Optimal. Leaf size=109 \[ \frac {\sqrt {a+c x^2}}{c e}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}-\frac {d^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \sqrt {c d^2+a e^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1668, 12, 858,
223, 212, 739} \begin {gather*} -\frac {d^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \sqrt {a e^2+c d^2}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}+\frac {\sqrt {a+c x^2}}{c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 223
Rule 739
Rule 858
Rule 1668
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx &=\frac {\sqrt {a+c x^2}}{c e}-\frac {\int \frac {c d e x}{(d+e x) \sqrt {a+c x^2}} \, dx}{c e^2}\\ &=\frac {\sqrt {a+c x^2}}{c e}-\frac {d \int \frac {x}{(d+e x) \sqrt {a+c x^2}} \, dx}{e}\\ &=\frac {\sqrt {a+c x^2}}{c e}-\frac {d \int \frac {1}{\sqrt {a+c x^2}} \, dx}{e^2}+\frac {d^2 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^2}\\ &=\frac {\sqrt {a+c x^2}}{c e}-\frac {d \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{e^2}-\frac {d^2 \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^2}\\ &=\frac {\sqrt {a+c x^2}}{c e}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}-\frac {d^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \sqrt {c d^2+a e^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 116, normalized size = 1.06 \begin {gather*} \frac {\frac {e \sqrt {a+c x^2}}{c}-\frac {2 d^2 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 172, normalized size = 1.58
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}+a}}{c e}-\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(172\) |
risch | \(\frac {\sqrt {c \,x^{2}+a}}{c e}-\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 89, normalized size = 0.82 \begin {gather*} \frac {d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-3\right )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e^{\left (-1\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.62, size = 721, normalized size = 6.61 \begin {gather*} \left [\frac {\sqrt {c d^{2} + a e^{2}} c d^{2} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + {\left (c d^{3} + a d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} c d^{2} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (c d^{3} + a d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}, \frac {\sqrt {c d^{2} + a e^{2}} c d^{2} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (c d^{3} + a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}, \frac {\sqrt {-c d^{2} - a e^{2}} c d^{2} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (c d^{3} + a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} e^{2} + a c e^{4}}\right ] \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^{2}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 105, normalized size = 0.96 \begin {gather*} \frac {2 \, d^{2} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {d e^{\left (-2\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e^{\left (-1\right )}}{c} \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^2}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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