Optimal. Leaf size=145 \[ -\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),\frac{1}{2}\right )}{15 d e^{7/2} \sqrt{(c+d x)^2+1}}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}-\frac{4 b \sqrt{(c+d x)^2+1}}{15 d e^2 (e (c+d x))^{3/2}} \]
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Rubi [A] time = 0.135654, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5865, 5661, 325, 329, 220} \[ -\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}-\frac{4 b \sqrt{(c+d x)^2+1}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )|\frac{1}{2}\right )}{15 d e^{7/2} \sqrt{(c+d x)^2+1}} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5661
Rule 325
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}(x)}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{(e x)^{5/2} \sqrt{1+x^2}} \, dx,x,c+d x\right )}{5 d e}\\ &=-\frac{4 b \sqrt{1+(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e x} \sqrt{1+x^2}} \, dx,x,c+d x\right )}{15 d e^3}\\ &=-\frac{4 b \sqrt{1+(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{15 d e^4}\\ &=-\frac{4 b \sqrt{1+(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}-\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 )}{15 d e^{7/2} \sqrt{1+(c+d x)^2}}\\ \end{align*}
Mathematica [C] time = 0.0356996, size = 61, normalized size = 0.42 \[ \frac{-4 b (c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{4},\frac{1}{2},\frac{1}{4},-(c+d x)^2\right )-6 \left (a+b \sinh ^{-1}(c+d x)\right )}{15 d e (e (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 176, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{de} \left ( -1/5\,{\frac{a}{ \left ( dex+ce \right ) ^{5/2}}}+b \left ( -1/5\,{\frac{1}{ \left ( dex+ce \right ) ^{5/2}}{\it Arcsinh} \left ({\frac{dex+ce}{e}} \right ) }+2/5\,{\frac{1}{e} \left ( -1/3\,{\frac{1}{ \left ( dex+ce \right ) ^{3/2}}\sqrt{{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}-1/3\,{\frac{1}{{e}^{2}}\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 ) \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^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, 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 \frac{b \operatorname{arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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