Optimal. Leaf size=189 \[ \frac {15 (a+b x) \sqrt {1+(a+b x)^2}}{64 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {9 \sinh ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac {\sinh ^{-1}(a+b x)^3}{8 b} \]
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Rubi [A]
time = 0.13, 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 = {5860, 5786,
5785, 5783, 5776, 327, 221, 5798, 201} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5860
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 {\text {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 {\text {Subst}\left (\int x \left (1+x^2\right ) \sinh ^{-1}(x) \, dx,x,a+b x\right )}{2 b}+\frac {3 \text {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 {\text {Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,a+b x\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {3 \text {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 \text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,a+b x\right )}{32 b}+\frac {3 \text {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 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{64 b}-\frac {3 \text {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 = 0.11, size = 211, normalized size = 1.12 \begin {gather*} \frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (17 a+2 a^3+17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right )-\left (17+40 a^2+8 a^4\right ) \sinh ^{-1}(a+b x)-8 b x \left (10 a+4 a^3+5 b x+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right ) \sinh ^{-1}(a+b x)+8 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (5 a+2 a^3+5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \sinh ^{-1}(a+b x)^2+8 \sinh ^{-1}(a+b x)^3}{64 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.93, size = 139, normalized size = 0.74 \[-\frac {-8 \arcsinh \left (b x +a \right )^{2} \cosh \left (2 \arcsinh \left (b x +a \right )\right ) \sinh \left (2 \arcsinh \left (b x +a \right )\right )+4 \arcsinh \left (b x +a \right ) \left (\cosh ^{2}\left (2 \arcsinh \left (b x +a \right )\right )\right )-16 \arcsinh \left (b x +a \right )^{3}-32 \arcsinh \left (b x +a \right )^{2} \sinh \left (2 \arcsinh \left (b x +a \right )\right )+32 \arcsinh \left (b x +a \right ) \cosh \left (2 \arcsinh \left (b x +a \right )\right )-\sinh \left (2 \arcsinh \left (b x +a \right )\right ) \cosh \left (2 \arcsinh \left (b x +a \right )\right )-2 \arcsinh \left (b x +a \right )-16 \sinh \left (2 \arcsinh \left (b x +a \right )\right )}{128 b}\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 259, normalized size = 1.37 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 568 vs.
\(2 (173) = 346\).
time = 0.99, size = 568, normalized size = 3.01 \begin {gather*} \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}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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