Optimal. Leaf size=97 \[ \frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 e}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \sinh ^{-1}(c x)}{4 e}-\frac {b \sqrt {c^2 x^2+1} (d+e x)}{4 c}-\frac {3 b d \sqrt {c^2 x^2+1}}{4 c} \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5801, 743, 641, 215} \[ \frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 e}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \sinh ^{-1}(c x)}{4 e}-\frac {b \sqrt {c^2 x^2+1} (d+e x)}{4 c}-\frac {3 b d \sqrt {c^2 x^2+1}}{4 c} \]
Antiderivative was successfully verified.
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Rule 215
Rule 641
Rule 743
Rule 5801
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 e}-\frac {(b c) \int \frac {(d+e x)^2}{\sqrt {1+c^2 x^2}} \, dx}{2 e}\\ &=-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {2 c^2 d^2-e^2+3 c^2 d e x}{\sqrt {1+c^2 x^2}} \, dx}{4 c e}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2}}{4 c}-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 e}-\frac {1}{4} \left (b \left (\frac {2 c d^2}{e}-\frac {e}{c}\right )\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {3 b d \sqrt {1+c^2 x^2}}{4 c}-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \sinh ^{-1}(c x)}{4 e}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 0.94 \[ a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {c^2 x^2+1}}{c}-\frac {b e x \sqrt {c^2 x^2+1}}{4 c}+\frac {b e \sinh ^{-1}(c x)}{4 c^2}+b d x \sinh ^{-1}(c x)+\frac {1}{2} b e x^2 \sinh ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 87, normalized size = 0.90 \[ \frac {2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x + {\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x + b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c e x + 4 \, b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 124, normalized size = 1.28 \[ {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} b d + a d x + \frac {1}{4} \, {\left (2 \, a x^{2} + {\left (2 \, x^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} + \frac {\log \left (-x {\left | c \right |} + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} {\left | c \right |}}\right )}\right )} b\right )} e \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 0.99 \[ \frac {\frac {a \left (\frac {1}{2} c^{2} x^{2} e +c^{2} d x \right )}{c}+\frac {b \left (\frac {\arcsinh \left (c x \right ) c^{2} x^{2} e}{2}+\arcsinh \left (c x \right ) c^{2} x d -\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )}{2}-c d \sqrt {c^{2} x^{2}+1}\right )}{c}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 82, normalized size = 0.85 \[ \frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 78, normalized size = 0.80 \[ \frac {a\,x\,\left (2\,d+e\,x\right )}{2}-\frac {b\,d\,\left (\sqrt {c^2\,x^2+1}-c\,x\,\mathrm {asinh}\left (c\,x\right )\right )}{c}-\frac {b\,e\,x\,\sqrt {c^2\,x^2+1}}{4\,c}+b\,e\,x\,\mathrm {asinh}\left (c\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,c^2\,x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 99, normalized size = 1.02 \[ \begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x \sqrt {c^{2} x^{2} + 1}}{4 c} + \frac {b e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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