Optimal. Leaf size=128 \[ -\frac {8 b d^2 \sqrt {1+c^2 x^2}}{15 c}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{45 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {200, 5784, 12,
1261, 712} \begin {gather*} \frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^2 x \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^2 \left (c^2 x^2+1\right )^{5/2}}{25 c}-\frac {4 b d^2 \left (c^2 x^2+1\right )^{3/2}}{45 c}-\frac {8 b d^2 \sqrt {c^2 x^2+1}}{15 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 712
Rule 1261
Rule 5784
Rubi steps
\begin {align*} \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1+c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{15} \left (b c d^2\right ) \int \frac {x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} \left (b c d^2\right ) \text {Subst}\left (\int \frac {15+10 c^2 x+3 c^4 x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} \left (b c d^2\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1+c^2 x}}+4 \sqrt {1+c^2 x}+3 \left (1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {8 b d^2 \sqrt {1+c^2 x^2}}{15 c}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{45 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} c^2 d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 95, normalized size = 0.74 \begin {gather*} \frac {d^2 \left (15 a c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+15 b c x \left (15+10 c^2 x^2+3 c^4 x^4\right ) \sinh ^{-1}(c x)\right )}{225 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 119, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \arcsinh \left (c x \right ) c^{3} x^{3}}{3}+\arcsinh \left (c x \right ) c x -\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(119\) |
default | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \arcsinh \left (c x \right ) c^{3} x^{3}}{3}+\arcsinh \left (c x \right ) c x -\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 194, normalized size = 1.52 \begin {gather*} \frac {1}{5} \, a c^{4} d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{4} d^{2} + \frac {2}{3} \, a c^{2} d^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 133, normalized size = 1.04 \begin {gather*} \frac {45 \, a c^{5} d^{2} x^{5} + 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} + 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (9 \, b c^{4} d^{2} x^{4} + 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{225 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 165, normalized size = 1.29 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{5}}{5} + \frac {2 a c^{2} d^{2} x^{3}}{3} + a d^{2} x + \frac {b c^{4} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {b c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {2 b c^{2} d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {38 b c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{225} + b d^{2} x \operatorname {asinh}{\left (c x \right )} - \frac {149 b d^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} & \text {for}\: c \neq 0 \\a d^{2} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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