Optimal. Leaf size=294 \[ \frac{d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{c^2 x^2+1}}+\frac{1}{4} x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{8} d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d \left (c^2 x^2+1\right )^{3/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac{3 b c d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{c^2 x^2+1}}+\frac{15}{64} b^2 d x \sqrt{c^2 d x^2+d}+\frac{1}{32} b^2 d x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}-\frac{9 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.245864, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{c^2 x^2+1}}+\frac{1}{4} x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{8} d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d \left (c^2 x^2+1\right )^{3/2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac{3 b c d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{c^2 x^2+1}}+\frac{15}{64} b^2 d x \sqrt{c^2 d x^2+d}+\frac{1}{32} b^2 d x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}-\frac{9 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c \sqrt{c^2 x^2+1}} \]
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 )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} (3 d) \int \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (b c d \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}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b d \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (3 d \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}{8 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c d \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{32} b^2 d x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{3 b c d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}-\frac{b d \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{32 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=\frac{15}{64} b^2 d x \sqrt{d+c^2 d x^2}+\frac{1}{32} b^2 d x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{3 b c d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}-\frac{b d \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{64 \sqrt{1+c^2 x^2}}-\frac{\left (3 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}\\ &=\frac{15}{64} b^2 d x \sqrt{d+c^2 d x^2}+\frac{1}{32} b^2 d x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{9 b^2 d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{64 c \sqrt{1+c^2 x^2}}-\frac{3 b c d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}-\frac{b d \left (1+c^2 x^2\right )^{3/2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.93777, size = 329, normalized size = 1.12 \[ \frac{288 a^2 d^{3/2} \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+96 a^2 c d x \sqrt{c^2 x^2+1} \left (2 c^2 x^2+5\right ) \sqrt{c^2 d x^2+d}-192 a b d \sqrt{c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-12 a b d \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )+32 b^2 d \sqrt{c^2 d x^2+d} \left (4 \sinh ^{-1}(c x)^3+\left (6 \sinh ^{-1}(c x)^2+3\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )-6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )\right )-b^2 d \sqrt{c^2 d x^2+d} \left (32 \sinh ^{-1}(c x)^3-3 \left (8 \sinh ^{-1}(c x)^2+1\right ) \sinh \left (4 \sinh ^{-1}(c x)\right )+12 \sinh ^{-1}(c x) \cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{768 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.22, size = 709, normalized size = 2.4 \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^{2} d x^{2} + a^{2} d +{\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} + a b d\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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