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

Optimal. Leaf size=380 \[ -4 \text {RootSum}\left [-\text {$\#$1}^4+\text {$\#$1}^3 a+2 \text {$\#$1}^2 c-\text {$\#$1} a c+b-c^2\& ,\frac {\text {$\#$1}^3 a^3 \left (-\log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )\right )-\text {$\#$1}^3 a c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^2 a^2 c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^2 b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+\text {$\#$1} a^3 c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+a^2 c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-a^2 b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+\text {$\#$1} a c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1} a b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{4 \text {$\#$1}^3-3 \text {$\#$1}^2 a-4 \text {$\#$1} c+a c}\& \right ]+\frac {4}{3} \left (3 a^2+c\right ) \sqrt {\sqrt {a x+b}+c}+\sqrt {a x+b} \left (\frac {4}{3} \sqrt {\sqrt {a x+b}+c}+2 a\right )+x \]

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Rubi [F]  time = 1.80, 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 {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*(a^2 - c)*Sqrt[b + a*x])/a + 4*a^2*Sqrt[c + Sqrt[b + a*x]] + (4*(c + Sqrt[b + a*x])^(3/2))/3 + (c + Sqrt[b
+ a*x])^2/a + a*(a^2 + c)*Log[b - c^2 - a*c*Sqrt[c + Sqrt[b + a*x]] + 2*c*(c + Sqrt[b + a*x]) + a*(c + Sqrt[b
+ a*x])^(3/2) - (c + Sqrt[b + a*x])^2] - a^2*(4*b - c*(a^2 + 5*c))*Defer[Subst][Defer[Int][(b - c^2 - a*c*x +
2*c*x^2 + a*x^3 - x^4)^(-1), x], x, Sqrt[c + Sqrt[b + a*x]]] - 4*a*b*Defer[Subst][Defer[Int][x/(b - c^2 - a*c*
x + 2*c*x^2 + a*x^3 - x^4), x], x, Sqrt[c + Sqrt[b + a*x]]] + (3*a^4 + 4*b + 7*a^2*c)*Defer[Subst][Defer[Int][
x^2/(-b + c^2 + a*c*x - 2*c*x^2 - a*x^3 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]]

Rubi steps

\begin {align*} \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (b-x^2\right )}{b+x \left (-x+a \sqrt {c+x}\right )} \, 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+(c+(a-x) x) \left (-c+x^2\right )} \, 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 c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (a^3+\left (a^2-c\right ) x+a x^2+x^3-\frac {a^3 \left (b-c^2\right )+a^2 \left (b-c \left (a^2+c\right )\right ) x+a \left (b+a^2 c\right ) x^2+a^2 \left (a^2+c\right ) x^3}{b-c^2-a c x+2 c x^2+a x^3-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 \left (a^2-c\right ) \sqrt {b+a x}}{a}+4 a^2 \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}-\frac {4 \operatorname {Subst}\left (\int \frac {a^3 \left (b-c^2\right )+a^2 \left (b-c \left (a^2+c\right )\right ) x+a \left (b+a^2 c\right ) x^2+a^2 \left (a^2+c\right ) x^3}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 \left (a^2-c\right ) \sqrt {b+a x}}{a}+4 a^2 \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+a \left (a^2+c\right ) \log \left (b-c^2-a c \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )+a \left (c+\sqrt {b+a x}\right )^{3/2}-\left (c+\sqrt {b+a x}\right )^2\right )+\frac {\operatorname {Subst}\left (\int \frac {-a^3 \left (4 b-c \left (a^2+5 c\right )\right )-4 a^2 b x-a \left (3 a^4+4 b+7 a^2 c\right ) x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 \left (a^2-c\right ) \sqrt {b+a x}}{a}+4 a^2 \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+a \left (a^2+c\right ) \log \left (b-c^2-a c \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )+a \left (c+\sqrt {b+a x}\right )^{3/2}-\left (c+\sqrt {b+a x}\right )^2\right )+\frac {\operatorname {Subst}\left (\int \left (\frac {a^3 \left (-4 b+c \left (a^2+5 c\right )\right )}{b-c^2-a c x+2 c x^2+a x^3-x^4}-\frac {4 a^2 b x}{b-c^2-a c x+2 c x^2+a x^3-x^4}+\frac {a \left (3 a^4+4 b+7 a^2 c\right ) x^2}{-b+c^2+a c x-2 c x^2-a x^3+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 \left (a^2-c\right ) \sqrt {b+a x}}{a}+4 a^2 \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+a \left (a^2+c\right ) \log \left (b-c^2-a c \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )+a \left (c+\sqrt {b+a x}\right )^{3/2}-\left (c+\sqrt {b+a x}\right )^2\right )-(4 a b) \operatorname {Subst}\left (\int \frac {x}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\left (3 a^4+4 b+7 a^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{-b+c^2+a c x-2 c x^2-a x^3+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (a^2 \left (4 b-c \left (a^2+5 c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b-c^2-a c x+2 c x^2+a x^3-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ \end {align*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 0.29, size = 389, normalized size = 1.02 \begin {gather*} \frac {4}{3} \sqrt {c+\sqrt {b+a x}} \left (3 a^2+c+\sqrt {b+a x}\right )+\frac {b+2 a^2 c-c^2+a x+2 a^2 \sqrt {b+a x}}{a}-4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-a^2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-a b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a^3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-a c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c + Sqrt[b + a*x]]*(3*a^2 + c + Sqrt[b + a*x]))/3 + (b + 2*a^2*c - c^2 + a*x + 2*a^2*Sqrt[b + a*x])/a
- 4*RootSum[b - c^2 - a*c*#1 + 2*c*#1^2 + a*#1^3 - #1^4 & , (-(a^2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]) + a^2*
c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - a*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a^3*c*Log[Sqrt[c + Sqrt[b +
a*x]] - #1]*#1 + a*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 - b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*
c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - a*c*Log[Sqrt[c + Sqrt[
b + a*x]] - #1]*#1^3)/(a*c - 4*c*#1 - 3*a*#1^2 + 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-(a*x+b)^(1/2)*(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}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.18, size = 203, normalized size = 0.53

method	result	size
derivativedivides	\(-\frac {2 \left (-2 a^{3} \sqrt {c +\sqrt {a x +b}}-a^{2} \left (c +\sqrt {a x +b}\right )-\frac {2 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )+2 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (a \left (-a^{2}-c \right ) \textit {\_R}^{3}+\left (-a^{2} c -b \right ) \textit {\_R}^{2}+a \left (a^{2} c +c^{2}-b \right ) \textit {\_R} +a^{2} c^{2}-a^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )\right )}{a}\)	\(203\)
default	\(-\frac {2 \left (-2 a^{3} \sqrt {c +\sqrt {a x +b}}-a^{2} \left (c +\sqrt {a x +b}\right )-\frac {2 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )+2 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (a \left (-a^{2}-c \right ) \textit {\_R}^{3}+\left (-a^{2} c -b \right ) \textit {\_R}^{2}+a \left (a^{2} c +c^{2}-b \right ) \textit {\_R} +a^{2} c^{2}-a^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )\right )}{a}\)	\(203\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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sympy [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(x/(x-(a*x+b)**(1/2)*(c+(a*x+b)**(1/2))**(1/2)),x)

[Out]

Timed out

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