Optimal. Leaf size=189 \[ \frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{32 b}+\frac {15 (a+b x) \sqrt {(a+b x)^2+1}}{64 b}+\frac {\sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{8 b}-\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {\left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {9 \sinh ^{-1}(a+b x)}{64 b} \]
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Rubi [A] time = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5867, 5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{32 b}+\frac {15 (a+b x) \sqrt {(a+b x)^2+1}}{64 b}+\frac {\sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{8 b}-\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {\left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {9 \sinh ^{-1}(a+b x)}{64 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5682
Rule 5684
Rule 5717
Rule 5867
Rubi steps
\begin {align*} \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right )^{3/2} \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}-\frac {\operatorname {Subst}\left (\int x \left (1+x^2\right ) \sinh ^{-1}(x) \, dx,x,a+b x\right )}{2 b}+\frac {3 \operatorname {Subst}\left (\int \sqrt {1+x^2} \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}\\ &=-\frac {\left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {\operatorname {Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,a+b x\right )}{8 b}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {3 \operatorname {Subst}\left (\int x \sinh ^{-1}(x) \, dx,x,a+b x\right )}{4 b}\\ &=\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {\sinh ^{-1}(a+b x)^3}{8 b}+\frac {3 \operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,a+b x\right )}{32 b}+\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=\frac {15 (a+b x) \sqrt {1+(a+b x)^2}}{64 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {\sinh ^{-1}(a+b x)^3}{8 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{64 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=\frac {15 (a+b x) \sqrt {1+(a+b x)^2}}{64 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {9 \sinh ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {\sinh ^{-1}(a+b x)^3}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 211, normalized size = 1.12 \[ \frac {-\left (8 a^4+40 a^2+17\right ) \sinh ^{-1}(a+b x)+\sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2+17 a+2 b^3 x^3+17 b x\right )+8 \sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2+5 a+2 b^3 x^3+5 b x\right ) \sinh ^{-1}(a+b x)^2-8 b x \left (4 a^3+6 a^2 b x+4 a b^2 x^2+10 a+b^3 x^3+5 b x\right ) \sinh ^{-1}(a+b x)+8 \sinh ^{-1}(a+b x)^3}{64 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 259, normalized size = 1.37 \[ \frac {8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 5\right )} b x + 5 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 8 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} + 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} + 5 \, a\right )} b x + 40 \, a^{2} + 17\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 17\right )} b x + 17 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 479, normalized size = 2.53 \[ \frac {16 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x^{3} b^{3}-8 \arcsinh \left (b x +a \right ) x^{4} b^{4}+48 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x^{2} a \,b^{2}-32 \arcsinh \left (b x +a \right ) x^{3} a \,b^{3}+48 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x \,a^{2} b +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x^{3} b^{3}-48 \arcsinh \left (b x +a \right ) x^{2} a^{2} b^{2}+16 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}+6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x^{2} a \,b^{2}-32 \arcsinh \left (b x +a \right ) x \,a^{3} b +40 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x b +6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x \,a^{2} b -40 \arcsinh \left (b x +a \right ) x^{2} b^{2}-8 \arcsinh \left (b x +a \right ) a^{4}+40 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-80 \arcsinh \left (b x +a \right ) x a b +17 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x b +8 \arcsinh \left (b x +a \right )^{3}-40 \arcsinh \left (b x +a \right ) a^{2}+17 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -17 \arcsinh \left (b x +a \right )}{64 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {asinh}\left (a+b\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.35, size = 568, normalized size = 3.01 \[ \begin {cases} - \frac {a^{4} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {a^{3} x \operatorname {asinh}{\left (a + b x \right )}}{2} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32 b} - \frac {3 a^{2} b x^{2} \operatorname {asinh}{\left (a + b x \right )}}{4} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32} - \frac {5 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {a b^{2} x^{3} \operatorname {asinh}{\left (a + b x \right )}}{2} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32} - \frac {5 a x \operatorname {asinh}{\left (a + b x \right )}}{4} + \frac {5 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8 b} + \frac {17 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{64 b} - \frac {b^{3} x^{4} \operatorname {asinh}{\left (a + b x \right )}}{8} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32} - \frac {5 b x^{2} \operatorname {asinh}{\left (a + b x \right )}}{8} + \frac {5 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} + \frac {17 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{64} + \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{8 b} - \frac {17 \operatorname {asinh}{\left (a + b x \right )}}{64 b} & \text {for}\: b \neq 0 \\x \left (a^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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