Optimal. Leaf size=267 \[ -\frac{2 b c^3 d x^5 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{c^2 x^2+1}}-\frac{4 b c d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{c^2 x^2+1}}-\frac{2 b d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{c^2 x^2+1}}+\frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{2 b^2 d \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}}{125 c^2}+\frac{16 b^2 d \sqrt{c^2 d x^2+d}}{75 c^2}+\frac{8 b^2 d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{225 c^2} \]
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Rubi [A] time = 0.21852, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5717, 194, 5679, 12, 1247, 698} \[ -\frac{2 b c^3 d x^5 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{c^2 x^2+1}}-\frac{4 b c d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{c^2 x^2+1}}-\frac{2 b d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{c^2 x^2+1}}+\frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{2 b^2 d \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}}{125 c^2}+\frac{16 b^2 d \sqrt{c^2 d x^2+d}}{75 c^2}+\frac{8 b^2 d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{225 c^2} \]
Antiderivative was successfully verified.
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Rule 5717
Rule 194
Rule 5679
Rule 12
Rule 1247
Rule 698
Rubi steps
\begin{align*} \int x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (2 b d \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{5 c \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{1+c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1+c^2 x^2}} \, dx}{5 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{1+c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1+c^2 x^2}} \, dx}{75 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{1+c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{15+10 c^2 x+3 c^4 x^2}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{75 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{1+c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1+c^2 x}}+4 \sqrt{1+c^2 x}+3 \left (1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt{1+c^2 x^2}}\\ &=\frac{16 b^2 d \sqrt{d+c^2 d x^2}}{75 c^2}+\frac{8 b^2 d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{225 c^2}+\frac{2 b^2 d \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}}{125 c^2}-\frac{2 b d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c \sqrt{1+c^2 x^2}}-\frac{4 b c d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.36113, size = 198, normalized size = 0.74 \[ \frac{d \sqrt{c^2 d x^2+d} \left (225 a^2 \left (c^2 x^2+1\right )^3-30 a b c x \left (3 c^4 x^4+10 c^2 x^2+15\right ) \sqrt{c^2 x^2+1}+30 b \sinh ^{-1}(c x) \left (15 a \left (c^2 x^2+1\right )^3-b c x \sqrt{c^2 x^2+1} \left (3 c^4 x^4+10 c^2 x^2+15\right )\right )+2 b^2 \left (9 c^6 x^6+47 c^4 x^4+187 c^2 x^2+149\right )+225 b^2 \left (c^2 x^2+1\right )^3 \sinh ^{-1}(c x)^2\right )}{1125 c^2 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.266, size = 1149, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28852, size = 311, normalized size = 1.16 \begin{align*} \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}} b^{2} \operatorname{arsinh}\left (c x\right )^{2}}{5 \, c^{2} d} + \frac{2}{1125} \, b^{2}{\left (\frac{9 \, \sqrt{c^{2} x^{2} + 1} c^{2} d^{\frac{5}{2}} x^{4} + 38 \, \sqrt{c^{2} x^{2} + 1} d^{\frac{5}{2}} x^{2} + \frac{149 \, \sqrt{c^{2} x^{2} + 1} d^{\frac{5}{2}}}{c^{2}}}{d} - \frac{15 \,{\left (3 \, c^{4} d^{\frac{5}{2}} x^{5} + 10 \, c^{2} d^{\frac{5}{2}} x^{3} + 15 \, d^{\frac{5}{2}} x\right )} \operatorname{arsinh}\left (c x\right )}{c d}\right )} + \frac{2 \,{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}} a b \operatorname{arsinh}\left (c x\right )}{5 \, c^{2} d} + \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}} a^{2}}{5 \, c^{2} d} - \frac{2 \,{\left (3 \, c^{4} d^{\frac{5}{2}} x^{5} + 10 \, c^{2} d^{\frac{5}{2}} x^{3} + 15 \, d^{\frac{5}{2}} x\right )} a b}{75 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.05506, size = 743, normalized size = 2.78 \begin{align*} \frac{225 \,{\left (b^{2} c^{6} d x^{6} + 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} + b^{2} d\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 30 \,{\left (15 \, a b c^{6} d x^{6} + 45 \, a b c^{4} d x^{4} + 45 \, a b c^{2} d x^{2} + 15 \, a b d -{\left (3 \, b^{2} c^{5} d x^{5} + 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (9 \,{\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} d x^{6} +{\left (675 \, a^{2} + 94 \, b^{2}\right )} c^{4} d x^{4} +{\left (675 \, a^{2} + 374 \, b^{2}\right )} c^{2} d x^{2} +{\left (225 \, a^{2} + 298 \, b^{2}\right )} d - 30 \,{\left (3 \, a b c^{5} d x^{5} + 10 \, a b c^{3} d x^{3} + 15 \, a b c d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{1125 \,{\left (c^{4} x^{2} + c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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