Optimal. Leaf size=198 \[ -\frac{1}{18} b c d x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{b d x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^2}{24 c^4}+\frac{1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{108} b^2 c^2 d x^6-\frac{b^2 d x^2}{24 c^2}+\frac{1}{72} b^2 d x^4 \]
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Rubi [A] time = 0.568175, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5744, 5661, 5758, 5675, 30, 5742} \[ -\frac{1}{18} b c d x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{b d x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^2}{24 c^4}+\frac{1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{108} b^2 c^2 d x^6-\frac{b^2 d x^2}{24 c^2}+\frac{1}{72} b^2 d x^4 \]
Antiderivative was successfully verified.
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Rule 5744
Rule 5661
Rule 5758
Rule 5675
Rule 30
Rule 5742
Rubi steps
\begin{align*} \int x^3 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} d \int x^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{3} (b c d) \int x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{18} b c d x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{1}{18} (b c d) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{6} (b c d) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{18} \left (b^2 c^2 d\right ) \int x^5 \, dx\\ &=\frac{1}{108} b^2 c^2 d x^6-\frac{b d x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{1}{18} b c d x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{72} \left (b^2 d\right ) \int x^3 \, dx+\frac{1}{24} \left (b^2 d\right ) \int x^3 \, dx+\frac{(b d) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{24 c}+\frac{(b d) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{8 c}\\ &=\frac{1}{72} b^2 d x^4+\frac{1}{108} b^2 c^2 d x^6+\frac{b d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac{b d x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{1}{18} b c d x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{(b d) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{48 c^3}-\frac{(b d) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{16 c^3}-\frac{\left (b^2 d\right ) \int x \, dx}{48 c^2}-\frac{\left (b^2 d\right ) \int x \, dx}{16 c^2}\\ &=-\frac{b^2 d x^2}{24 c^2}+\frac{1}{72} b^2 d x^4+\frac{1}{108} b^2 c^2 d x^6+\frac{b d x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac{b d x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{1}{18} b c d x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{d \left (a+b \sinh ^{-1}(c x)\right )^2}{24 c^4}+\frac{1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.259205, size = 186, normalized size = 0.94 \[ \frac{d \left (c x \left (18 a^2 c^3 x^3 \left (2 c^2 x^2+3\right )-6 a b \sqrt{c^2 x^2+1} \left (2 c^4 x^4+2 c^2 x^2-3\right )+b^2 c x \left (2 c^4 x^4+3 c^2 x^2-9\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (4 c^6 x^6+6 c^4 x^4-1\right )+b c x \sqrt{c^2 x^2+1} \left (-2 c^4 x^4-2 c^2 x^2+3\right )\right )+9 b^2 \left (4 c^6 x^6+6 c^4 x^4-1\right ) \sinh ^{-1}(c x)^2\right )}{216 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 298, normalized size = 1.5 \begin{align*}{\frac{1}{{c}^{4}} \left ( d{a}^{2} \left ({\frac{{c}^{6}{x}^{6}}{6}}+{\frac{{c}^{4}{x}^{4}}{4}} \right ) +d{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 ) }{12}}-{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{12}}-{\frac{{\it Arcsinh} \left ( cx \right ) cx}{18} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{{\it Arcsinh} \left ( cx \right ) cx}{18} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{108}}-{\frac{{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{216}}-{\frac{5\,{c}^{2}{x}^{2}}{108}}-{\frac{5}{108}}+{\frac{{\it Arcsinh} \left ( cx \right ) cx}{12}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{24}} \right ) +2\,dab \left ( 1/6\,{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}+1/4\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}-1/36\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}-1/36\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+1/24\,cx\sqrt{{c}^{2}{x}^{2}+1}-1/24\,{\it Arcsinh} \left ( cx \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.25314, size = 680, normalized size = 3.43 \begin{align*} \frac{1}{6} \, b^{2} c^{2} d x^{6} \operatorname{arsinh}\left (c x\right )^{2} + \frac{1}{6} \, a^{2} c^{2} d x^{6} + \frac{1}{4} \, b^{2} d x^{4} \operatorname{arsinh}\left (c x\right )^{2} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{144} \,{\left (48 \, x^{6} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac{10 \, \sqrt{c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} + 1} x}{c^{6}} - \frac{15 \, \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} a b c^{2} d + \frac{1}{864} \,{\left ({\left (\frac{8 \, x^{6}}{c^{2}} - \frac{15 \, x^{4}}{c^{4}} + \frac{45 \, x^{2}}{c^{6}} - \frac{45 \, \log \left (\frac{c^{2} x}{\sqrt{c^{2}}} + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{8}}\right )} c^{2} - 6 \,{\left (\frac{8 \, \sqrt{c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac{10 \, \sqrt{c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} + 1} x}{c^{6}} - \frac{15 \, \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c \operatorname{arsinh}\left (c x\right )\right )} b^{2} c^{2} d + \frac{1}{16} \,{\left (8 \, x^{4} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac{3 \, \sqrt{c^{2} x^{2} + 1} x}{c^{4}} + \frac{3 \, \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b d + \frac{1}{32} \,{\left ({\left (\frac{x^{4}}{c^{2}} - \frac{3 \, x^{2}}{c^{4}} + \frac{3 \, \log \left (\frac{c^{2} x}{\sqrt{c^{2}}} + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \,{\left (\frac{2 \, \sqrt{c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac{3 \, \sqrt{c^{2} x^{2} + 1} x}{c^{4}} + \frac{3 \, \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c \operatorname{arsinh}\left (c x\right )\right )} b^{2} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69882, size = 525, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (18 \, a^{2} + b^{2}\right )} c^{6} d x^{6} + 3 \,{\left (18 \, a^{2} + b^{2}\right )} c^{4} d x^{4} - 9 \, b^{2} c^{2} d x^{2} + 9 \,{\left (4 \, b^{2} c^{6} d x^{6} + 6 \, b^{2} c^{4} d x^{4} - b^{2} d\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (12 \, a b c^{6} d x^{6} + 18 \, a b c^{4} d x^{4} - 3 \, a b d -{\left (2 \, b^{2} c^{5} d x^{5} + 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 6 \,{\left (2 \, a b c^{5} d x^{5} + 2 \, a b c^{3} d x^{3} - 3 \, a b c d x\right )} \sqrt{c^{2} x^{2} + 1}}{216 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.7605, size = 332, normalized size = 1.68 \begin{align*} \begin{cases} \frac{a^{2} c^{2} d x^{6}}{6} + \frac{a^{2} d x^{4}}{4} + \frac{a b c^{2} d x^{6} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{a b c d x^{5} \sqrt{c^{2} x^{2} + 1}}{18} + \frac{a b d x^{4} \operatorname{asinh}{\left (c x \right )}}{2} - \frac{a b d x^{3} \sqrt{c^{2} x^{2} + 1}}{18 c} + \frac{a b d x \sqrt{c^{2} x^{2} + 1}}{12 c^{3}} - \frac{a b d \operatorname{asinh}{\left (c x \right )}}{12 c^{4}} + \frac{b^{2} c^{2} d x^{6} \operatorname{asinh}^{2}{\left (c x \right )}}{6} + \frac{b^{2} c^{2} d x^{6}}{108} - \frac{b^{2} c d x^{5} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{18} + \frac{b^{2} d x^{4} \operatorname{asinh}^{2}{\left (c x \right )}}{4} + \frac{b^{2} d x^{4}}{72} - \frac{b^{2} d x^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{18 c} - \frac{b^{2} d x^{2}}{24 c^{2}} + \frac{b^{2} d x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{12 c^{3}} - \frac{b^{2} d \operatorname{asinh}^{2}{\left (c x \right )}}{24 c^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} d x^{4}}{4} & \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 )}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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