Optimal. Leaf size=223 \[ -\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),\frac{1}{2}\right )}{d \sqrt{e} \sqrt{(c+d x)^2+1}}+\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\frac{4 b \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{d e (c+d x+1)}+\frac{4 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{d \sqrt{e} \sqrt{(c+d x)^2+1}} \]
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Rubi [A] time = 0.21161, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5865, 5661, 329, 305, 220, 1196} \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\frac{4 b \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{d e (c+d x+1)}-\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{d \sqrt{e} \sqrt{(c+d x)^2+1}}+\frac{4 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{d \sqrt{e} \sqrt{(c+d x)^2+1}} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5661
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c+d x)}{\sqrt{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}(x)}{\sqrt{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{d e^2}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\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}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{e}}{\sqrt{1+\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{d e}\\ &=-\frac{4 b \sqrt{e (c+d x)} \sqrt{1+(c+d x)^2}}{d e (1+c+d x)}+\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}+\frac{4 b (1+c+d x) \sqrt{\frac{1+(c+d x)^2}{(1+c+d x)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{d \sqrt{e} \sqrt{1+(c+d x)^2}}-\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 \sqrt{e} \sqrt{1+(c+d x)^2}}\\ \end{align*}
Mathematica [C] time = 0.0303084, size = 61, normalized size = 0.27 \[ -\frac{2 \sqrt{e (c+d x)} \left (2 b (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-(c+d x)^2\right )-3 \left (a+b \sinh ^{-1}(c+d x)\right )\right )}{3 d e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 161, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{de} \left ( a\sqrt{dex+ce}+b \left ( \sqrt{dex+ce}{\it Arcsinh} \left ({\frac{dex+ce}{e}} \right ) -{2\,i\sqrt{1-{\frac{i \left ( dex+ce \right ) }{e}}}\sqrt{1+{\frac{i \left ( dex+ce \right ) }{e}}} \left ({\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{\frac{i}{e}}},i \right ) -{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{{\frac{i}{e}}},i \right ) \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{b \operatorname{arsinh}\left (d x + c\right ) + a}{\sqrt{d e x + c e}}, 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 )}}{\sqrt{e \left (c + d x\right )}}\, 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}{\sqrt{d e x + c e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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