3.24.4 \(\int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx\)

Optimal. Leaf size=176 \[ \frac {4 \text {RootSum}\left [-\text {$\#$1}^4+2 \text {$\#$1}^2 c-\text {$\#$1} a+b-c^2\& ,\frac {-\text {$\#$1}^2 a c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+\text {$\#$1} a^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+a c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-a b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{4 \text {$\#$1}^3-4 \text {$\#$1} c+a}\& \right ]}{a}-4 \sqrt {\sqrt {a x+b}+c}+x \]

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Rubi [F]  time = 0.92, 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 {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*c*Sqrt[b + a*x])/a - 4*Sqrt[c + Sqrt[b + a*x]] + (c + Sqrt[b + a*x])^2/a - 4*(b - c^2)*Defer[Subst][Defer[
Int][(-b + c^2 + a*x - 2*c*x^2 + x^4)^(-1), x], x, Sqrt[c + Sqrt[b + a*x]]] + 4*a*Defer[Subst][Defer[Int][x/(-
b + c^2 + a*x - 2*c*x^2 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]] - 4*c*Defer[Subst][Defer[Int][x^2/(-b + c^2 +
a*x - 2*c*x^2 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]]

Rubi steps

\begin {align*} \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-b+x^2\right )}{-b+x^2+a \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right ) \left (-b+\left (c-x^2\right )^2\right )}{-b+a x+\left (c-x^2\right )^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x \left (c-x^2\right ) \left (-b+c^2-2 c x^2+x^4\right )}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (-a-c x+x^3+\frac {a \left (b-c^2\right )-a^2 x+a c x^2}{b-c^2-a x+2 c x^2-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=-\frac {2 c \sqrt {b+a x}}{a}-4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \operatorname {Subst}\left (\int \frac {a \left (b-c^2\right )-a^2 x+a c x^2}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=-\frac {2 c \sqrt {b+a x}}{a}-4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \operatorname {Subst}\left (\int \left (-\frac {a \left (b-c^2\right )}{-b+c^2+a x-2 c x^2+x^4}+\frac {a^2 x}{-b+c^2+a x-2 c x^2+x^4}-\frac {a c x^2}{-b+c^2+a x-2 c x^2+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=-\frac {2 c \sqrt {b+a x}}{a}-4 \sqrt {c+\sqrt {b+a x}}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+(4 a) \operatorname {Subst}\left (\int \frac {x}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(4 c) \operatorname {Subst}\left (\int \frac {x^2}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 \left (b-c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ \end {align*}

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Mathematica [F]  time = 4.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x+\sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 183, normalized size = 1.04 \begin {gather*} \frac {b-c^2+a x}{a}-4 \sqrt {c+\sqrt {b+a x}}+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-a+4 c \text {$\#$1}-4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x + \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 118, normalized size = 0.67

method	result	size
derivativedivides	\(\frac {\left (c +\sqrt {a x +b}\right )^{2}-2 c \left (c +\sqrt {a x +b}\right )-4 a \sqrt {c +\sqrt {a x +b}}+4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c +\textit {\_R} a +c^{2}-b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\)	\(118\)
default	\(\frac {\left (c +\sqrt {a x +b}\right )^{2}-2 c \left (c +\sqrt {a x +b}\right )-4 a \sqrt {c +\sqrt {a x +b}}+4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c +\textit {\_R} a +c^{2}-b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x+(c+(a*x+b)^(1/2))^(1/2)),x,method=_RETURNVERBOSE)

[Out]

2/a*(1/2*(c+(a*x+b)^(1/2))^2-c*(c+(a*x+b)^(1/2))-2*a*(c+(a*x+b)^(1/2))^(1/2)+2*a*sum((-_R^2*c+_R*a+c^2-b)/(4*_
R^3-4*_R*c+a)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^2*c+_Z*a+c^2-b)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x + \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{x+\sqrt {c+\sqrt {b+a\,x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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