Optimal. Leaf size=153 \[ \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}-\frac {b \sqrt {d} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e} \]
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Rubi [A] time = 0.30, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6299, 517, 446, 105, 63, 217, 203, 93, 207} \[ \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}-\frac {b \sqrt {d} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 203
Rule 207
Rule 217
Rule 446
Rule 517
Rule 6299
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x^2}}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx}{e}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x^2}}{x \sqrt {1-c^2 x^2}} \, dx}{e}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 e}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {1}{2} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )+\frac {\left (b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c^2}+\frac {\left (b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{e}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \sqrt {d} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{c^2}\\ &=\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}-\frac {b \sqrt {d} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 239, normalized size = 1.56 \[ \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {b \sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {e} \sqrt {c^2 (-d)-e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {c^2 (-d)-e}}\right )+c^3 \sqrt {d} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{c^3 e (c x-1) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 1102, normalized size = 7.20 \[ \left [\frac {4 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + b c \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, {\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, \sqrt {e x^{2} + d} a c - b \sqrt {-e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{4} e x^{3} + {\left (c^{4} d - c^{2} e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {-e} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + e^{2}\right )}{4 \, c e}, \frac {4 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + b c \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, {\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, \sqrt {e x^{2} + d} a c - 2 \, b \sqrt {e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{3} + {\left (c^{2} d - e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {e} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e - e^{2}\right )} x^{2} - d e\right )}}\right )}{4 \, c e}, -\frac {2 \, b c \sqrt {-d} \arctan \left (-\frac {{\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - 4 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 4 \, \sqrt {e x^{2} + d} a c + b \sqrt {-e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{4} e x^{3} + {\left (c^{4} d - c^{2} e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {-e} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + e^{2}\right )}{4 \, c e}, -\frac {b c \sqrt {-d} \arctan \left (-\frac {{\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - 2 \, \sqrt {e x^{2} + d} b c \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, \sqrt {e x^{2} + d} a c + b \sqrt {e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{3} + {\left (c^{2} d - e\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {e} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e - e^{2}\right )} x^{2} - d e\right )}}\right )}{2 \, c e}\right ] \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{\sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.73, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]
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 \[ b {\left (\frac {\sqrt {e x^{2} + d} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{e} - \int \frac {2 \, {\left (c^{2} e x^{2} - e\right )} x \log \left (\sqrt {x}\right ) + {\left (c^{2} e x^{2} \log \relax (c) - e \log \relax (c)\right )} x + {\left (2 \, {\left (c^{2} e x^{2} - e\right )} x \log \left (\sqrt {x}\right ) + {\left ({\left (e \log \relax (c) + e\right )} c^{2} x^{2} + c^{2} d - e \log \relax (c)\right )} x\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{{\left (c^{2} e x^{2} + {\left (c^{2} e x^{2} - e\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} - e\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} + \frac {\sqrt {e x^{2} + d} a}{e} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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