Optimal. Leaf size=169 \[ -\frac{20 b e^{5/2} \sqrt{-c-d x+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),-1\right )}{147 d \sqrt{c+d x-1}}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{20 b e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{e (c+d x)}}{147 d}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{5/2}}{49 d} \]
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Rubi [A] time = 0.133221, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 5662, 102, 12, 117, 116} \[ \frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{20 b e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{e (c+d x)}}{147 d}-\frac{20 b e^{5/2} \sqrt{-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{147 d \sqrt{c+d x-1}}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{5/2}}{49 d} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 5662
Rule 102
Rule 12
Rule 117
Rule 116
Rubi steps
\begin{align*} \int (c e+d e x)^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{5/2} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{7/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{5/2} \sqrt{1+c+d x}}{49 d}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{5 e^2 (e x)^{3/2}}{2 \sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{49 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{5/2} \sqrt{1+c+d x}}{49 d}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{(10 b e) \operatorname{Subst}\left (\int \frac{(e x)^{3/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{49 d}\\ &=-\frac{20 b e^2 \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{147 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{5/2} \sqrt{1+c+d x}}{49 d}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{(20 b e) \operatorname{Subst}\left (\int \frac{e^2}{2 \sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{147 d}\\ &=-\frac{20 b e^2 \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{147 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{5/2} \sqrt{1+c+d x}}{49 d}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{\left (10 b e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{147 d}\\ &=-\frac{20 b e^2 \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{147 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{5/2} \sqrt{1+c+d x}}{49 d}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{\left (10 b e^3 \sqrt{1-c-d x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{147 d \sqrt{-1+c+d x}}\\ &=-\frac{20 b e^2 \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}{147 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{5/2} \sqrt{1+c+d x}}{49 d}+\frac{2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac{20 b e^{5/2} \sqrt{1-c-d x} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{147 d \sqrt{-1+c+d x}}\\ \end{align*}
Mathematica [C] time = 0.26376, size = 180, normalized size = 1.07 \[ \frac{2 (e (c+d x))^{5/2} \left (-10 b \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},(c+d x)^2\right )+21 a \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3-6 b (c+d x)^4-4 b (c+d x)^2+21 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3 \cosh ^{-1}(c+d x)+10 b\right )}{147 d \sqrt{\frac{c+d x-1}{c+d x}} (c+d x)^{5/2} \sqrt{c+d x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 218, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{de} \left ( 1/7\, \left ( dex+ce \right ) ^{7/2}a+b \left ( 1/7\, \left ( dex+ce \right ) ^{7/2}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )-{\frac{2}{147\,e} \left ( 3\,\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{9/2}+2\,\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{5/2}{e}^{2}+5\,{e}^{4}\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) -5\,\sqrt{-{e}^{-1}}\sqrt{dex+ce}{e}^{4} \right ){\frac{1}{\sqrt{-{e}^{-1}}}}{\frac{1}{\sqrt{{\frac{dex+ce+e}{e}}}}}{\frac{1}{\sqrt{{\frac{dex+ce-e}{e}}}}}} \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^{2} e^{2} x^{2} + 2 \, a c d e^{2} x + a c^{2} e^{2} +{\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + b c^{2} e^{2}\right )} \operatorname{arcosh}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}}{\left (b \operatorname{arcosh}\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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