Optimal. Leaf size=105 \[ \frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac {b x \sqrt {c^2 d x^2+d}}{3 c \sqrt {c^2 x^2+1}}-\frac {b c x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5717} \[ \frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac {b c x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {b x \sqrt {c^2 d x^2+d}}{3 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5717
Rubi steps
\begin {align*} \int x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac {\left (b \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 92, normalized size = 0.88 \[ \frac {\sqrt {c^2 d x^2+d} \left (3 a \left (c^2 x^2+1\right )^2-b c x \left (c^2 x^2+3\right ) \sqrt {c^2 x^2+1}+3 b \left (c^2 x^2+1\right )^2 \sinh ^{-1}(c x)\right )}{9 c^2 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 127, normalized size = 1.21 \[ \frac {3 \, {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 3 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{9 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]
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 [B] time = 0.14, size = 321, normalized size = 3.06 \[ \frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \arcsinh \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\arcsinh \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+\arcsinh \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+3 \arcsinh \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 73, normalized size = 0.70 \[ \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, c^{2} d} - \frac {{\left (c^{2} d^{\frac {3}{2}} x^{3} + 3 \, d^{\frac {3}{2}} x\right )} b}{9 \, c d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, c^{2} d} \]
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 \[ \int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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