Optimal. Leaf size=455 \[ -4 a \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c-2 \text {$\#$1}^4 b+6 \text {$\#$1}^4 c^2-\text {$\#$1}^3 a^2+4 \text {$\#$1}^2 b c-4 \text {$\#$1}^2 c^3+\text {$\#$1} a^2 c+b^2-2 b c^2+c^4\& ,\frac {\text {$\#$1}^6 (-c) \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+3 \text {$\#$1}^4 c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-2 \text {$\#$1}^4 b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^3 a^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-3 \text {$\#$1}^2 c^3 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+4 \text {$\#$1}^2 b c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+\text {$\#$1} a^2 c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{8 \text {$\#$1}^7-24 \text {$\#$1}^5 c-8 \text {$\#$1}^3 b+24 \text {$\#$1}^3 c^2-3 \text {$\#$1}^2 a^2+8 \text {$\#$1} b c-8 \text {$\#$1} c^3+a^2 c}\& \right ]+4 a \sqrt {\sqrt {a x+b}+c}+\frac {2 (a x+b)^{3/2}}{3 a} \]
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Rubi [F] time = 5.08, 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 \sqrt {b+a x}}{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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\begin {align*} \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (-b x+x^3\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 {2 \operatorname {Subst}\left (\int \frac {x^2 \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 )^2 \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 )^2 \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 (a^2+c^2 x-2 c x^3+x^5-\frac {a^2 \left (b-c^2\right )^2+a^4 c x+a^2 c \left (4 b-3 c^2\right ) x^2-a^4 x^3-a^2 \left (2 b-3 c^2\right ) x^4-a^2 c x^6}{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^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\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 \operatorname {Subst}\left (\int \frac {a^2 \left (b-c^2\right )^2+a^4 c x+a^2 c \left (4 b-3 c^2\right ) x^2-a^4 x^3-a^2 \left (2 b-3 c^2\right ) x^4-a^2 c x^6}{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^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\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 \operatorname {Subst}\left (\int \frac {a^2 \left (b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {3 c^2}{4 b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{2 b}\right ) x^4-c x^6\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^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-(4 a) \operatorname {Subst}\left (\int \frac {b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {3 c^2}{4 b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{2 b}\right ) x^4-c x^6}{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 )\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-(4 a) \operatorname {Subst}\left (\int \left (\frac {a^2 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}+\frac {c 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}+\frac {\left (b-c^2\right )^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^2 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 {c \left (4 b-3 c^2\right ) 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 {\left (-2 b+3 c^2\right ) 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}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-\left (4 a^3\right ) \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^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 )-\left (4 a^3 c\right ) \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 )+\left (4 a \left (2 b-3 c^2\right )\right ) \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 \left (4 b-3 c^2\right )\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 )-\left (4 a \left (b-c^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{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^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-\left (4 a^3\right ) \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^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 )-\left (4 a^3 c\right ) \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 )+\left (4 a \left (2 b-3 c^2\right )\right ) \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 \left (4 b-3 c^2\right )\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 )-\left (4 a \left (b-c^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{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 = 3.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \sqrt {b+a x}}{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 = 0.85, size = 459, normalized size = 1.01 \begin {gather*} 4 a \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+4 b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-3 c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$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 {\sqrt {a x + b} 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 = 0.23, size = 264, normalized size = 0.58
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}-c^{2} \left (c +\sqrt {a x +b}\right )-2 a^{2} \sqrt {c +\sqrt {a x +b}}+2 a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{6} c +\left (3 c^{2}-2 b \right ) \textit {\_R}^{4}-\textit {\_R}^{3} a^{2}+c \left (-3 c^{2}+4 b \right ) \textit {\_R}^{2}+a^{2} c \textit {\_R} +c^{4}-2 b \,c^{2}+b^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\) | \(264\) |
default | \(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}-c^{2} \left (c +\sqrt {a x +b}\right )-2 a^{2} \sqrt {c +\sqrt {a x +b}}+2 a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{6} c +\left (3 c^{2}-2 b \right ) \textit {\_R}^{4}-\textit {\_R}^{3} a^{2}+c \left (-3 c^{2}+4 b \right ) \textit {\_R}^{2}+a^{2} c \textit {\_R} +c^{4}-2 b \,c^{2}+b^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\) | \(264\) |
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 {\sqrt {a x + b} 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 {b+a\,x}}{\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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