3.29.30 \(\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{(b+a^2 x^2)^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\)

Optimal. Leaf size=287 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c+6 \text {$\#$1}^4 c^2-4 \text {$\#$1}^2 c^3+b+c^4\& ,\frac {\text {$\#$1}^2 \left (-\log \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\text {$\#$1}\right )\right )-c \log \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\text {$\#$1}\right )}{\text {$\#$1} c-\text {$\#$1}^3}\& \right ]}{4 a b}+\frac {a x \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+\sqrt {a^2 x^2+b} \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{a^2 b x \sqrt {a^2 x^2+b}+a b \left (a^2 x^2+b\right )} \]

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Rubi [F]  time = 1.11, 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 {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]

[Out]

Defer[Int][Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ \end {align*}

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Mathematica [F]  time = 14.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]

[Out]

Integrate[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]), x]

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IntegrateAlgebraic [A]  time = 1.00, size = 287, normalized size = 1.00 \begin {gather*} \frac {a x \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} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a^2 b x \sqrt {b+a^2 x^2}+a b \left (b+a^2 x^2\right )}+\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c \text {$\#$1}-\text {$\#$1}^3}\&\right ]}{4 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]])
,x]

[Out]

(a*x*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]*Sqrt[a*x + Sqrt
[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(a^2*b*x*Sqrt[b + a^2*x^2] + a*b*(b + a^2*x^2)) + Root
Sum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] -
#1]) - Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1^2)/(c*#1 - #1^3) & ]/(4*a*b)

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fricas [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((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x, algorithm
="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x, algorithm
="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(4*a^2*x^2-4*a*x+4*b+1)]Warning, need to choose a branch for the root of a polynomial with parameters. This m
ight be wrong.The choice was done assuming [c]=[51,-96,32]sym2poly/r2sym(const gen & e,const index_m & i,const
vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with para
meters. This might be wrong.The choice was done assuming [c]=[-32,-18,-85]schur row 3 -9.50039e-09sym2poly/r2s
ym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, choosing root of [1,0,
%%%{-2,[2,0,2]%%%}+%%%{2,[1,0,1]%%%}+%%%{-2,[0,1,0]%%%},%%%{4,[2,0,2]%%%}+%%%{4,[0,1,0]%%%},%%%{1,[4,0,4]%%%}+
%%%{2,[3,0,3]%%%}+%%%{2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{1,[0,2,0]%%%}+%%%{-1,[0,1,0]%%%}] at parameters valu
es [23,15,-30]Warning, choosing root of [1,0,%%%{-2,[2,0,2]%%%}+%%%{2,[1,0,1]%%%}+%%%{-2,[0,1,0]%%%},%%%{4,[2,
0,2]%%%}+%%%{4,[0,1,0]%%%},%%%{1,[4,0,4]%%%}+%%%{2,[3,0,3]%%%}+%%%{2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{1,[0,2,
0]%%%}+%%%{-1,[0,1,0]%%%}] at parameters values [26,40,-54]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[89,80,72]schur ro
w 3 -1.62329e-07Warning, need to choose a branch for the root of a polynomial with parameters. This might be w
rong.The choice was done assuming [a,b,t_nostep]=[-43,-85,11]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,t_nostep]=[62,-15,-10]schu
r row 3 4.84558e-08Warning, need to choose a branch for the root of a polynomial with parameters. This might b
e wrong.The choice was done assuming [a,b,t_nostep]=[92,90,-98]schur row 3 -1.16593e-07Discontinuities at zero
es of 4*a^2*x^2-4*a*x+4*b+1 were not checkedEvaluation time: 40.41Done

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\left (a^{2} x^{2}+b \right )^{\frac {3}{2}} \sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x)

[Out]

int((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(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 {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{{\left (a^{2} x^{2} + b\right )}^{\frac {3}{2}} \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((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),x, algorithm
="maxima")

[Out]

integrate(sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))/((a^2*x^2 + b)^(3/2)*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 {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}}}{\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,{\left (a^2\,x^2+b\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2)/(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(b + a^2*x^2)^(3/2)),x)

[Out]

int((c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2)/(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(b + a^2*x^2)^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b}} \left (a^{2} x^{2} + b\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+(a*x+(a**2*x**2+b)**(1/2))**(1/2))**(1/2)/(a**2*x**2+b)**(3/2)/(a*x+(a**2*x**2+b)**(1/2))**(1/2),
x)

[Out]

Integral(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 + b)))/(sqrt(a*x + sqrt(a**2*x**2 + b))*(a**2*x**2 + b)**(3/2)), x
)

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