Optimal. Leaf size=59 \[ -\frac{a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b \sqrt{(c+d x)^2+1}}{2 d e^3 (c+d x)} \]
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Rubi [A] time = 0.0509626, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5865, 12, 5661, 264} \[ -\frac{a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b \sqrt{(c+d x)^2+1}}{2 d e^3 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5661
Rule 264
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac{a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x^2}} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac{b \sqrt{1+(c+d x)^2}}{2 d e^3 (c+d x)}-\frac{a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}\\ \end{align*}
Mathematica [A] time = 0.049776, size = 47, normalized size = 0.8 \[ -\frac{a+b (c+d x) \sqrt{(c+d x)^2+1}+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 60, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{a}{2\,{e}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{b}{{e}^{3}} \left ( -{\frac{{\it Arcsinh} \left ( dx+c \right ) }{2\, \left ( dx+c \right ) ^{2}}}-{\frac{1}{2\,dx+2\,c}\sqrt{1+ \left ( dx+c \right ) ^{2}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13941, size = 158, normalized size = 2.68 \begin{align*} -\frac{1}{2} \, b{\left (\frac{\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac{\operatorname{arsinh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac{a}{2 \,{\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.63617, size = 257, normalized size = 4.36 \begin{align*} \frac{a d^{2} x^{2} + 2 \, a c d x - b c^{2} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) -{\left (b c^{2} d x + b c^{3}\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{2 \,{\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{b \operatorname{asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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