Optimal. Leaf size=67 \[ -\frac {a e^{2 \sinh ^{-1}(a+b x)}}{4 b^2}-\frac {a \sinh ^{-1}(a+b x)}{2 b^2}+\frac {e^{-\sinh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \sinh ^{-1}(a+b x)}}{12 b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5898, 2282, 12, 1628} \[ -\frac {a e^{2 \sinh ^{-1}(a+b x)}}{4 b^2}-\frac {a \sinh ^{-1}(a+b x)}{2 b^2}+\frac {e^{-\sinh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \sinh ^{-1}(a+b x)}}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1628
Rule 2282
Rule 5898
Rubi steps
\begin {align*} \int e^{\sinh ^{-1}(a+b x)} x \, dx &=\frac {\operatorname {Subst}\left (\int e^x \cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1+2 a x-x^2\right )}{4 b x^2} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1+2 a x-x^2\right )}{x^2} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{4 b^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^2}-\frac {2 a}{x}-2 a x+x^2\right ) \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{4 b^2}\\ &=\frac {e^{-\sinh ^{-1}(a+b x)}}{4 b^2}-\frac {a e^{2 \sinh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \sinh ^{-1}(a+b x)}}{12 b^2}-\frac {a \sinh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 73, normalized size = 1.09 \[ \frac {1}{6} \left (\frac {\sqrt {a^2+2 a b x+b^2 x^2+1} \left (-a^2+a b x+2 b^2 x^2+2\right )}{b^2}-\frac {3 a \sinh ^{-1}(a+b x)}{b^2}+3 a x^2+2 b x^3\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 93, normalized size = 1.39 \[ \frac {2 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (2 \, b^{2} x^{2} + a b x - a^{2} + 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 106, normalized size = 1.58 \[ \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {1}{6} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (2 \, x + \frac {a}{b}\right )} x - \frac {a^{2} b - 2 \, b}{b^{3}}\right )} + \frac {a \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 138, normalized size = 2.06 \[ \frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b}-\frac {a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b \sqrt {b^{2}}}+\frac {b \,x^{3}}{3}+\frac {a \,x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 175, normalized size = 2.61 \[ \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b} - \frac {{\left (a^{2} + 1\right )} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{2}} \]
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 \[ \int x\,\left (a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x\right ) \,d x \]
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 \[ \int x \left (a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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