Optimal. Leaf size=131 \[ -\frac{3 (a+b x)^2}{8 b}+\frac{\sinh ^{-1}(a+b x)^4}{8 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^3}{2 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{4 b}-\frac{3 \sinh ^{-1}(a+b x)^2}{8 b}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{4 b} \]
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Rubi [A] time = 0.188993, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5867, 5682, 5675, 5661, 5758, 30} \[ -\frac{3 (a+b x)^2}{8 b}+\frac{\sinh ^{-1}(a+b x)^4}{8 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^3}{2 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{4 b}-\frac{3 \sinh ^{-1}(a+b x)^2}{8 b}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 5867
Rule 5682
Rule 5675
Rule 5661
Rule 5758
Rule 30
Rubi steps
\begin{align*} \int \sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{1+x^2} \sinh ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)^3}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{2 b}-\frac{3 \operatorname{Subst}\left (\int x \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{4 b}+\frac{(a+b x) \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{2 b}+\frac{\sinh ^{-1}(a+b x)^4}{8 b}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2 \sinh ^{-1}(x)}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=\frac{3 (a+b x) \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{4 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{4 b}+\frac{(a+b x) \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{2 b}+\frac{\sinh ^{-1}(a+b x)^4}{8 b}-\frac{3 \operatorname{Subst}(\int x \, dx,x,a+b x)}{4 b}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{4 b}\\ &=-\frac{3 (a+b x)^2}{8 b}+\frac{3 (a+b x) \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{4 b}-\frac{3 \sinh ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{4 b}+\frac{(a+b x) \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{2 b}+\frac{\sinh ^{-1}(a+b x)^4}{8 b}\\ \end{align*}
Mathematica [A] time = 0.108373, size = 127, normalized size = 0.97 \[ \frac{4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^3-3 \left (2 a^2+4 a b x+2 b^2 x^2+1\right ) \sinh ^{-1}(a+b x)^2+6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)-3 b x (2 a+b x)+\sinh ^{-1}(a+b x)^4}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 204, normalized size = 1.6 \begin{align*}{\frac{1}{8\,b} \left ( 4\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}xb-6\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}{x}^{2}{b}^{2}+4\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}a-12\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}xab+ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{4}-6\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}{a}^{2}+6\,{\it Arcsinh} \left ( bx+a \right ) \sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}xb-3\,{b}^{2}{x}^{2}+6\,{\it Arcsinh} \left ( bx+a \right ) \sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}a-6\,xab-3\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}-3\,{a}^{2}-3 \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 = 2.71438, size = 495, normalized size = 3.78 \begin{align*} \frac{4 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (b x + a\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, b^{2} x^{2} + \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{4} - 6 \, a b x - 3 \,{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 6 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (b x + a\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{8 \, b} \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 \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a + b x \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 \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname{arsinh}\left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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