Optimal. Leaf size=420 \[ \frac{5 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt{c^2 x^2+1}}+\frac{5}{16} d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d^2 \left (c^2 x^2+1\right )^{5/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{5 b c d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt{c^2 x^2+1}}+\frac{1}{6} x \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{108} b^2 d^2 x \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}+\frac{245 b^2 d^2 x \sqrt{c^2 d x^2+d}}{1152}+\frac{65 b^2 d^2 x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{1728}-\frac{115 b^2 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.403579, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac{5 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt{c^2 x^2+1}}+\frac{5}{16} d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d^2 \left (c^2 x^2+1\right )^{5/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{5 b c d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt{c^2 x^2+1}}+\frac{1}{6} x \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{108} b^2 d^2 x \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}+\frac{245 b^2 d^2 x \sqrt{c^2 d x^2+d}}{1152}+\frac{65 b^2 d^2 x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{1728}-\frac{115 b^2 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5661
Rule 321
Rule 215
Rule 5717
Rule 195
Rubi steps
\begin{align*} \int \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} (5 d) \int \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} \left (5 d^2\right ) \int \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{18 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{12 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}-\frac{5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (5 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{108 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{48 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=\frac{65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{1728}+\frac{1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}-\frac{5 b c d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt{1+c^2 x^2}}-\frac{5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{144 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{64 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}\\ &=\frac{245 b^2 d^2 x \sqrt{d+c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{1728}+\frac{1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}-\frac{5 b c d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt{1+c^2 x^2}}-\frac{5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{288 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{128 \sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{32 \sqrt{1+c^2 x^2}}\\ &=\frac{245 b^2 d^2 x \sqrt{d+c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{1728}+\frac{1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}-\frac{115 b^2 d^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c \sqrt{1+c^2 x^2}}-\frac{5 b c d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt{1+c^2 x^2}}-\frac{5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{5}{16} d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.53273, size = 499, normalized size = 1.19 \[ \frac{d^2 \left (2304 a^2 c^5 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+7488 a^2 c^3 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+9504 a^2 c x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+4320 a^2 \sqrt{d} \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+72 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (60 a+45 b \sinh \left (2 \sinh ^{-1}(c x)\right )+9 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )-3240 a b \sqrt{c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-324 a b \sqrt{c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b \sqrt{c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )-12 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left (-540 a \sinh \left (2 \sinh ^{-1}(c x)\right )-108 a \sinh \left (4 \sinh ^{-1}(c x)\right )-12 a \sinh \left (6 \sinh ^{-1}(c x)\right )+270 b \cosh \left (2 \sinh ^{-1}(c x)\right )+27 b \cosh \left (4 \sinh ^{-1}(c x)\right )+2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )+1440 b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3+1620 b^2 \sqrt{c^2 d x^2+d} \sinh \left (2 \sinh ^{-1}(c x)\right )+81 b^2 \sqrt{c^2 d x^2+d} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 \sqrt{c^2 d x^2+d} \sinh \left (6 \sinh ^{-1}(c x)\right )\right )}{13824 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.286, size = 966, normalized size = 2.3 \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 [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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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