Optimal. Leaf size=87 \[ \frac {d \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2}-\frac {b d x \left (c^2 x^2+1\right )^{3/2}}{16 c}-\frac {3 b d x \sqrt {c^2 x^2+1}}{32 c}-\frac {3 b d \sinh ^{-1}(c x)}{32 c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5717, 195, 215} \[ \frac {d \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2}-\frac {b d x \left (c^2 x^2+1\right )^{3/2}}{16 c}-\frac {3 b d x \sqrt {c^2 x^2+1}}{32 c}-\frac {3 b d \sinh ^{-1}(c x)}{32 c^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 5717
Rubi steps
\begin {align*} \int x \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {d \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2}-\frac {(b d) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{4 c}\\ &=-\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}+\frac {d \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2}-\frac {(3 b d) \int \sqrt {1+c^2 x^2} \, dx}{16 c}\\ &=-\frac {3 b d x \sqrt {1+c^2 x^2}}{32 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}+\frac {d \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2}-\frac {(3 b d) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 c}\\ &=-\frac {3 b d x \sqrt {1+c^2 x^2}}{32 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}-\frac {3 b d \sinh ^{-1}(c x)}{32 c^2}+\frac {d \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.89 \[ \frac {d \left (c x \left (8 a c x \left (c^2 x^2+2\right )-b \sqrt {c^2 x^2+1} \left (2 c^2 x^2+5\right )\right )+b \left (8 c^4 x^4+16 c^2 x^2+5\right ) \sinh ^{-1}(c x)\right )}{32 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 98, normalized size = 1.13 \[ \frac {8 \, a c^{4} d x^{4} + 16 \, a c^{2} d x^{2} + {\left (8 \, b c^{4} d x^{4} + 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{32 \, c^{2}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 1.08 \[ \frac {d a \left (\frac {1}{4} c^{4} x^{4}+\frac {1}{2} c^{2} x^{2}\right )+d b \left (\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {\arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {5 c x \sqrt {c^{2} x^{2}+1}}{32}+\frac {5 \arcsinh \left (c x \right )}{32}\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 127, normalized size = 1.46 \[ \frac {1}{4} \, a c^{2} d x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d \]
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 \[ \int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 117, normalized size = 1.34 \[ \begin {cases} \frac {a c^{2} d x^{4}}{4} + \frac {a d x^{2}}{2} + \frac {b c^{2} d x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b c d x^{3} \sqrt {c^{2} x^{2} + 1}}{16} + \frac {b d x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {5 b d x \sqrt {c^{2} x^{2} + 1}}{32 c} + \frac {5 b d \operatorname {asinh}{\left (c x \right )}}{32 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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