Optimal. Leaf size=119 \[ \frac{x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d}-\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.145791, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5758, 5677, 5675, 30} \[ \frac{x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 d}-\frac{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5758
Rule 5677
Rule 5675
Rule 30
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\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 c^2 d}-\frac{\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+c^2 d x^2}} \, dx}{2 c^2}-\frac{\left (b \sqrt{1+c^2 x^2}\right ) \int x \, dx}{2 c \sqrt{d+c^2 d x^2}}\\ &=-\frac{b x^2 \sqrt{1+c^2 x^2}}{4 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 c^2 d}-\frac{\sqrt{1+c^2 x^2} \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{2 c^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b x^2 \sqrt{1+c^2 x^2}}{4 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 c^2 d}-\frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.723047, size = 121, normalized size = 1.02 \[ -\frac{-\frac{4 a c x \sqrt{c^2 d x^2+d}}{d}+\frac{4 a \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )}{\sqrt{d}}+\frac{b \sqrt{c^2 x^2+1} \left (2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-\sinh \left (2 \sinh ^{-1}(c x)\right )\right )+\cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{\sqrt{c^2 d x^2+d}}}{8 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 247, normalized size = 2.1 \begin{align*}{\frac{ax}{2\,{c}^{2}d}\sqrt{{c}^{2}d{x}^{2}+d}}-{\frac{a}{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 \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{4\,{c}^{3}d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ){x}^{3}}{2\,d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{x}^{2}}{4\,cd}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ) x}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{b}{8\,{c}^{3}d}\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 x^{2} \operatorname{arsinh}\left (c x\right ) + a 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 )}{\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 )} 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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