3.25.10 \(\int \frac {1}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx\)

Optimal. Leaf size=193 \[ 4 a \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c-2 \text {$\#$1}^4 b+6 \text {$\#$1}^4 c^2-\text {$\#$1}^3 a^2+4 \text {$\#$1}^2 b c-4 \text {$\#$1}^2 c^3+\text {$\#$1} a^2 c+b^2-2 b c^2+c^4\& ,\frac {\text {$\#$1} c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^3 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{-8 \text {$\#$1}^7+24 \text {$\#$1}^5 c+8 \text {$\#$1}^3 b-24 \text {$\#$1}^3 c^2+3 \text {$\#$1}^2 a^2-8 \text {$\#$1} b c+8 \text {$\#$1} c^3-a^2 c}\& \right ] \]

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Rubi [F]  time = 0.81, 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^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

4*a*c*Defer[Subst][Defer[Int][x/(-b^2 + 2*b*(c - x^2)^2 - (c - x^2)*(c^3 + a^2*x - 3*c^2*x^2 + 3*c*x^4 - x^6))
, x], x, Sqrt[c + Sqrt[b + a*x]]] + 4*a*Defer[Subst][Defer[Int][x^3/(b^2 - 2*b*(c - x^2)^2 + (c - x^2)*(c^3 +
a^2*x - 3*c^2*x^2 + 3*c*x^4 - x^6)), x], x, Sqrt[c + Sqrt[b + a*x]]]

Rubi steps

\begin {align*} \int \frac {1}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=(2 a) \operatorname {Subst}\left (\int \frac {x}{b^2-2 b x^2+x^4-a^2 x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right )}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \left (\frac {c x}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8}+\frac {x^3}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x^3}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+(4 a c) \operatorname {Subst}\left (\int \frac {x}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x^3}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+(4 a c) \operatorname {Subst}\left (\int \frac {x}{-b^2+2 b \left (c-x^2\right )^2-\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ \end {align*}

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Mathematica [F]  time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]

[Out]

Integrate[(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 193, normalized size = 1.00 \begin {gather*} 4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-a^2 c-8 b c \text {$\#$1}+8 c^3 \text {$\#$1}+3 a^2 \text {$\#$1}^2+8 b \text {$\#$1}^3-24 c^2 \text {$\#$1}^3+24 c \text {$\#$1}^5-8 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]

[Out]

4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*
#1^6 + #1^8 & , (c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 - Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(-(a^2*c) -
8*b*c*#1 + 8*c^3*#1 + 3*a^2*#1^2 + 8*b*#1^3 - 24*c^2*#1^3 + 24*c*#1^5 - 8*#1^7) & ]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 1.66, size = 149, normalized size = 0.77

method	result	size
derivativedivides	\(-4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+\textit {\_R} c \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\)	\(149\)
default	\(\text {Expression too large to display}\)	\(7120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*a*sum((-_R^3+_R*c)/(8*_R^7-24*_R^5*c+24*_R^3*c^2-8*_R^3*b-3*_R^2*a^2-8*_R*c^3+8*_R*b*c+a^2*c)*ln((c+(a*x+b)
^(1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*c*_Z^6+(6*c^2-2*b)*_Z^4-a^2*_Z^3+(-4*c^3+4*b*c)*_Z^2+a^2*c*_Z+c^4-2*b*c^2+b
^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^2 - sqrt(a*x + b)*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 {1}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}-x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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