Optimal. Leaf size=192 \[ \frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}+\frac{3 \sqrt{1-a^2 x^2}}{256 a c^2 (c-a c x)^{3/2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{256 \sqrt{2} a c^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{64 a c (c-a c x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.159156, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6127, 663, 673, 661, 208} \[ \frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}+\frac{3 \sqrt{1-a^2 x^2}}{256 a c^2 (c-a c x)^{3/2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{256 \sqrt{2} a c^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{64 a c (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6127
Rule 663
Rule 673
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{13/2}} \, dx\\ &=\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}-\frac{1}{8} (3 c) \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{9/2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}+\frac{\int \frac{1}{(c-a c x)^{5/2} \sqrt{1-a^2 x^2}} \, dx}{16 c}\\ &=-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{64 a c (c-a c x)^{5/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}+\frac{3 \int \frac{1}{(c-a c x)^{3/2} \sqrt{1-a^2 x^2}} \, dx}{128 c^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{64 a c (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{256 a c^2 (c-a c x)^{3/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}+\frac{3 \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{512 c^3}\\ &=-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{64 a c (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{256 a c^2 (c-a c x)^{3/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )}{256 c^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{8 a (c-a c x)^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{64 a c (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{256 a c^2 (c-a c x)^{3/2}}+\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{4 a (c-a c x)^{11/2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{256 \sqrt{2} a c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0238101, size = 57, normalized size = 0.3 \[ \frac{(a x+1)^{5/2} (c-a c x)^{3/2} \text{Hypergeometric2F1}\left (\frac{5}{2},5,\frac{7}{2},\frac{1}{2} (a x+1)\right )}{80 a c^5 (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.109, size = 258, normalized size = 1.3 \begin{align*} -{\frac{1}{512\, \left ( ax-1 \right ) ^{5}a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{4}{a}^{4}c-12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{3}{a}^{3}c-6\,{x}^{3}{a}^{3}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+18\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c+26\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac+158\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c+78\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \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 x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (-a c x + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.33018, size = 944, normalized size = 4.92 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \,{\left (3 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 79 \, a x - 39\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{1024 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac{3 \, \sqrt{2}{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + 2 \,{\left (3 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 79 \, a x - 39\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{512 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x + 1\right )^{3}}{\left (- c \left (a x - 1\right )\right )^{\frac{7}{2}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.4621, size = 142, normalized size = 0.74 \begin{align*} -\frac{\frac{3 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{2 \,{\left (3 \,{\left (a c x + c\right )}^{\frac{7}{2}} - 22 \,{\left (a c x + c\right )}^{\frac{5}{2}} c - 44 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{2} + 24 \, \sqrt{a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{2}}}{512 \, a{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]