Optimal. Leaf size=252 \[ \frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^2 \sqrt{c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt{c^2 x^2}}-\frac{b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{1680 c^6 \sqrt{c^2 x^2}}-\frac{b e x^4 \sqrt{c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt{c^2 x^2}}-\frac{b e^2 x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.235731, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {270, 5238, 12, 1267, 459, 321, 217, 206} \[ \frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^2 \sqrt{c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt{c^2 x^2}}-\frac{b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{1680 c^6 \sqrt{c^2 x^2}}-\frac{b e x^4 \sqrt{c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt{c^2 x^2}}-\frac{b e^2 x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 5238
Rule 12
Rule 1267
Rule 459
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{105 \sqrt{c^2 x^2}}\\ &=-\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b x) \int \frac{x^2 \left (210 c^2 d^2+3 e \left (84 c^2 d+25 e\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{630 c \sqrt{c^2 x^2}}\\ &=-\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}-\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac{\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{2520 c^3 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1+c^2 x^2}}{1680 c^5 \sqrt{c^2 x^2}}-\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}-\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac{\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{5040 c^5 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1+c^2 x^2}}{1680 c^5 \sqrt{c^2 x^2}}-\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}-\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac{\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{5040 c^5 \sqrt{c^2 x^2}}\\ &=-\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1+c^2 x^2}}{1680 c^5 \sqrt{c^2 x^2}}-\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}-\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{1680 c^6 \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.355263, size = 186, normalized size = 0.74 \[ \frac{c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b \sqrt{1-\frac{1}{c^2 x^2}} \left (8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )+2 c^2 e \left (126 d+25 e x^2\right )+75 e^2\right )\right )-b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )+16 b c^7 x^3 \sec ^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{1680 c^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.167, size = 494, normalized size = 2. \begin{align*}{\frac{a{e}^{2}{x}^{7}}{7}}+{\frac{2\,aed{x}^{5}}{5}}+{\frac{a{x}^{3}{d}^{2}}{3}}+{\frac{b{\rm arcsec} \left (cx\right ){e}^{2}{x}^{7}}{7}}+{\frac{2\,b{\rm arcsec} \left (cx\right )ed{x}^{5}}{5}}+{\frac{b{\rm arcsec} \left (cx\right ){x}^{3}{d}^{2}}{3}}-{\frac{b{x}^{6}{e}^{2}}{42\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{x}^{4}{e}^{2}}{168\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{x}^{4}ed}{10\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{be{x}^{2}d}{20\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{d}^{2}{x}^{2}}{6\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{d}^{2}}{6\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{d}^{2}}{6\,{c}^{4}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{5\,b{x}^{2}{e}^{2}}{336\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,bed}{20\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{3\,bed}{20\,{c}^{6}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b{e}^{2}}{112\,{c}^{7}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{5\,b{e}^{2}}{112\,{c}^{8}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02231, size = 547, normalized size = 2.17 \begin{align*} \frac{1}{7} \, a e^{2} x^{7} + \frac{2}{5} \, a d e x^{5} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arcsec}\left (c x\right ) - \frac{\frac{2 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac{1}{40} \,{\left (16 \, x^{5} \operatorname{arcsec}\left (c x\right ) + \frac{\frac{2 \,{\left (3 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arcsec}\left (c x\right ) - \frac{\frac{2 \,{\left (15 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac{15 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac{15 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 4.6337, size = 641, normalized size = 2.54 \begin{align*} \frac{240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} + 16 \,{\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \operatorname{arcsec}\left (c x\right ) + 32 \,{\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (40 \, b c^{5} e^{2} x^{5} + 2 \,{\left (84 \, b c^{5} d e + 25 \, b c^{3} e^{2}\right )} x^{3} +{\left (280 \, b c^{5} d^{2} + 252 \, b c^{3} d e + 75 \, b c e^{2}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{1680 \, c^{7}} \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 x^{2} \left (a + b \operatorname{asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \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{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]