Optimal. Leaf size=127 \[ \frac {\left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^3}+\frac {d \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3}-\frac {\sqrt {a+c x^2} (2 d-e x)}{2 e^2} \]
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Rubi [A] time = 0.11, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {815, 844, 217, 206, 725} \[ \frac {\left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^3}+\frac {d \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3}-\frac {\sqrt {a+c x^2} (2 d-e x)}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {x \sqrt {a+c x^2}}{d+e x} \, dx &=-\frac {(2 d-e x) \sqrt {a+c x^2}}{2 e^2}+\frac {\int \frac {-a c d e+c \left (2 c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c e^2}\\ &=-\frac {(2 d-e x) \sqrt {a+c x^2}}{2 e^2}-\frac {\left (d \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^3}+\frac {\left (2 c d^2+a e^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 e^3}\\ &=-\frac {(2 d-e x) \sqrt {a+c x^2}}{2 e^2}+\frac {\left (d \left (c d^2+a e^2\right )\right ) \operatorname {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^3}+\frac {\left (2 c d^2+a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 e^3}\\ &=-\frac {(2 d-e x) \sqrt {a+c x^2}}{2 e^2}+\frac {\left (2 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^3}+\frac {d \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 175, normalized size = 1.38 \[ \frac {\frac {a^{3/2} e^2 \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {c} \sqrt {a+c x^2}}+2 d \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )+2 \sqrt {c} d^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-2 d e \sqrt {a+c x^2}+e^2 x \sqrt {a+c x^2}}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 684, normalized size = 5.39 \[ \left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} c d \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + {\left (2 \, c d^{2} + a e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (c e^{2} x - 2 \, c d e\right )} \sqrt {c x^{2} + a}}{4 \, c e^{3}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} c d \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (2 \, c d^{2} + a e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (c e^{2} x - 2 \, c d e\right )} \sqrt {c x^{2} + a}}{4 \, c e^{3}}, \frac {\sqrt {c d^{2} + a e^{2}} c d \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (2 \, c d^{2} + a e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (c e^{2} x - 2 \, c d e\right )} \sqrt {c x^{2} + a}}{2 \, c e^{3}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} c d \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (2 \, c d^{2} + a e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (c e^{2} x - 2 \, c d e\right )} \sqrt {c x^{2} + a}}{2 \, c e^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 135, normalized size = 1.06 \[ -\frac {2 \, {\left (c d^{3} + a d e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-3\right )}}{\sqrt {-c d^{2} - a e^{2}}} - \frac {{\left (2 \, c^{\frac {3}{2}} d^{2} + a \sqrt {c} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c} + \frac {1}{2} \, \sqrt {c x^{2} + a} {\left (x e^{\left (-1\right )} - 2 \, d e^{\left (-2\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 423, normalized size = 3.33 \[ \frac {a d \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{2}}+\frac {c \,d^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{4}}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}\, e}+\frac {\sqrt {c}\, d^{2} \ln \left (\frac {-\frac {c d}{e}+\left (x +\frac {d}{e}\right ) c}{\sqrt {c}}+\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e^{3}}+\frac {\sqrt {c \,x^{2}+a}\, x}{2 e}-\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 122, normalized size = 0.96 \[ \frac {\sqrt {c x^{2} + a} x}{2 \, e} + \frac {\sqrt {c} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{e^{3}} + \frac {a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c} e} - \frac {\sqrt {a + \frac {c d^{2}}{e^{2}}} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{e^{2}} - \frac {\sqrt {c x^{2} + a} d}{e^{2}} \]
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 \[ \int \frac {x\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \]
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 \[ \int \frac {x \sqrt {a + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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