Optimal. Leaf size=353 \[ \frac {3}{8} d f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {c^2 x^2+1}}+\frac {d g \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}-\frac {5 b c d f x^2 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {b d g x \sqrt {c^2 d x^2+d}}{5 c \sqrt {c^2 x^2+1}}-\frac {2 b c d g x^3 \sqrt {c^2 d x^2+d}}{15 \sqrt {c^2 x^2+1}}-\frac {b c^3 d f x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {b c^3 d g x^5 \sqrt {c^2 d x^2+d}}{25 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.34, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 5717, 194} \[ \frac {3}{8} d f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {c^2 x^2+1}}+\frac {d g \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}-\frac {b c^3 d f x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {5 b c d f x^2 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {b c^3 d g x^5 \sqrt {c^2 d x^2+d}}{25 \sqrt {c^2 x^2+1}}-\frac {2 b c d g x^3 \sqrt {c^2 d x^2+d}}{15 \sqrt {c^2 x^2+1}}-\frac {b d g x \sqrt {c^2 d x^2+d}}{5 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 194
Rule 5675
Rule 5682
Rule 5684
Rule 5717
Rule 5821
Rule 5835
Rubi steps
\begin {align*} \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int (f+g x) \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \left (f \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+g x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d f \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} d f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {\left (3 d 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}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b d g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {1+c^2 x^2}}\\ &=\frac {3}{8} d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {\left (3 d f \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d f \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b d g \sqrt {d+c^2 d x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b d g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {2 b c d g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {3 d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.23, size = 392, normalized size = 1.11 \[ \frac {3600 a c d^{3/2} f \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+240 a d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (5 c^2 f x \left (2 c^2 x^2+5\right )+8 g \left (c^2 x^2+1\right )^2\right )+2400 b c d f \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )-1200 b c d f \sqrt {c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-75 b c d f \sqrt {c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )-640 b c d g x \left (c^2 x^2+3\right ) \sqrt {c^2 d x^2+d}+3200 b d g \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)+640 b d g \left (c^2 x^2+1\right )^{3/2} \left (3 c^2 x^2-2\right ) \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)-128 b c^3 d g x^3 \left (3 c^2 x^2+5\right ) \sqrt {c^2 d x^2+d}}{9600 c^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{2} d g x^{3} + a c^{2} d f x^{2} + a d g x + a d f + {\left (b c^{2} d g x^{3} + b c^{2} d f x^{2} + b d g x + b d f\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}, x\right ) \]
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 [A] time = 0.50, size = 601, normalized size = 1.70 \[ \frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+\frac {a f x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right )^{2} d}{16 \sqrt {c^{2} x^{2}+1}\, c}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d \,c^{4} \arcsinh \left (c x \right ) x^{6}}{5 c^{2} x^{2}+5}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d \,c^{3} x^{5}}{25 \sqrt {c^{2} x^{2}+1}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d \,c^{2} \arcsinh \left (c x \right ) x^{4}}{5 \left (c^{2} x^{2}+1\right )}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d c \,x^{3}}{15 \sqrt {c^{2} x^{2}+1}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d \arcsinh \left (c x \right ) x^{2}}{5 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d x}{5 c \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f d \,c^{4} \arcsinh \left (c x \right ) x^{5}}{4 c^{2} x^{2}+4}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f d \,c^{3} x^{4}}{16 \sqrt {c^{2} x^{2}+1}}+\frac {7 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f d \,c^{2} \arcsinh \left (c x \right ) x^{3}}{8 \left (c^{2} x^{2}+1\right )}-\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f d c \,x^{2}}{16 \sqrt {c^{2} x^{2}+1}}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f d \arcsinh \left (c x \right ) x}{8 \left (c^{2} x^{2}+1\right )}-\frac {17 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f d}{128 c \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g d \arcsinh \left (c x \right )}{5 c^{2} \left (c^{2} x^{2}+1\right )} \]
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 \[ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
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 \[ \int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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