Optimal. Leaf size=165 \[ -\frac {4}{3} c \sqrt {c+\sqrt {b+a x}}+\sqrt {b+a x} \left (\frac {2 (b+a x)}{3 a}-\frac {4}{3} \sqrt {c+\sqrt {b+a x}}\right )+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\& ,\frac {-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a-4 c \text {$\#$1}+4 \text {$\#$1}^3}\& \right ] \]
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Rubi [F]
time = 1.11, 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 \sqrt {b+a x}}{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 \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^2 \left (-b+x^2\right )}{-b+x^2+a \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )^2 \left (-b+\left (c-x^2\right )^2\right )}{-b+a x+\left (c-x^2\right )^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )^2 \left (b-c^2+2 c x^2-x^4\right )}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \text {Subst}\left (\int \left (c^2 x-a x^2-2 c x^3+x^5+\frac {x^2 \left (a b-a^2 x\right )}{b-c^2-a x+2 c x^2-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+\frac {4 \text {Subst}\left (\int \frac {x^2 \left (a b-a^2 x\right )}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+a \log \left (b-c^2-a \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )-\left (c+\sqrt {b+a x}\right )^2\right )-\frac {\text {Subst}\left (\int \frac {-a^3+4 a^2 c x-4 a b x^2}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+a \log \left (b-c^2-a \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )-\left (c+\sqrt {b+a x}\right )^2\right )-\frac {\text {Subst}\left (\int \left (\frac {a^3}{-b+c^2+a x-2 c x^2+x^4}-\frac {4 a^2 c x}{-b+c^2+a x-2 c x^2+x^4}+\frac {4 a b x^2}{-b+c^2+a x-2 c x^2+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+a \log \left (b-c^2-a \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )-\left (c+\sqrt {b+a x}\right )^2\right )-a^2 \text {Subst}\left (\int \frac {1}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(4 b) \text {Subst}\left (\int \frac {x^2}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+(4 a c) \text {Subst}\left (\int \frac {x}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 142, normalized size = 0.86 \begin {gather*} -\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a-4 c \text {$\#$1}+4 \text {$\#$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 = 0.10, size = 129, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (c +\sqrt {a x +b}\right )^{3}}{3}-2 c \left (c +\sqrt {a x +b}\right )^{2}-\frac {4 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 c^{2} \left (c +\sqrt {a x +b}\right )+4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\) | \(129\) |
default | \(\frac {\frac {2 \left (c +\sqrt {a x +b}\right )^{3}}{3}-2 c \left (c +\sqrt {a x +b}\right )^{2}-\frac {4 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 c^{2} \left (c +\sqrt {a x +b}\right )+4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\) | \(129\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {a x + b}}{x + \sqrt {c + \sqrt {a x + b}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {x\,\sqrt {b+a\,x}}{x+\sqrt {c+\sqrt {b+a\,x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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