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 = 2.84, 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.
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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 = 2.21, 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.
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IntegrateAlgebraic [A] time = 0.00, 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.
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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.
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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.
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maple [B] time = 0.14, 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.
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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.
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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.
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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.
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