2])^2*(c + Sqrt[a*x + Sqrt[b + a^2*x^2]])*(16*c^3 - 24*c^2*Sqrt
[a*x + Sqrt[b + a^2*x^2]] + 30*c*(a*x + Sqrt[b + a^2*x^2]) - 35*(a*x + Sqrt[b + a^2*x^2])^(3/2)) - 105*b^2*(48
*c^3 + 8*c^2*Sqrt[a*x + Sqrt[b + a^2*x^2]] - 10*c*(a*x + Sqrt[b + a^2*x^2]) + 15*(a*x + Sqrt[b + a^2*x^2])^(3/
2))) + 315*b*(5*b - 256*c^4)*(a*x + Sqrt[b + a^2*x^2])^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[
c]]))/(80640*a*c^(7/2)*Sqrt[b + a^2*x^2]*(a*x + Sqrt[b + a^2*x^2])^3)

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IntegrateAlgebraic [A]  time = 1.20, size = 515, normalized size = 1.00 \begin {gather*} \frac {\left (80080 b^2 c^3-2048 b c^7+1050 a b^2 c x-2304 a b c^5 x+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2-3072 a^3 c^5 x^3+35840 a^4 c^3 x^4\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x+2048 a^2 c^6 x^2+2560 a^3 c^4 x^3\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (1050 b^2 c-768 b c^5+179200 a b c^3 x-4096 a c^7 x-3072 a^2 c^5 x^2+35840 a^3 c^3 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-1575 b^2+640 b c^4+2048 a c^6 x+2560 a^2 c^4 x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{80640 a^2 c^3 x \sqrt {b+a^2 x^2}+40320 a c^3 \left (b+2 a^2 x^2\right )}+\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{128 a c^{7/2}}-\frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b + a^2*x^2]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]

[Out]

((80080*b^2*c^3 - 2048*b*c^7 + 1050*a*b^2*c*x - 2304*a*b*c^5*x + 197120*a^2*b*c^3*x^2 - 4096*a^2*c^7*x^2 - 307
2*a^3*c^5*x^3 + 35840*a^4*c^3*x^4)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-840*b^2*c^2 + 1024*b*c^6 - 1575
*a*b^2*x + 1920*a*b*c^4*x + 2048*a^2*c^6*x^2 + 2560*a^3*c^4*x^3)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a
*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((1050*b^2*c - 768*b*c^5 + 179200*a*b*c^3*x - 4096*a*c^7*x - 3072
*a^2*c^5*x^2 + 35840*a^3*c^3*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-1575*b^2 + 640*b*c^4 + 2048*a*c^
6*x + 2560*a^2*c^4*x^2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(80640*a^2*c^3
*x*Sqrt[b + a^2*x^2] + 40320*a*c^3*(b + 2*a^2*x^2)) + (5*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/S
qrt[c]])/(128*a*c^(7/2)) - (2*b*Sqrt[c]*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/a

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fricas [A]  time = 0.58, size = 554, normalized size = 1.08 \begin {gather*} \left [\frac {315 \, {\left (256 \, b c^{4} - 5 \, b^{2}\right )} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) - 2 \, {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 80080 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} - 9520 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} - 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} + 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{80640 \, a c^{4}}, \frac {315 \, {\left (256 \, b c^{4} - 5 \, b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) - {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 80080 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} - 9520 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} - 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} + 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{40320 \, a c^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/80640*(315*(256*b*c^4 - 5*b^2)*sqrt(c)*log(2*(a*sqrt(c)*x - sqrt(a^2*x^2 + b)*sqrt(c))*sqrt(a*x + sqrt(a^2*
x^2 + b))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))) - 2*(a*c*x - sqrt(a^2*x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2 +
b)) + b) - 2*(2048*c^8 + 1120*a^2*c^4*x^2 - 80080*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 - 9520*a*
c^4*x - 525*b*c^2)*sqrt(a^2*x^2 + b) - (1024*c^7 - 1680*a^2*c^3*x^2 - 840*b*c^3 + 5*(128*a*c^5 + 315*a*b*c)*x
+ 5*(128*c^5 + 336*a*c^3*x - 315*b*c)*sqrt(a^2*x^2 + b))*sqrt(a*x + sqrt(a^2*x^2 + b)))*sqrt(c + sqrt(a*x + sq
rt(a^2*x^2 + b))))/(a*c^4), 1/40320*(315*(256*b*c^4 - 5*b^2)*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt
(a^2*x^2 + b)))/c) - (2048*c^8 + 1120*a^2*c^4*x^2 - 80080*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 -
9520*a*c^4*x - 525*b*c^2)*sqrt(a^2*x^2 + b) - (1024*c^7 - 1680*a^2*c^3*x^2 - 840*b*c^3 + 5*(128*a*c^5 + 315*a
*b*c)*x + 5*(128*c^5 + 336*a*c^3*x - 315*b*c)*sqrt(a^2*x^2 + b))*sqrt(a*x + sqrt(a^2*x^2 + b)))*sqrt(c + sqrt(
a*x + sqrt(a^2*x^2 + b))))/(a*c^4)]

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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((a^2*x^2+b)^(1/2)*(c+(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 \[\int \sqrt {a^{2} x^{2}+b}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*x^2+b)^(1/2)*(c+(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 \sqrt {a^{2} x^{2} + b} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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