Optimal. Leaf size=204 \[ -\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{c^2 d x^2+d}}+\frac{x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{b x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt{c^2 d x^2+d}}+\frac{b^2 x \left (c^2 x^2+1\right )}{4 c^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^3 \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.275487, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5758, 5677, 5675, 5661, 321, 215} \[ -\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{c^2 d x^2+d}}+\frac{x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{b x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt{c^2 d x^2+d}}+\frac{b^2 x \left (c^2 x^2+1\right )}{4 c^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^3 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5758
Rule 5677
Rule 5675
Rule 5661
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx &=\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx}{2 c^2}-\frac{\left (b \sqrt{1+c^2 x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c \sqrt{d+c^2 d x^2}}\\ &=-\frac{b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt{d+c^2 d x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{d+c^2 d x^2}}-\frac{\sqrt{1+c^2 x^2} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{2 c^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 x \left (1+c^2 x^2\right )}{4 c^2 \sqrt{d+c^2 d x^2}}-\frac{b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt{d+c^2 d x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d+c^2 d x^2}}-\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{4 c^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 x \left (1+c^2 x^2\right )}{4 c^2 \sqrt{d+c^2 d x^2}}-\frac{b^2 \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^3 \sqrt{d+c^2 d x^2}}-\frac{b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt{d+c^2 d x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.880818, size = 198, normalized size = 0.97 \[ \frac{12 a^2 c x \left (c^2 d x^2+d\right )-12 a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )-6 a b d \sqrt{c^2 x^2+1} \left (2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-\sinh \left (2 \sinh ^{-1}(c x)\right )\right )+\cosh \left (2 \sinh ^{-1}(c x)\right )\right )-b^2 d \sqrt{c^2 x^2+1} \left (4 \sinh ^{-1}(c x)^3-3 \left (2 \sinh ^{-1}(c x)^2+1\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )+6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 c^3 d \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.262, size = 530, normalized size = 2.6 \begin{align*}{\frac{{a}^{2}x}{2\,{c}^{2}d}\sqrt{{c}^{2}d{x}^{2}+d}}-{\frac{{a}^{2}}{2\,{c}^{2}}\ln \left ({{c}^{2}dx{\frac{1}{\sqrt{{c}^{2}d}}}}+\sqrt{{c}^{2}d{x}^{2}+d} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{{b}^{2}{x}^{3}}{4\,d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{{b}^{2}x}{4\,{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{6\,d{c}^{3}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{x}^{3}}{2\,d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}x}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{{b}^{2}{\it Arcsinh} \left ( cx \right ) }{4\,d{c}^{3}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{{b}^{2}{\it Arcsinh} \left ( cx \right ){x}^{2}}{2\,cd}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{ab \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2\,d{c}^{3}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{ab{\it Arcsinh} \left ( cx \right ){x}^{3}}{d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{ab{x}^{2}}{2\,cd}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{ab{\it Arcsinh} \left ( cx \right ) x}{{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{ab}{4\,d{c}^{3}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \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 (\frac{b^{2} x^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{2}}{\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 \frac{x^{2} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{\sqrt{d \left (c^{2} x^{2} + 1\right )}}\, dx \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 \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{\sqrt{c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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