3.18.12 \(\int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx\) [1712]

Optimal. Leaf size=115 \[ 4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\& ,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.42, 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 {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]

[Out]

Log[b + Sqrt[b + a*x]*(c + Sqrt[c + Sqrt[b + a*x]]*(a - Sqrt[c + Sqrt[b + a*x]]))] - 3*a*Defer[Subst][Defer[In
t][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]]] - a*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]]]

Rubi steps

\begin {align*} \int \frac {1}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x}{b-x^2+a x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )\right )\\ &=-\left (4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )}{-b+(c+(a-x) x) \left (c-x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right )\\ &=\log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \frac {a c-3 a x^2}{-b+(c+(a-x) x) \left (c-x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=\log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \frac {-a c+3 a 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 )\\ &=\log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-\text {Subst}\left (\int \left (\frac {3 a x^2}{b-c^2-a c x+2 c x^2+a x^3-x^4}+\frac {a c}{-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 )\\ &=\log \left (b+\sqrt {b+a x} \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )\right )-(3 a) \text {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 )-(a c) \text {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 [A]
time = 0.10, size = 115, normalized size = 1.00 \begin {gather*} 4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.62, size = 3177, normalized size = 27.63

method	result	size
derivativedivides	\(-4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{3}+\textit {\_R} c \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )\)	\(80\)
default	\(\text {Expression too large to display}\)	\(3177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*sum((_R^2-_R*c)/(3*_R^2*a^2-2*_R*a^2*c+4*_R^3-4*_R*b)*ln((a*x+b)^(1/2)-_R),_R=RootOf(_Z^4+a^2*_Z^3+(-a^2*c
-2*b)*_Z^2+b^2))+a^2*sum((_R^2+_R*c)/(-3*_R^2*a^2-2*_R*a^2*c+4*_R^3-4*_R*b)*ln((a*x+b)^(1/2)-_R),_R=RootOf(_Z^
4-a^2*_Z^3+(-a^2*c-2*b)*_Z^2+b^2))-1/b*c*sum((-_R^2*a*c+(-a^2*c-b)*_R+c^2*a-a*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))+1/b*sum(((a^2*c+b)*_R^3+a*(-a^2*c
-2*b)*_R^2+(-a^2*c^2+a^2*b-b*c)*_R)/(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))-b*c*sum(_R/(-3*_R^2*a^3+2*_R*a^2*c^2-6*_R^2*a*c-6*_R*a^2*b+2*a*b*c^2+4*_R^3-4*
_R*b*c-3*a*b^2)*ln(x-_R),_R=RootOf(_Z^4+(-a^3-2*a*c)*_Z^3+(a^2*c^2-3*a^2*b-2*b*c)*_Z^2+(2*a*b*c^2-3*a*b^2)*_Z+
b^2*c^2-b^3))+a*sum((_R^6-3*_R^4*c+2*_R^2*c^2)/(4*_R^7+3*_R^5*a^2-12*_R^5*c-8*_R^3*a^2*c+12*_R^3*c^2+5*_R*a^2*
c^2-4*_R^3*b-4*_R*c^3+4*_R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^8+(a^2-4*c)*_Z^6+(-4*a^2*c+6*c^2-2
*b)*_Z^4+(5*a^2*c^2-4*c^3+4*b*c)*_Z^2-2*c^3*a^2+c^4-2*b*c^2+b^2))-a*c*sum(_R^2/(-3*_R^2*a^3+2*_R*a^2*c^2-6*_R^
2*a*c-6*_R*a^2*b+2*a*b*c^2+4*_R^3-4*_R*b*c-3*a*b^2)*ln(x-_R),_R=RootOf(_Z^4+(-a^3-2*a*c)*_Z^3+(a^2*c^2-3*a^2*b
-2*b*c)*_Z^2+(2*a*b*c^2-3*a*b^2)*_Z+b^2*c^2-b^3))+sum(_R^3/(-3*_R^2*a^3+2*_R*a^2*c^2-6*_R^2*a*c-6*_R*a^2*b+2*a
*b*c^2+4*_R^3-4*_R*b*c-3*a*b^2)*ln(x-_R),_R=RootOf(_Z^4+(-a^3-2*a*c)*_Z^3+(a^2*c^2-3*a^2*b-2*b*c)*_Z^2+(2*a*b*
c^2-3*a*b^2)*_Z+b^2*c^2-b^3))-sum((_R^3+_R^2*a^2+(-a^2*c-b)*_R)/(3*_R^2*a^2-2*_R*a^2*c+4*_R^3-4*_R*b)*ln((a*x+
b)^(1/2)-_R),_R=RootOf(_Z^4+a^2*_Z^3+(-a^2*c-2*b)*_Z^2+b^2))-1/b*sum(((a^2*c+b)*_R^3+a*(a^2*c+2*b)*_R^2+(-a^2*
c^2+a^2*b-b*c)*_R)/(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))+sum(_R^3/(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))-2*a/b*sum((_R^3*a*c+(-a^2*c-b)*_R^2+a*(-c^2+b)*_R)/(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))-a^2/b*sum((_R^3*c+_R^2*a*c+(-c^2+b)*_R)/(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))-sum(_R^
3/(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))+a^
3*sum((_R^4-2*_R^2*c)/(4*_R^7+3*_R^5*a^2-12*_R^5*c-8*_R^3*a^2*c+12*_R^3*c^2+5*_R*a^2*c^2-4*_R^3*b-4*_R*c^3+4*_
R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^8+(a^2-4*c)*_Z^6+(-4*a^2*c+6*c^2-2*b)*_Z^4+(5*a^2*c^2-4*c^3
+4*b*c)*_Z^2-2*c^3*a^2+c^4-2*b*c^2+b^2))+sum((_R^3-_R^2*a^2+(-a^2*c-b)*_R)/(-3*_R^2*a^2-2*_R*a^2*c+4*_R^3-4*_R
*b)*ln((a*x+b)^(1/2)-_R),_R=RootOf(_Z^4-a^2*_Z^3+(-a^2*c-2*b)*_Z^2+b^2))+a*sum((_R^6+(a^2-2*c)*_R^4+(-2*a^2*c+
c^2-2*b)*_R^2)/(4*_R^7+3*_R^5*a^2-12*_R^5*c-8*_R^3*a^2*c+12*_R^3*c^2+5*_R*a^2*c^2-4*_R^3*b-4*_R*c^3+4*_R*b*c)*
ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^8+(a^2-4*c)*_Z^6+(-4*a^2*c+6*c^2-2*b)*_Z^4+(5*a^2*c^2-4*c^3+4*b*c)
*_Z^2-2*c^3*a^2+c^4-2*b*c^2+b^2))-1/b*c*sum((-_R^2*a*c+(a^2*c+b)*_R+c^2*a-a*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))-a*c/b*sum((_R^2*c+_R*a*c-c^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))-a*c/b*s
um((-c*_R^6+(-a^2*c+3*c^2-b)*_R^4+3*c*(a^2*c-c^2+b)*_R^2-2*c^3*a^2+c^4-2*b*c^2+b^2)/(4*_R^7+3*_R^5*a^2-12*_R^5
*c-8*_R^3*a^2*c+12*_R^3*c^2+5*_R*a^2*c^2-4*_R^3*b-4*_R*c^3+4*_R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(
_Z^8+(a^2-4*c)*_Z^6+(-4*a^2*c+6*c^2-2*b)*_Z^4+(5*a^2*c^2-4*c^3+4*b*c)*_Z^2-2*c^3*a^2+c^4-2*b*c^2+b^2))-a/b*c*s
um((c*_R^6+c*(a^2-3*c)*_R^4+c*(-3*a^2*c+3*c^2-2*b)*_R^2+2*c^3*a^2-c^4+2*b*c^2-b^2)/(4*_R^7+3*_R^5*a^2-12*_R^5*
c-8*_R^3*a^2*c+12*_R^3*c^2+5*_R*a^2*c^2-4*_R^3*b-4*_R*c^3+4*_R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_
Z^8+(a^2-4*c)*_Z^6+(-4*a^2*c+6*c^2-2*b)*_Z^4+(5*a^2*c^2-4*c^3+4*b*c)*_Z^2-2*c^3*a^2+c^4-2*b*c^2+b^2))+2*a/b*su
m((_R^3*a*c+(a^2*c+b)*_R^2+a*(-c^2+b)*_R)/(4*_R^3+3*_R^2*a-4*_R*c-a*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootO
f(_Z^4+_Z^3*a-2*_Z^2*c-_Z*a*c+c^2-b))-2*a*sum((_R^6+(a^2-2*c)*_R^4+(-2*a^2*c+c^2-b)*_R^2)/(4*_R^7+3*_R^5*a^2-1
2*_R^5*c-8*_R^3*a^2*c+12*_R^3*c^2+5*_R*a^2*c^2-4*_R^3*b-4*_R*c^3+4*_R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=R
ootOf(_Z^8+(a^2-4*c)*_Z^6+(-4*a^2*c+6*c^2-2*b)*_Z^4+(5*a^2*c^2-4*c^3+4*b*c)*_Z^2-2*c^3*a^2+c^4-2*b*c^2+b^2))+a
^2/b*sum((_R^3*c-_R^2*a*c+(-c^2+b)*_R)/(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))-a*c/b*sum((_R^2*c-_R*a*c-c^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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1/(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x - \sqrt {c + \sqrt {a x + b}} \sqrt {a x + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x - sqrt(c + sqrt(a*x + b))*sqrt(a*x + b)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-1/(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 = -0.01 \begin {gather*} \int \frac {1}{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(1/(x - (c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2)),x)

[Out]

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

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