Optimal. Leaf size=161 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^4}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}+\frac{c x \sqrt{c-a^2 c x^2}}{8 a^3}-\frac{(105 a x+88) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.335875, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6151, 1809, 833, 780, 195, 217, 203} \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^4}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}+\frac{c x \sqrt{c-a^2 c x^2}}{8 a^3}-\frac{(105 a x+88) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6151
Rule 1809
Rule 833
Rule 780
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x^3 (1+a x)^2 \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{\int x^3 \left (-11 a^2 c-14 a^3 c x\right ) \sqrt{c-a^2 c x^2} \, dx}{7 a^2}\\ &=-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}+\frac{\int x^2 \left (42 a^3 c^2+66 a^4 c^2 x\right ) \sqrt{c-a^2 c x^2} \, dx}{42 a^4 c}\\ &=-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{\int x \left (-132 a^4 c^3-210 a^5 c^3 x\right ) \sqrt{c-a^2 c x^2} \, dx}{210 a^6 c^2}\\ &=-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac{c \int \sqrt{c-a^2 c x^2} \, dx}{4 a^3}\\ &=\frac{c x \sqrt{c-a^2 c x^2}}{8 a^3}-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac{c^2 \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{8 a^3}\\ &=\frac{c x \sqrt{c-a^2 c x^2}}{8 a^3}-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3}\\ &=\frac{c x \sqrt{c-a^2 c x^2}}{8 a^3}-\frac{11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac{1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac{(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac{c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.136877, size = 113, normalized size = 0.7 \[ \frac{c \left (120 a^6 x^6+280 a^5 x^5+144 a^4 x^4-70 a^3 x^3-88 a^2 x^2-105 a x-176\right ) \sqrt{c-a^2 c x^2}-105 c^{3/2} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{840 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.046, size = 268, normalized size = 1.7 \begin{align*}{\frac{{x}^{2}}{7\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{16}{35\,c{a}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{x}{3\,{a}^{3}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,x}{12\,{a}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{7\,cx}{8\,{a}^{3}}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{7\,{c}^{2}}{8\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{2}{3\,{a}^{4}} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{cx}{{a}^{3}}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{{c}^{2}}{{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.75058, size = 566, normalized size = 3.52 \begin{align*} \left [\frac{105 \, \sqrt{-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{1680 \, a^{4}}, -\frac{105 \, c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) -{\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{840 \, a^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.9152, size = 420, normalized size = 2.61 \begin{align*} a^{2} c \left (\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{16 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{16 a^{5}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17976, size = 158, normalized size = 0.98 \begin{align*} \frac{1}{840} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (3 \, a^{2} c x + 7 \, a c\right )} x + 18 \, c\right )} x - \frac{35 \, c}{a}\right )} x - \frac{44 \, c}{a^{2}}\right )} x - \frac{105 \, c}{a^{3}}\right )} x - \frac{176 \, c}{a^{4}}\right )} - \frac{c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{3} \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]