Optimal. Leaf size=106 \[ \frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),\frac{1}{2}\right )}{d e^{3/2} \sqrt{(c+d x)^2+1}}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}} \]
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Rubi [A] time = 0.112615, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5865, 5661, 329, 220} \[ \frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{d e^{3/2} \sqrt{(c+d x)^2+1}}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5661
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}(x)}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e x} \sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{d e^2}\\ &=-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}}+\frac{2 b (1+c+d x) \sqrt{\frac{1+(c+d x)^2}{(1+c+d x)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{d e^{3/2} \sqrt{1+(c+d x)^2}}\\ \end{align*}
Mathematica [C] time = 0.0253449, size = 56, normalized size = 0.53 \[ -\frac{2 \left (-2 b (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-(c+d x)^2\right )+a+b \sinh ^{-1}(c+d x)\right )}{d e \sqrt{e (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.009, size = 140, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{de} \left ( -{\frac{a}{\sqrt{dex+ce}}}+b \left ( -{\frac{1}{\sqrt{dex+ce}}{\it Arcsinh} \left ({\frac{dex+ce}{e}} \right ) }+2\,{\frac{1}{e}\sqrt{1-{\frac{i \left ( dex+ce \right ) }{e}}}\sqrt{1+{\frac{i \left ( dex+ce \right ) }{e}}}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{\frac{i}{e}}},i \right ){\frac{1}{\sqrt{{\frac{i}{e}}}}}{\frac{1}{\sqrt{{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d e x + c e}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\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*} \int \frac{a + b \operatorname{asinh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac{3}{2}}}\, dx \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 )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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