Optimal. Leaf size=142 \[ \frac{x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac{2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}-\frac{b x^3 \sqrt{c^2 x^2+1}}{9 c \sqrt{c^2 d x^2+d}}+\frac{2 b x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.158251, antiderivative size = 142, 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, 5717, 8, 30} \[ \frac{x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac{2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}-\frac{b x^3 \sqrt{c^2 x^2+1}}{9 c \sqrt{c^2 d x^2+d}}+\frac{2 b x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5758
Rule 5717
Rule 8
Rule 30
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+c^2 d x^2}} \, dx &=\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac{2 \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+c^2 d x^2}} \, dx}{3 c^2}-\frac{\left (b \sqrt{1+c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt{d+c^2 d x^2}}\\ &=-\frac{b x^3 \sqrt{1+c^2 x^2}}{9 c \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}+\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt{d+c^2 d x^2}}\\ &=\frac{2 b x \sqrt{1+c^2 x^2}}{3 c^3 \sqrt{d+c^2 d x^2}}-\frac{b x^3 \sqrt{1+c^2 x^2}}{9 c \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}+\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.13315, size = 93, normalized size = 0.65 \[ \frac{3 a \left (c^4 x^4-c^2 x^2-2\right )+b c x \sqrt{c^2 x^2+1} \left (6-c^2 x^2\right )+3 b \left (c^4 x^4-c^2 x^2-2\right ) \sinh ^{-1}(c x)}{9 c^4 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.158, size = 358, normalized size = 2.5 \begin{align*} a \left ({\frac{{x}^{2}}{3\,{c}^{2}d}\sqrt{{c}^{2}d{x}^{2}+d}}-{\frac{2}{3\,d{c}^{4}}\sqrt{{c}^{2}d{x}^{2}+d}} \right ) +b \left ({\frac{-1+3\,{\it Arcsinh} \left ( cx \right ) }{72\,d{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ( 4\,{c}^{4}{x}^{4}+4\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+5\,{c}^{2}{x}^{2}+3\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }-{\frac{-3+3\,{\it Arcsinh} \left ( cx \right ) }{8\,d{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}+cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }-{\frac{3+3\,{\it Arcsinh} \left ( cx \right ) }{8\,d{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}-cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{1+3\,{\it Arcsinh} \left ( cx \right ) }{72\,d{c}^{4} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ( 4\,{c}^{4}{x}^{4}-4\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+5\,{c}^{2}{x}^{2}-3\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) } \right ) \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 [A] time = 2.514, size = 278, normalized size = 1.96 \begin{align*} \frac{3 \,{\left (b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (3 \, a c^{4} x^{4} - 3 \, a c^{2} x^{2} -{\left (b c^{3} x^{3} - 6 \, b c x\right )} \sqrt{c^{2} x^{2} + 1} - 6 \, a\right )} \sqrt{c^{2} d x^{2} + d}}{9 \,{\left (c^{6} d x^{2} + c^{4} d\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^{3} \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^{3}}{\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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