Optimal. Leaf size=251 \[ \frac{7 \sqrt{a x+1} (1-a x)^{7/2}}{4 a^4 x^3 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{(1-a x)^{7/2}}{4 a^3 x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{(1-a x)^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 (1-a x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{4 \sqrt{2} a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{(1-a x)^{5/2}}{2 a^2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.221361, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6134, 6129, 98, 150, 154, 157, 54, 215, 93, 206} \[ \frac{7 \sqrt{a x+1} (1-a x)^{7/2}}{4 a^4 x^3 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{(1-a x)^{7/2}}{4 a^3 x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{(1-a x)^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 (1-a x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{4 \sqrt{2} a^{9/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{(1-a x)^{5/2}}{2 a^2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6134
Rule 6129
Rule 98
Rule 150
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{7/2}} \, dx &=\frac{(1-a x)^{7/2} \int \frac{e^{-3 \tanh ^{-1}(a x)} x^{7/2}}{(1-a x)^{7/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{7/2} \int \frac{x^{7/2}}{(1-a x)^2 (1+a x)^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{5/2}}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}-\frac{(1-a x)^{7/2} \int \frac{x^{3/2} \left (\frac{5}{2}+3 a x\right )}{(1-a x) (1+a x)^{3/2}} \, dx}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{5/2}}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}-\frac{(1-a x)^{7/2}}{4 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}-\frac{(1-a x)^{7/2} \int \frac{\sqrt{x} \left (-\frac{3 a}{4}+\frac{7 a^2 x}{2}\right )}{(1-a x) \sqrt{1+a x}} \, dx}{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{5/2}}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}-\frac{(1-a x)^{7/2}}{4 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}+\frac{7 (1-a x)^{7/2} \sqrt{1+a x}}{4 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}+\frac{(1-a x)^{7/2} \int \frac{-\frac{7 a^2}{4}-a^3 x}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{2 a^6 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{5/2}}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}-\frac{(1-a x)^{7/2}}{4 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}+\frac{7 (1-a x)^{7/2} \sqrt{1+a x}}{4 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}+\frac{(1-a x)^{7/2} \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}-\frac{\left (11 (1-a x)^{7/2}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{8 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{5/2}}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}-\frac{(1-a x)^{7/2}}{4 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}+\frac{7 (1-a x)^{7/2} \sqrt{1+a x}}{4 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}+\frac{(1-a x)^{7/2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}-\frac{\left (11 (1-a x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{4 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac{(1-a x)^{5/2}}{2 a^2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}-\frac{(1-a x)^{7/2}}{4 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}+\frac{7 (1-a x)^{7/2} \sqrt{1+a x}}{4 a^4 \left (c-\frac{c}{a x}\right )^{7/2} x^3}+\frac{(1-a x)^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{9/2} \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}-\frac{11 (1-a x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{4 \sqrt{2} a^{9/2} \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.599073, size = 234, normalized size = 0.93 \[ \frac{-40 a^{7/2} x^{7/2} (a x-1) \sqrt{a x+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{7}{2},\frac{9}{2},-a x\right )-56 a^{5/2} x^{5/2} (a x-1) \sqrt{a x+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},-a x\right )+35 \left (\sqrt{a} \sqrt{x} \left (2 a^3 x^3+13 a^2 x^2+2 a x-25\right )+19 (a x-1) \sqrt{a x+1} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-22 (a x-1) \sqrt{2 a x+2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{560 a^{3/2} c^3 \sqrt{x} \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.157, size = 315, normalized size = 1.3 \begin{align*} -{\frac{x\sqrt{2}}{16\,{c}^{4} \left ( ax+1 \right ) \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 8\,\sqrt{- \left ( ax+1 \right ) x}{a}^{7/2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}+2\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x-4\,{a}^{3}\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}+11\,{a}^{5/2}\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ){x}^{2}-14\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}+4\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}-11\,\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ) \sqrt{a} \right ) \sqrt{-{a}^{2}{x}^{2}+1}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-{a}^{-1}}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]