Optimal. Leaf size=380 \[ x+\frac {4}{3} \left (3 a^2+c\right ) \sqrt {c+\sqrt {b+a x}}+\sqrt {b+a x} \left (2 a+\frac {4}{3} \sqrt {c+\sqrt {b+a x}}\right )-4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\& ,\frac {-a^2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-a b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a^3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-a c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\& \right ] \]
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Rubi [F]
time = 1.30, 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.
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Rubi steps
\begin {align*} \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \text {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 \text {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 \text {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 \text {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 \text {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 {\text {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 {\text {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) \text {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 ) \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 )-\left (a^2 \left (4 b-c \left (a^2+5 c\right )\right )\right ) \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 = 0.26, size = 383, normalized size = 1.01 \begin {gather*} \frac {b}{a}-\frac {c^2}{a}+x+2 a \left (c+\sqrt {b+a x}\right )+\frac {4}{3} \sqrt {c+\sqrt {b+a x}} \left (3 a^2+c+\sqrt {b+a x}\right )-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.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.03, 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.
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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.
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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.
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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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}{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.
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