Optimal. Leaf size=261 \[ \frac{6 b e^{3/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),\frac{1}{2}\right )}{25 d \sqrt{(c+d x)^2+1}}+\frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}-\frac{12 b e^{3/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{25 d \sqrt{(c+d x)^2+1}}-\frac{4 b \sqrt{(c+d x)^2+1} (e (c+d x))^{3/2}}{25 d}+\frac{12 b e \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{25 d (c+d x+1)} \]
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Rubi [A] time = 0.24261, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5865, 5661, 321, 329, 305, 220, 1196} \[ \frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}+\frac{6 b e^{3/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{25 d \sqrt{(c+d x)^2+1}}-\frac{12 b e^{3/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{25 d \sqrt{(c+d x)^2+1}}-\frac{4 b \sqrt{(c+d x)^2+1} (e (c+d x))^{3/2}}{25 d}+\frac{12 b e \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{25 d (c+d x+1)} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5661
Rule 321
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int (c e+d e x)^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{3/2} \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{5 d e}\\ &=-\frac{4 b (e (c+d x))^{3/2} \sqrt{1+(c+d x)^2}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}+\frac{(6 b e) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{4 b (e (c+d x))^{3/2} \sqrt{1+(c+d x)^2}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}+\frac{(12 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{25 d}\\ &=-\frac{4 b (e (c+d x))^{3/2} \sqrt{1+(c+d x)^2}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}+\frac{(12 b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{25 d}-\frac{(12 b e) \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 )}{25 d}\\ &=-\frac{4 b (e (c+d x))^{3/2} \sqrt{1+(c+d x)^2}}{25 d}+\frac{12 b e \sqrt{e (c+d x)} \sqrt{1+(c+d x)^2}}{25 d (1+c+d x)}+\frac{2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e}-\frac{12 b e^{3/2} (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 )}{25 d \sqrt{1+(c+d x)^2}}+\frac{6 b e^{3/2} (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 )}{25 d \sqrt{1+(c+d x)^2}}\\ \end{align*}
Mathematica [C] time = 0.0465343, size = 87, normalized size = 0.33 \[ \frac{2 (e (c+d x))^{3/2} \left (2 b \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-(c+d x)^2\right )+5 a c+5 a d x-2 b \sqrt{(c+d x)^2+1}+5 b c \sinh ^{-1}(c+d x)+5 b d x \sinh ^{-1}(c+d x)\right )}{25 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 205, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{de} \left ( 1/5\, \left ( dex+ce \right ) ^{5/2}a+b \left ( 1/5\, \left ( dex+ce \right ) ^{5/2}{\it Arcsinh} \left ({\frac{dex+ce}{e}} \right ) -2/5\,{\frac{1}{e} \left ( 1/5\,{e}^{2} \left ( dex+ce \right ) ^{3/2}\sqrt{{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}-{3/5\,i{e}^{3}\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 ) \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 ({\left (a d e x + a c e +{\left (b d e x + b c e\right )} \operatorname{arsinh}\left (d x + c\right )\right )} \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 \left (e \left (c + d x\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asinh}{\left (c + d x \right )}\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{\left (d e x + c e\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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