Optimal. Leaf size=204 \[ -\frac{b d^2 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b d^2 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{5 b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}+\frac{d^2 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}-\frac{5 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (c^2 x^2+1\right )^3}{108 c^2}+\frac{25}{288} b^2 d^2 x^2 \]
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Rubi [A] time = 0.205952, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5717, 5684, 5682, 5675, 30, 14, 261} \[ -\frac{b d^2 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b d^2 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{5 b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}+\frac{d^2 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}-\frac{5 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (c^2 x^2+1\right )^3}{108 c^2}+\frac{25}{288} b^2 d^2 x^2 \]
Antiderivative was successfully verified.
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Rule 5717
Rule 5684
Rule 5682
Rule 5675
Rule 30
Rule 14
Rule 261
Rubi steps
\begin{align*} \int x \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}-\frac{\left (b d^2\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c}\\ &=-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{18} \left (b^2 d^2\right ) \int x \left (1+c^2 x^2\right )^2 \, dx-\frac{\left (5 b d^2\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{18 c}\\ &=\frac{b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac{5 b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{72} \left (5 b^2 d^2\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac{\left (5 b d^2\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{24 c}\\ &=\frac{b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac{5 b d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{5 b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{72} \left (5 b^2 d^2\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{48} \left (5 b^2 d^2\right ) \int x \, dx-\frac{\left (5 b d^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{48 c}\\ &=\frac{25}{288} b^2 d^2 x^2+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac{5 b d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{5 b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{96 c^2}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}\\ \end{align*}
Mathematica [A] time = 0.487445, size = 208, normalized size = 1.02 \[ \frac{d^2 \left (c x \left (144 a^2 c x \left (c^4 x^4+3 c^2 x^2+3\right )-6 a b \sqrt{c^2 x^2+1} \left (8 c^4 x^4+26 c^2 x^2+33\right )+b^2 c x \left (8 c^4 x^4+39 c^2 x^2+99\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right )-b c x \sqrt{c^2 x^2+1} \left (8 c^4 x^4+26 c^2 x^2+33\right )\right )+9 b^2 \left (16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right ) \sinh ^{-1}(c x)^2\right )}{864 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 324, normalized size = 1.6 \begin{align*}{\frac{1}{{c}^{2}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{6}{x}^{6}}{6}}+{\frac{{c}^{4}{x}^{4}}{2}}+{\frac{{c}^{2}{x}^{2}}{2}} \right ) +{d}^{2}{b}^{2} \left ({\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{6}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{6}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{6}}-{\frac{{\it Arcsinh} \left ( cx \right ) cx}{18} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{\it Arcsinh} \left ( cx \right ) cx}{72} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{\it Arcsinh} \left ( cx \right ) cx}{48}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{5\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{96}}+{\frac{{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{108}}+{\frac{23\,{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{864}}+{\frac{17\,{c}^{2}{x}^{2}}{216}}+{\frac{17}{216}} \right ) +2\,{d}^{2}ab \left ( 1/6\,{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}+1/2\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}+1/2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}-1/36\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}-{\frac{13\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}}{144}}-{\frac{11\,cx\sqrt{{c}^{2}{x}^{2}+1}}{96}}+{\frac{11\,{\it Arcsinh} \left ( cx \right ) }{96}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23769, size = 964, normalized size = 4.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.73979, size = 663, normalized size = 3.25 \begin{align*} \frac{8 \,{\left (18 \, a^{2} + b^{2}\right )} c^{6} d^{2} x^{6} + 3 \,{\left (144 \, a^{2} + 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \,{\left (48 \, a^{2} + 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \,{\left (16 \, b^{2} c^{6} d^{2} x^{6} + 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} + 11 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (48 \, a b c^{6} d^{2} x^{6} + 144 \, a b c^{4} d^{2} x^{4} + 144 \, a b c^{2} d^{2} x^{2} + 33 \, a b d^{2} -{\left (8 \, b^{2} c^{5} d^{2} x^{5} + 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, b^{2} c d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 6 \,{\left (8 \, a b c^{5} d^{2} x^{5} + 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}}{864 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.8526, size = 430, normalized size = 2.11 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{6}}{6} + \frac{a^{2} c^{2} d^{2} x^{4}}{2} + \frac{a^{2} d^{2} x^{2}}{2} + \frac{a b c^{4} d^{2} x^{6} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{a b c^{3} d^{2} x^{5} \sqrt{c^{2} x^{2} + 1}}{18} + a b c^{2} d^{2} x^{4} \operatorname{asinh}{\left (c x \right )} - \frac{13 a b c d^{2} x^{3} \sqrt{c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname{asinh}{\left (c x \right )} - \frac{11 a b d^{2} x \sqrt{c^{2} x^{2} + 1}}{48 c} + \frac{11 a b d^{2} \operatorname{asinh}{\left (c x \right )}}{48 c^{2}} + \frac{b^{2} c^{4} d^{2} x^{6} \operatorname{asinh}^{2}{\left (c x \right )}}{6} + \frac{b^{2} c^{4} d^{2} x^{6}}{108} - \frac{b^{2} c^{3} d^{2} x^{5} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{18} + \frac{b^{2} c^{2} d^{2} x^{4} \operatorname{asinh}^{2}{\left (c x \right )}}{2} + \frac{13 b^{2} c^{2} d^{2} x^{4}}{288} - \frac{13 b^{2} c d^{2} x^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{72} + \frac{b^{2} d^{2} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{2} + \frac{11 b^{2} d^{2} x^{2}}{96} - \frac{11 b^{2} d^{2} x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{48 c} + \frac{11 b^{2} d^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{96 c^{2}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{2} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} + d\right )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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