Optimal. Leaf size=235 \[ -\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}+\frac {45 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)}{32 b}-\frac {27 \sinh ^{-1}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b} \]
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Rubi [A]
time = 0.23, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps
used = 15, 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, 5812, 30, 5798, 14} \begin {gather*} -\frac {3 (a+b x)^4}{128 b}-\frac {51 (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {\left ((a+b x)^2+1\right )^{3/2} (a+b x) \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^3}{8 b}+\frac {3 \left ((a+b x)^2+1\right )^{3/2} (a+b x) \sinh ^{-1}(a+b x)}{32 b}+\frac {45 \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{64 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b}-\frac {3 \left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {27 \sinh ^{-1}(a+b x)^2}{128 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5812
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)^3 \, dx &=\frac {\text {Subst}\left (\int \left (1+x^2\right )^{3/2} \sinh ^{-1}(x)^3 \, 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)^3}{4 b}-\frac {3 \text {Subst}\left (\int x \left (1+x^2\right ) \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}+\frac {3 \text {Subst}\left (\int \sqrt {1+x^2} \sinh ^{-1}(x)^3 \, dx,x,a+b x\right )}{4 b}\\ &=-\frac {3 \left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \text {Subst}\left (\int \left (1+x^2\right )^{3/2} \sinh ^{-1}(x) \, dx,x,a+b x\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\sinh ^{-1}(x)^3}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {9 \text {Subst}\left (\int x \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{8 b}\\ &=\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b}-\frac {3 \text {Subst}\left (\int x \left (1+x^2\right ) \, dx,x,a+b x\right )}{32 b}+\frac {9 \text {Subst}\left (\int \sqrt {1+x^2} \sinh ^{-1}(x) \, dx,x,a+b x\right )}{32 b}+\frac {9 \text {Subst}\left (\int \frac {x^2 \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=\frac {45 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b}-\frac {3 \text {Subst}\left (\int \left (x+x^3\right ) \, dx,x,a+b x\right )}{32 b}-\frac {9 \text {Subst}(\int x \, dx,x,a+b x)}{64 b}+\frac {9 \text {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{64 b}-\frac {9 \text {Subst}(\int x \, dx,x,a+b x)}{16 b}-\frac {9 \text {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=-\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}+\frac {45 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)}{32 b}-\frac {27 \sinh ^{-1}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 266, normalized size = 1.13 \begin {gather*} -\frac {6 a \left (17+2 a^2\right ) b x+3 \left (17+6 a^2\right ) b^2 x^2+12 a b^3 x^3+3 b^4 x^4-6 \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 ) \sinh ^{-1}(a+b x)+3 \left (17+8 a^4+32 a^3 b x+40 b^2 x^2+8 b^4 x^4+16 a b x \left (5+2 b^2 x^2\right )+8 a^2 \left (5+6 b^2 x^2\right )\right ) \sinh ^{-1}(a+b x)^2-16 \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)^3-12 \sinh ^{-1}(a+b x)^4}{128 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}} \arcsinh \left (b x +a \right )^{3}\, dx\]
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 = 0.42, size = 332, normalized size = 1.41 \begin {gather*} -\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} + 17\right )} b^{2} x^{2} - 16 \, {\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 )^{3} - 12 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{4} + 6 \, {\left (2 \, a^{3} + 17 \, a\right )} b x + 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 )^{2} - 6 \, {\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} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{128 \, 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. 694 vs.
\(2 (223) = 446\).
time = 1.62, size = 694, normalized size = 2.95 \begin {gather*} \begin {cases} - \frac {3 a^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} - \frac {3 a^{3} x \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} - \frac {3 a^{3} x}{32} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4 b} + \frac {3 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32 b} - \frac {9 a^{2} b x^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} - \frac {9 a^{2} b x^{2}}{64} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {9 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 a^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} - \frac {3 a b^{2} x^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} - \frac {3 a b^{2} x^{3}}{32} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {9 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 a x \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} - \frac {51 a x}{64} + \frac {5 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{8 b} + \frac {51 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{64 b} - \frac {3 b^{3} x^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16} - \frac {3 b^{3} x^{4}}{128} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {3 b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 b x^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16} - \frac {51 b x^{2}}{128} + \frac {5 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{8} + \frac {51 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{64} + \frac {3 \operatorname {asinh}^{4}{\left (a + b x \right )}}{32 b} - \frac {51 \operatorname {asinh}^{2}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \left (a^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{3}{\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 = -0.00 \begin {gather*} \int {\mathrm {asinh}\left (a+b\,x\right )}^3\,{\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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