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

Optimal. Leaf size=228 \[ 4 a \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c-2 \text {$\#$1}^4 b+6 \text {$\#$1}^4 c^2-\text {$\#$1}^3 a^2+4 \text {$\#$1}^2 b c-4 \text {$\#$1}^2 c^3+\text {$\#$1} a^2 c+b^2-2 b c^2+c^4\& ,\frac {\text {$\#$1}^6 \left (-\log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )\right )+2 \text {$\#$1}^4 c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^2 c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{-8 \text {$\#$1}^7+24 \text {$\#$1}^5 c+8 \text {$\#$1}^3 b-24 \text {$\#$1}^3 c^2+3 \text {$\#$1}^2 a^2-8 \text {$\#$1} b c+8 \text {$\#$1} c^3-a^2 c}\& \right ]+x \]

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Rubi [F]  time = 3.41, 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^2}{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/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

(-2*c*Sqrt[b + a*x])/a + (c + Sqrt[b + a*x])^2/a + 8*a*c*Defer[Subst][Defer[Int][x^4/(-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*c^2*Defer[Subst][
Defer[Int][x^2/(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^6/(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 {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (b-x^2\right )^2}{b^2-2 b x^2+x^4-a^2 x \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 )^2}{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 )}{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 )^2}{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 )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (-c x+x^3+\frac {x^2 \left (a^2 c^2-2 a^2 c x^2+a^2 x^4\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 )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=-\frac {2 c \sqrt {b+a x}}{a}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \operatorname {Subst}\left (\int \frac {x^2 \left (a^2 c^2-2 a^2 c x^2+a^2 x^4\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 )}{a}\\ &=-\frac {2 c \sqrt {b+a x}}{a}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \operatorname {Subst}\left (\int \frac {x^2 \left (-a^2 c+a^2 x^2\right )^2}{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 )}{a^3}\\ &=-\frac {2 c \sqrt {b+a x}}{a}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+\frac {4 \operatorname {Subst}\left (\int \left (\frac {2 a^4 c x^4}{-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 {a^4 c^2 x^2}{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 {a^4 x^6}{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 )}{a^3}\\ &=-\frac {2 c \sqrt {b+a x}}{a}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+(4 a) \operatorname {Subst}\left (\int \frac {x^6}{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 )+(8 a c) \operatorname {Subst}\left (\int \frac {x^4}{-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 )+\left (4 a c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{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 )\\ &=-\frac {2 c \sqrt {b+a x}}{a}+\frac {\left (c+\sqrt {b+a x}\right )^2}{a}+(4 a) \operatorname {Subst}\left (\int \frac {x^6}{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 )+(8 a c) \operatorname {Subst}\left (\int \frac {x^4}{-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 )+\left (4 a c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{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 = 2.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{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/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.62, size = 239, normalized size = 1.05 \begin {gather*} \frac {b-c^2+a x}{a}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-a^2 c-8 b c \text {$\#$1}+8 c^3 \text {$\#$1}+3 a^2 \text {$\#$1}^2+8 b \text {$\#$1}^3-24 c^2 \text {$\#$1}^3+24 c \text {$\#$1}^5-8 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b - c^2 + a*x)/a - 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^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 2*c*Log[Sqrt[c + Sqrt[b + a*
x]] - #1]*#1^4 + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(-(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(x^2/(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 {x^{2}}{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(x^2/(x^2-(a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.15, size = 189, normalized size = 0.83

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )-2 a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4} c +\textit {\_R}^{2} c^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\)	\(189\)
default	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )-2 a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4} c +\textit {\_R}^{2} c^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^2-(a*x+b)^(1/2)*(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^2*sum((_R^6-2*_R^4*c+_R^2*c^2)/(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 {x^{2}}{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(x^2/(x^2-(a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

integrate(x^2/(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 = -0.00 \begin {gather*} -\int \frac {x^2}{\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(-x^2/((c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2) - x^2),x)

[Out]

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

[Out]

Timed out

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