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

Optimal. Leaf size=294 \[ \frac {2 \sqrt {\sqrt {x \left (\sqrt {a^2 x^2-b}+a x\right )}+c}}{a}-\frac {\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c} \left (\sqrt {2} \sqrt {a} c-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {x \left (\sqrt {a^2 x^2-b}+a x\right )}+c}}{\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c}}\right )}{a^{5/4} \left (2 \sqrt {a} c-\sqrt {2} \sqrt {b}\right )}-\frac {\left (\sqrt {2} \sqrt {a} c+\sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {x \left (\sqrt {a^2 x^2-b}+a x\right )}+c}}{\sqrt {2 \sqrt {a} c+\sqrt {2} \sqrt {b}}}\right )}{a^{5/4} \sqrt {2 \sqrt {a} c+\sqrt {2} \sqrt {b}}} \]

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Rubi [F]  time = 0.44, 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^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 4.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Could not integrate

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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^2+x*(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/(a^2*x^2-b)^(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^2+x*(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/(a^2*x^2-b)^(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^2+x^2-4*b)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argu
ment Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluatio
n time: 1.4gen.cc:simplify/tmp.type!=_EXT Error: Bad Argument Value

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c +\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b}}}}{\sqrt {a^{2} x^{2}-b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(c + sqrt(a*x^2 + sqrt(a^2*x^2 - b)*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 {x\,\sqrt {a^2\,x^2-b}+a\,x^2}}}{\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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