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

Optimal. Leaf size=235 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c+6 \text {$\#$1}^4 c^2-4 \text {$\#$1}^2 c^3+b+c^4\& ,\frac {3 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\text {$\#$1}\right )-c \log \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\text {$\#$1}\right )}{-\text {$\#$1}^7+3 \text {$\#$1}^5 c-3 \text {$\#$1}^3 c^2+\text {$\#$1} c^3}\& \right ]}{4 a}-\frac {\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{a^2 x \sqrt {a^2 x^2+b}+a \left (a^2 x^2+b\right )} \]

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Rubi [F]  time = 0.66, 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 {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 341, normalized size = 1.45 \begin {gather*} -\frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a^2 x \sqrt {b+a^2 x^2}+a \left (b+a^2 x^2\right )}+\frac {2 \text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-c^3+3 c^2 \text {$\#$1}^2-3 c \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]}{a}-\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )-5 \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c^3 \text {$\#$1}-3 c^2 \text {$\#$1}^3+3 c \text {$\#$1}^5-\text {$\#$1}^7}\&\right ]}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(a^2*x*Sqrt[b + a^2*x^2] + a*(b + a^
2*x^2))) + (2*RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b
+ a^2*x^2]]] - #1]*#1)/(-c^3 + 3*c^2*#1^2 - 3*c*#1^4 + #1^6) & ])/a - RootSum[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]) - 5*Log[Sqrt[c + Sqrt[a*x + Sq
rt[b + a^2*x^2]]] - #1]*#1^2)/(c^3*#1 - 3*c^2*#1^3 + 3*c*#1^5 - #1^7) & ]/(4*a)

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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((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)/(a^2*x^2+b)^(3/2),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((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)/(a^2*x^2+b)^(3/2),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]=[14,-38,-55]schur row 3 5.50999e-08sym2poly/r2sym(const gen & e
,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the root o
f a polynomial with parameters. This might be wrong.The choice was done assuming [c]=[3,57,80]sym2poly/r2sym(c
onst 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 values [
-43,4,18]schur row 3 2.66498e-08Warning, 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 [-82,-64,-74]Warning, choosing root of [1,0,
%%%{-10,[2,0,2]%%%}+%%%{8,[1,0,1]%%%}+%%%{-10,[0,1,0]%%%}+%%%{-2,[0,0,0]%%%},0,%%%{9,[4,0,4]%%%}+%%%{-24,[3,0,
3]%%%}+%%%{18,[2,1,2]%%%}+%%%{22,[2,0,2]%%%}+%%%{-24,[1,1,1]%%%}+%%%{-8,[1,0,1]%%%}+%%%{9,[0,2,0]%%%}+%%%{6,[0
,1,0]%%%}+%%%{1,[0,0,0]%%%}] at parameters values [-19,90,91]Warning, choosing root of [1,0,%%%{-10,[2,0,2]%%%
}+%%%{-8,[1,0,1]%%%}+%%%{-10,[0,1,0]%%%}+%%%{-2,[0,0,0]%%%},0,%%%{9,[4,0,4]%%%}+%%%{24,[3,0,3]%%%}+%%%{18,[2,1
,2]%%%}+%%%{22,[2,0,2]%%%}+%%%{24,[1,1,1]%%%}+%%%{8,[1,0,1]%%%}+%%%{9,[0,2,0]%%%}+%%%{6,[0,1,0]%%%}+%%%{1,[0,0
,0]%%%}] at parameters values [81,95,-98]Discontinuities at zeroes of 4*a^2*x^2-4*a*x+4*b+1 were not checkedEv
aluation time: 16.03Done

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,\sqrt {c+\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((((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2))/(b + a^2*x^2)^(3/2),x)

[Out]

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