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

Optimal. Leaf size=495 \[ -\frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{128 a c^{9/2}}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {\sqrt {a^2 x^2-b} \left (\left (-9216 a^2 c^5 x^2-12288 a c^7 x-2450 b^2 c+2304 b c^5\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (7680 a^2 c^4 x^2+6144 a c^6 x+3675 b^2-1920 b c^4\right )\right )+\left (-9216 a^3 c^5 x^3-12288 a^2 c^7 x^2-2450 a b^2 c x+6912 a b c^5 x-1680 b^2 c^3+6144 b c^7\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (7680 a^3 c^4 x^3+6144 a^2 c^6 x^2+3675 a b^2 x-5760 a b c^4 x+1960 b^2 c^2-3072 b c^6\right )}{13440 a c^4 \left (2 a^2 x^2-b\right )+26880 a^2 c^4 x \sqrt {a^2 x^2-b}} \]

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Rubi [F]  time = 0.33, 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 {\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 \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {\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 = 3.32, size = 536, normalized size = 1.08 \begin {gather*} \frac {\left (2 a x \left (\sqrt {a^2 x^2-b}+a x\right )-2 b\right ) \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+c\right )^{3/2} \left (\frac {\sqrt {c} \left (35 b^2 \left (56 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}-70 c \left (\sqrt {a^2 x^2-b}+a x\right )+105 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}-48 c^3\right )+1536 a c^4 x \left (\sqrt {a^2 x^2-b}+a x\right ) \left (4 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}+5 a x \sqrt {\sqrt {a^2 x^2-b}+a x}-6 a c x-8 c^3\right )-384 b c^4 \left (8 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}-6 c \left (\sqrt {a^2 x^2-b}+3 a x\right )+5 \sqrt {\sqrt {a^2 x^2-b}+a x} \left (\sqrt {a^2 x^2-b}+3 a x\right )-16 c^3\right )\right )}{\left (\sqrt {\sqrt {a^2 x^2-b}+a x}+c\right )^7}-\frac {105 b \left (35 b-256 c^4\right ) \left (\sqrt {a^2 x^2-b}+a x\right )^2 \tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}\right )}{\left (\sqrt {\sqrt {a^2 x^2-b}+a x}+c\right )^{15/2}}\right )}{13440 a c^{9/2} \sqrt {4 a^2 x^2-4 b} \left (\frac {c}{\sqrt {\sqrt {a^2 x^2-b}+a x}+c}-1\right )^6} \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]

((c + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(3/2)*(-2*b + 2*a*x*(a*x + Sqrt[-b + a^2*x^2]))*((Sqrt[c]*(1536*a*c^4*x*
(a*x + Sqrt[-b + a^2*x^2])*(-8*c^3 - 6*a*c*x + 4*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 5*a*x*Sqrt[a*x + Sqrt[-b
+ a^2*x^2]]) + 35*b^2*(-48*c^3 + 56*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 70*c*(a*x + Sqrt[-b + a^2*x^2]) + 10
5*(a*x + Sqrt[-b + a^2*x^2])^(3/2)) - 384*b*c^4*(-16*c^3 + 8*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 6*c*(3*a*x +
Sqrt[-b + a^2*x^2]) + 5*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(3*a*x + Sqrt[-b + a^2*x^2]))))/(c + Sqrt[a*x + Sqrt[-
b + a^2*x^2]])^7 - (105*b*(35*b - 256*c^4)*(a*x + Sqrt[-b + a^2*x^2])^2*ArcTanh[Sqrt[c]/Sqrt[c + Sqrt[a*x + Sq
rt[-b + a^2*x^2]]]])/(c + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(15/2)))/(13440*a*c^(9/2)*Sqrt[-4*b + 4*a^2*x^2]*(-1
+ c/(c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))^6)

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IntegrateAlgebraic [A]  time = 1.06, size = 495, normalized size = 1.00 \begin {gather*} \frac {\left (-1680 b^2 c^3+6144 b c^7-2450 a b^2 c x+6912 a b c^5 x-12288 a^2 c^7 x^2-9216 a^3 c^5 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (1960 b^2 c^2-3072 b c^6+3675 a b^2 x-5760 a b c^4 x+6144 a^2 c^6 x^2+7680 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 (-2450 b^2 c+2304 b c^5-12288 a c^7 x-9216 a^2 c^5 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (3675 b^2-1920 b c^4+6144 a c^6 x+7680 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 )}{26880 a^2 c^4 x \sqrt {-b+a^2 x^2}+13440 a c^4 \left (-b+2 a^2 x^2\right )}-\frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{128 a c^{9/2}}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a \sqrt {c}} \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]

((-1680*b^2*c^3 + 6144*b*c^7 - 2450*a*b^2*c*x + 6912*a*b*c^5*x - 12288*a^2*c^7*x^2 - 9216*a^3*c^5*x^3)*Sqrt[c
+ Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (1960*b^2*c^2 - 3072*b*c^6 + 3675*a*b^2*x - 5760*a*b*c^4*x + 6144*a^2*c^6*
x^2 + 7680*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]*((-2450*b^2*c + 2304*b*c^5 - 12288*a*c^7*x - 9216*a^2*c^5*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]
+ (3675*b^2 - 1920*b*c^4 + 6144*a*c^6*x + 7680*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]]]))/(26880*a^2*c^4*x*Sqrt[-b + a^2*x^2] + 13440*a*c^4*(-b + 2*a^2*x^2)) - (35*b^2*ArcTanh
[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(128*a*c^(9/2)) + (2*b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b
+ a^2*x^2]]]/Sqrt[c]])/(a*Sqrt[c])

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fricas [A]  time = 0.63, size = 581, normalized size = 1.17 \begin {gather*} \left [\frac {105 \, {\left (256 \, b c^{4} - 35 \, 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 (6144 \, c^{8} + 3360 \, a^{2} c^{4} x^{2} - 1680 \, b c^{4} + 2 \, {\left (1152 \, a c^{6} + 1225 \, a b c^{2}\right )} x + 2 \, {\left (1152 \, c^{6} - 1680 \, a c^{4} x - 1225 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (3072 \, c^{7} + 3920 \, a^{2} c^{3} x^{2} - 1960 \, b c^{3} + 15 \, {\left (128 \, a c^{5} + 245 \, a b c\right )} x + 5 \, {\left (384 \, c^{5} - 784 \, a c^{3} x - 735 \, 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}}}}{26880 \, a c^{5}}, -\frac {105 \, {\left (256 \, b c^{4} - 35 \, b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) + {\left (6144 \, c^{8} + 3360 \, a^{2} c^{4} x^{2} - 1680 \, b c^{4} + 2 \, {\left (1152 \, a c^{6} + 1225 \, a b c^{2}\right )} x + 2 \, {\left (1152 \, c^{6} - 1680 \, a c^{4} x - 1225 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (3072 \, c^{7} + 3920 \, a^{2} c^{3} x^{2} - 1960 \, b c^{3} + 15 \, {\left (128 \, a c^{5} + 245 \, a b c\right )} x + 5 \, {\left (384 \, c^{5} - 784 \, a c^{3} x - 735 \, 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}}}}{13440 \, a c^{5}}\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]

[1/26880*(105*(256*b*c^4 - 35*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*(6144*c^8 + 3360*a^2*c^4*x^2 - 1680*b*c^4 + 2*(1152*a*c^6 + 1225*a*b*c^2)*x + 2*(1152*c^6 - 1680
*a*c^4*x - 1225*b*c^2)*sqrt(a^2*x^2 - b) - (3072*c^7 + 3920*a^2*c^3*x^2 - 1960*b*c^3 + 15*(128*a*c^5 + 245*a*b
*c)*x + 5*(384*c^5 - 784*a*c^3*x - 735*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^5), -1/13440*(105*(256*b*c^4 - 35*b^2)*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*
x + sqrt(a^2*x^2 - b)))/c) + (6144*c^8 + 3360*a^2*c^4*x^2 - 1680*b*c^4 + 2*(1152*a*c^6 + 1225*a*b*c^2)*x + 2*(
1152*c^6 - 1680*a*c^4*x - 1225*b*c^2)*sqrt(a^2*x^2 - b) - (3072*c^7 + 3920*a^2*c^3*x^2 - 1960*b*c^3 + 15*(128*
a*c^5 + 245*a*b*c)*x + 5*(384*c^5 - 784*a*c^3*x - 735*b*c)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)))*s
qrt(c + sqrt(a*x + sqrt(a^2*x^2 - b))))/(a*c^5)]

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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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}}{\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]

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 \frac {\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 \frac {\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]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \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(a**2*x**2 - b)/sqrt(c + sqrt(a*x + sqrt(a**2*x**2 - b))), x)

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