Optimal. Leaf size=430 \[ \frac {f^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 d x^2+d}}+\frac {3 f^2 g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {c^2 d x^2+d}}+\frac {3 f g^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 \sqrt {c^2 d x^2+d}}+\frac {g^3 x^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \sqrt {c^2 d x^2+d}}-\frac {2 g^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \sqrt {c^2 d x^2+d}}-\frac {3 f g^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {c^2 d x^2+d}}-\frac {3 b f^2 g x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}-\frac {3 b f g^2 x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}-\frac {b g^3 x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}+\frac {2 b g^3 x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.58, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {5835, 5821, 5675, 5717, 8, 5758, 30} \[ \frac {3 f^2 g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {c^2 d x^2+d}}+\frac {f^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 d x^2+d}}-\frac {3 f g^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {c^2 d x^2+d}}+\frac {3 f g^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 \sqrt {c^2 d x^2+d}}-\frac {2 g^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \sqrt {c^2 d x^2+d}}+\frac {g^3 x^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \sqrt {c^2 d x^2+d}}-\frac {3 b f^2 g x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}-\frac {3 b f g^2 x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}-\frac {b g^3 x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}+\frac {2 b g^3 x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5675
Rule 5717
Rule 5758
Rule 5821
Rule 5835
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {(f+g x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\sqrt {1+c^2 x^2} \int \left (\frac {f^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3 f^2 g x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3 f g^2 x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {g^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\left (f^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}+\frac {\left (3 f^2 g \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}+\frac {\left (3 f g^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}+\frac {\left (g^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {3 f^2 g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {d+c^2 d x^2}}+\frac {3 f g^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 \sqrt {d+c^2 d x^2}}+\frac {g^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \sqrt {d+c^2 d x^2}}+\frac {f^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {\left (3 b f^2 g \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{c \sqrt {d+c^2 d x^2}}-\frac {\left (3 f g^2 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b f g^2 \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 c \sqrt {d+c^2 d x^2}}-\frac {\left (2 g^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 c^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b g^3 \sqrt {1+c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt {d+c^2 d x^2}}\\ &=-\frac {3 b f^2 g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {1+c^2 x^2}}{4 c \sqrt {d+c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}+\frac {3 f^2 g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {d+c^2 d x^2}}-\frac {2 g^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \sqrt {d+c^2 d x^2}}+\frac {3 f g^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 \sqrt {d+c^2 d x^2}}+\frac {g^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \sqrt {d+c^2 d x^2}}+\frac {f^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {3 f g^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b g^3 \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt {d+c^2 d x^2}}\\ &=-\frac {3 b f^2 g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {2 b g^3 x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {1+c^2 x^2}}{4 c \sqrt {d+c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}+\frac {3 f^2 g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {d+c^2 d x^2}}-\frac {2 g^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \sqrt {d+c^2 d x^2}}+\frac {3 f g^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 c^2 \sqrt {d+c^2 d x^2}}+\frac {g^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \sqrt {d+c^2 d x^2}}+\frac {f^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {3 f g^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 304, normalized size = 0.71 \[ \frac {4 \sqrt {d} g \left (3 a \left (c^2 x^2+1\right ) \left (c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )-4 g^2\right )-2 b c x \sqrt {c^2 x^2+1} \left (c^2 \left (27 f^2+g^2 x^2\right )-6 g^2\right )\right )+36 a c f \sqrt {c^2 d x^2+d} \left (2 c^2 f^2-3 g^2\right ) \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+12 b \sqrt {d} g \left (c^2 x^2+1\right ) \sinh ^{-1}(c x) \left (c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )-4 g^2\right )+18 b c \sqrt {d} f \sqrt {c^2 x^2+1} \left (2 c^2 f^2-3 g^2\right ) \sinh ^{-1}(c x)^2-27 b c \sqrt {d} f g^2 \sqrt {c^2 x^2+1} \cosh \left (2 \sinh ^{-1}(c x)\right )}{72 c^4 \sqrt {d} \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} + {\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d x^{2} + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 751, normalized size = 1.75 \[ \frac {a \,g^{3} x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 a \,g^{3} \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}+\frac {3 a f \,g^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {3 a f \,g^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}+\frac {3 a \,f^{2} g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a \,f^{3} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right )^{2} g^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g^{3} \arcsinh \left (c x \right ) x^{4}}{3 d \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g^{3} x^{3}}{9 c d \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g^{3} \arcsinh \left (c x \right ) x^{2}}{3 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g^{3} x}{3 c^{3} d \sqrt {c^{2} x^{2}+1}}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,g^{2}}{8 c^{3} d \sqrt {c^{2} x^{2}+1}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \arcsinh \left (c x \right ) f^{2}}{c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,g^{2} \arcsinh \left (c x \right ) x}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \arcsinh \left (c x \right ) x^{2} f^{2}}{d \left (c^{2} x^{2}+1\right )}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g x \,f^{2}}{c d \sqrt {c^{2} x^{2}+1}}+\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,g^{2} \arcsinh \left (c x \right ) x^{3}}{2 d \left (c^{2} x^{2}+1\right )}-\frac {3 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \,g^{2} x^{2}}{4 c d \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f^{3} \arcsinh \left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c d}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g^{3} \arcsinh \left (c x \right )}{3 c^{4} d \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 \[ \int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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