Optimal. Leaf size=227 \[ \frac {1}{2} f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {c^2 x^2+1}}+\frac {g \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2}-\frac {b c f x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}-\frac {b g x \sqrt {c^2 d x^2+d}}{3 c \sqrt {c^2 x^2+1}}-\frac {b c g x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.25, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5835, 5821, 5682, 5675, 30, 5717} \[ \frac {1}{2} f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {c^2 x^2+1}}+\frac {g \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2}-\frac {b c f x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}-\frac {b c g x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {b g x \sqrt {c^2 d x^2+d}}{3 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5675
Rule 5682
Rule 5717
Rule 5821
Rule 5835
Rubi steps
\begin {align*} \int (f+g x) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d+c^2 d x^2} \int (f+g x) \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \int \left (f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+g x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (f \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (g \sqrt {d+c^2 d x^2}\right ) \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{2} f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2}+\frac {\left (f \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c f \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b g \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 g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b c g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2}+\frac {f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 208, normalized size = 0.92 \[ \frac {1}{6} a \sqrt {c^2 d x^2+d} \left (\frac {2 g}{c^2}+x (3 f+2 g x)\right )+\frac {a \sqrt {d} f \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )}{2 c}+\frac {b f \sqrt {c^2 d x^2+d} \left (2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )-\cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{8 c \sqrt {c^2 x^2+1}}-\frac {b g \sqrt {c^2 d x^2+d} \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )}{9 c^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c^{2} d x^{2} + d} {\left (a g x + a f + {\left (b g x + b f\right )} \operatorname {arsinh}\left (c x\right )\right )}, x\right ) \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 423, normalized size = 1.86 \[ \frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+\frac {a f x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a f d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \arcsinh \left (c x \right )}{3 c^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f}{8 c \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \,c^{2} \arcsinh \left (c x \right ) x^{4}}{3 c^{2} x^{2}+3}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g c \,x^{3}}{9 \sqrt {c^{2} x^{2}+1}}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \arcsinh \left (c x \right ) x^{2}}{3 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g x}{3 c \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,c^{2} \arcsinh \left (c x \right ) x^{3}}{2 c^{2} x^{2}+2}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f c \,x^{2}}{4 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right ) x}{2 c^{2} x^{2}+2}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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 \[ \int \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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