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

Optimal. Leaf size=431 \[ \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{16 a c^{7/2}}+\frac {\sqrt {a^2 x^2-b} \left (\left (-2560 a^2 c^4 x^2-2048 a c^6 x-1575 b^2+640 b c^4\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (2240 a^2 c^3 x^2+1536 a c^5 x-10640 b c^3+2048 c^7\right )\right )+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (2240 a^3 c^3 x^3+1536 a^2 c^5 x^2-11760 a b c^3 x+2048 a c^7 x+1050 b^2 c-768 b c^5\right )+\left (-2560 a^3 c^4 x^3-2048 a^2 c^6 x^2-1575 a b^2 x+1920 a b c^4 x-840 b^2 c^2+1024 b c^6\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{5040 a c^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}} \]

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Rubi [F]  time = 1.09, 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 {a x+\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[a*x + 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[a*x + 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 {a x+\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 {a x+\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 [F]  time = 10.90, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.10, size = 431, normalized size = 1.00 \begin {gather*} \frac {\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 {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (1050 b^2 c-768 b c^5-11760 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 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 (-1575 b^2+640 b c^4-2048 a c^6 x-2560 a^2 c^4 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (-10640 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{5040 a c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}+\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-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[c + Sq
rt[a*x + Sqrt[-b + a^2*x^2]]] + (1050*b^2*c - 768*b*c^5 - 11760*a*b*c^3*x + 2048*a*c^7*x + 1536*a^2*c^5*x^2 +
2240*a^3*c^3*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]
*((-1575*b^2 + 640*b*c^4 - 2048*a*c^6*x - 2560*a^2*c^4*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-10640
*b*c^3 + 2048*c^7 + 1536*a*c^5*x + 2240*a^2*c^3*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-
b + a^2*x^2]]]))/(5040*a*c^3*(a*x + Sqrt[-b + a^2*x^2])^(3/2)) + (5*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b +
a^2*x^2]]]/Sqrt[c]])/(16*a*c^(7/2))

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fricas [A]  time = 0.63, size = 564, normalized size = 1.31 \begin {gather*} \left [\frac {1575 \, b^{2} \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} - 10640 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, 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}}}}{10080 \, a c^{4}}, -\frac {1575 \, b^{2} \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} - 10640 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, 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}}}}{5040 \, 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)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2),x, algorithm
="fricas")

[Out]

[1/10080*(1575*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*(2
048*c^8 + 1120*a^2*c^4*x^2 - 10640*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 + 560*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 - 3
36*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), -1/5040*(1575*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 - 10640*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 + 560*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)*(a*x+(a^2*x^2-b)^(1/2))^(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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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