Optimal. Leaf size=131 \[ -\frac{\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt{(a+b x)^2+1}}{96 b^4}-\frac{\left (8 a^4-24 a^2+3\right ) \sinh ^{-1}(a+b x)}{32 b^4}+\frac{7 a x^2 \sqrt{(a+b x)^2+1}}{48 b^2}-\frac{x^3 \sqrt{(a+b x)^2+1}}{16 b}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x) \]
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Rubi [A] time = 0.1722, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5865, 5801, 743, 833, 780, 215} \[ -\frac{\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt{(a+b x)^2+1}}{96 b^4}-\frac{\left (8 a^4-24 a^2+3\right ) \sinh ^{-1}(a+b x)}{32 b^4}+\frac{7 a x^2 \sqrt{(a+b x)^2+1}}{48 b^2}-\frac{x^3 \sqrt{(a+b x)^2+1}}{16 b}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5801
Rule 743
Rule 833
Rule 780
Rule 215
Rubi steps
\begin{align*} \int x^3 \sinh ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^3 \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{4} x^4 \sinh ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^4}{\sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac{x^3 \sqrt{1+(a+b x)^2}}{16 b}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x)-\frac{1}{16} \operatorname{Subst}\left (\int \frac{\left (-\frac{3-4 a^2}{b^2}-\frac{7 a x}{b^2}\right ) \left (-\frac{a}{b}+\frac{x}{b}\right )^2}{\sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=\frac{7 a x^2 \sqrt{1+(a+b x)^2}}{48 b^2}-\frac{x^3 \sqrt{1+(a+b x)^2}}{16 b}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x)-\frac{1}{48} \operatorname{Subst}\left (\int \frac{\left (\frac{a \left (23-12 a^2\right )}{b^3}-\frac{\left (9-26 a^2\right ) x}{b^3}\right ) \left (-\frac{a}{b}+\frac{x}{b}\right )}{\sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=\frac{7 a x^2 \sqrt{1+(a+b x)^2}}{48 b^2}-\frac{x^3 \sqrt{1+(a+b x)^2}}{16 b}-\frac{\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt{1+(a+b x)^2}}{96 b^4}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x)-\frac{\left (3-24 a^2+8 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{32 b^4}\\ &=\frac{7 a x^2 \sqrt{1+(a+b x)^2}}{48 b^2}-\frac{x^3 \sqrt{1+(a+b x)^2}}{16 b}-\frac{\left (4 a \left (16-19 a^2\right )-\left (9-26 a^2\right ) (a+b x)\right ) \sqrt{1+(a+b x)^2}}{96 b^4}-\frac{\left (3-24 a^2+8 a^4\right ) \sinh ^{-1}(a+b x)}{32 b^4}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x)\\ \end{align*}
Mathematica [A] time = 0.0817239, size = 95, normalized size = 0.73 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left (-26 a^2 b x+50 a^3+a \left (14 b^2 x^2-55\right )-6 b^3 x^3+9 b x\right )-3 \left (8 a^4-24 a^2-8 b^4 x^4+3\right ) \sinh ^{-1}(a+b x)}{96 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 200, normalized size = 1.5 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{{\it Arcsinh} \left ( bx+a \right ) \left ( bx+a \right ) ^{4}}{4}}-{\it Arcsinh} \left ( bx+a \right ) \left ( bx+a \right ) ^{3}a+{\frac{3\,{\it Arcsinh} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}{a}^{2}}{2}}-{\it Arcsinh} \left ( bx+a \right ) \left ( bx+a \right ){a}^{3}-{\frac{ \left ( bx+a \right ) ^{3}}{16}\sqrt{1+ \left ( bx+a \right ) ^{2}}}+{\frac{3\,bx+3\,a}{32}\sqrt{1+ \left ( bx+a \right ) ^{2}}}-{\frac{3\,{\it Arcsinh} \left ( bx+a \right ) }{32}}+a \left ({\frac{ \left ( bx+a \right ) ^{2}}{3}\sqrt{1+ \left ( bx+a \right ) ^{2}}}-{\frac{2}{3}\sqrt{1+ \left ( bx+a \right ) ^{2}}} \right ) -{\frac{3\,{a}^{2}}{2} \left ({\frac{bx+a}{2}\sqrt{1+ \left ( bx+a \right ) ^{2}}}-{\frac{{\it Arcsinh} \left ( bx+a \right ) }{2}} \right ) }+{a}^{3}\sqrt{1+ \left ( bx+a \right ) ^{2}} \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 = 2.8298, size = 262, normalized size = 2. \begin{align*} \frac{3 \,{\left (8 \, b^{4} x^{4} - 8 \, a^{4} + 24 \, a^{2} - 3\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) -{\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} +{\left (26 \, a^{2} - 9\right )} b x + 55 \, a\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{96 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.81439, size = 255, normalized size = 1.95 \begin{align*} \begin{cases} - \frac{a^{4} \operatorname{asinh}{\left (a + b x \right )}}{4 b^{4}} + \frac{25 a^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{4}} - \frac{13 a^{2} x \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{3}} + \frac{3 a^{2} \operatorname{asinh}{\left (a + b x \right )}}{4 b^{4}} + \frac{7 a x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{48 b^{2}} - \frac{55 a \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{96 b^{4}} + \frac{x^{4} \operatorname{asinh}{\left (a + b x \right )}}{4} - \frac{x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{16 b} + \frac{3 x \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32 b^{3}} - \frac{3 \operatorname{asinh}{\left (a + b x \right )}}{32 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{asinh}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29298, size = 219, normalized size = 1.67 \begin{align*} \frac{1}{4} \, x^{4} \log \left (b x + a + \sqrt{{\left (b x + a\right )}^{2} + 1}\right ) - \frac{1}{96} \,{\left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left ({\left (2 \, x{\left (\frac{3 \, x}{b^{2}} - \frac{7 \, a}{b^{3}}\right )} + \frac{26 \, a^{2} b^{3} - 9 \, b^{3}}{b^{7}}\right )} x - \frac{5 \,{\left (10 \, a^{3} b^{2} - 11 \, a b^{2}\right )}}{b^{7}}\right )} - \frac{3 \,{\left (8 \, a^{4} - 24 \, a^{2} + 3\right )} \log \left (-a b -{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{\left | b \right |}\right )}{b^{4}{\left | b \right |}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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