Optimal. Leaf size=187 \[ \frac{3 c^2 x \sqrt{c-a^2 c x^2}}{64 a^3}+\frac{3 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{64 a^4}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac{(315 a x+208) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37002, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{3 c^2 x \sqrt{c-a^2 c x^2}}{64 a^3}+\frac{3 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{64 a^4}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac{(315 a x+208) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4} \]
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 )^{5/2} \, dx &=c \int x^3 (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{\int x^3 \left (-13 a^2 c-18 a^3 c x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{9 a^2}\\ &=-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}+\frac{\int x^2 \left (54 a^3 c^2+104 a^4 c^2 x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{72 a^4 c}\\ &=-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{\int x \left (-208 a^4 c^3-378 a^5 c^3 x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{504 a^6 c^2}\\ &=-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac{c \int \left (c-a^2 c x^2\right )^{3/2} \, dx}{8 a^3}\\ &=\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac{\left (3 c^2\right ) \int \sqrt{c-a^2 c x^2} \, dx}{32 a^3}\\ &=\frac{3 c^2 x \sqrt{c-a^2 c x^2}}{64 a^3}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac{\left (3 c^3\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{64 a^3}\\ &=\frac{3 c^2 x \sqrt{c-a^2 c x^2}}{64 a^3}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{64 a^3}\\ &=\frac{3 c^2 x \sqrt{c-a^2 c x^2}}{64 a^3}+\frac{c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac{13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac{x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac{1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac{(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac{3 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{64 a^4}\\ \end{align*}
Mathematica [A] time = 0.214483, size = 131, normalized size = 0.7 \[ -\frac{c^2 \left (\left (2240 a^8 x^8+5040 a^7 x^7-320 a^6 x^6-7560 a^5 x^5-4416 a^4 x^4+630 a^3 x^3+832 a^2 x^2+945 a x+1664\right ) \sqrt{c-a^2 c x^2}+945 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )\right )}{20160 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.046, size = 330, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}}{9\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{20}{63\,c{a}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{x}{4\,{a}^{3}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,x}{8\,{a}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{15\,cx}{32\,{a}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{45\,x{c}^{2}}{64\,{a}^{3}}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{45\,{c}^{3}}{64\,{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}{5\,{a}^{4}} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{cx}{2\,{a}^{3}} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x{c}^{2}}{4\,{a}^{3}}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{3\,{c}^{3}}{4\,{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 = 2.88005, size = 726, normalized size = 3.88 \begin{align*} \left [\frac{945 \, \sqrt{-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{40320 \, a^{4}}, -\frac{945 \, c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) +{\left (2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{20160 \, a^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 25.4706, size = 763, normalized size = 4.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18195, size = 209, normalized size = 1.12 \begin{align*} \frac{1}{20160} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left ({\left (4 \,{\left (552 \, c^{2} + 5 \,{\left (189 \, a c^{2} + 2 \,{\left (4 \, a^{2} c^{2} - 7 \,{\left (4 \, a^{4} c^{2} x + 9 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac{315 \, c^{2}}{a}\right )} x - \frac{416 \, c^{2}}{a^{2}}\right )} x - \frac{945 \, c^{2}}{a^{3}}\right )} x - \frac{1664 \, c^{2}}{a^{4}}\right )} - \frac{3 \, c^{3} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{64 \, a^{3} \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]