Optimal. Leaf size=431 \[ \frac{1}{2} f^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{f^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt{c^2 x^2+1}}+\frac{2 f g \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2}+\frac{1}{4} g^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{g^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^2}-\frac{g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{c^2 x^2+1}}-\frac{b c f^2 x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}-\frac{2 b c f g x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}-\frac{2 b f g x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}}-\frac{b c g^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{b g^2 x^2 \sqrt{c^2 d x^2+d}}{16 c \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.532257, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5835, 5821, 5682, 5675, 30, 5717, 5742, 5758} \[ \frac{1}{2} f^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{f^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt{c^2 x^2+1}}+\frac{2 f g \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2}+\frac{1}{4} g^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{g^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^2}-\frac{g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{c^2 x^2+1}}-\frac{b c f^2 x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}-\frac{2 b c f g x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}-\frac{2 b f g x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}}-\frac{b c g^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{b g^2 x^2 \sqrt{c^2 d x^2+d}}{16 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5835
Rule 5821
Rule 5682
Rule 5675
Rule 30
Rule 5717
Rule 5742
Rule 5758
Rubi steps
\begin{align*} \int (f+g x)^2 \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)^2 \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^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+2 f g x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 f 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{\left (g^2 \sqrt{d+c^2 d x^2}\right ) \int x^2 \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^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 f 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^2 \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^2 \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (2 b f 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{\left (g^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{4 \sqrt{1+c^2 x^2}}-\frac{\left (b c g^2 \sqrt{d+c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b f g x \sqrt{d+c^2 d x^2}}{3 c \sqrt{1+c^2 x^2}}-\frac{b c f^2 x^2 \sqrt{d+c^2 d x^2}}{4 \sqrt{1+c^2 x^2}}-\frac{2 b c f g x^3 \sqrt{d+c^2 d x^2}}{9 \sqrt{1+c^2 x^2}}-\frac{b c g^2 x^4 \sqrt{d+c^2 d x^2}}{16 \sqrt{1+c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 f 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^2 \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}}-\frac{\left (g^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{8 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (b g^2 \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b f g x \sqrt{d+c^2 d x^2}}{3 c \sqrt{1+c^2 x^2}}-\frac{b c f^2 x^2 \sqrt{d+c^2 d x^2}}{4 \sqrt{1+c^2 x^2}}-\frac{b g^2 x^2 \sqrt{d+c^2 d x^2}}{16 c \sqrt{1+c^2 x^2}}-\frac{2 b c f g x^3 \sqrt{d+c^2 d x^2}}{9 \sqrt{1+c^2 x^2}}-\frac{b c g^2 x^4 \sqrt{d+c^2 d x^2}}{16 \sqrt{1+c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 f 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^2 \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}}-\frac{g^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.926528, size = 301, normalized size = 0.7 \[ \frac{48 a c \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (12 c^2 f^2 x+16 f \left (c^2 g x^2+g\right )+3 g^2 x \left (2 c^2 x^2+1\right )\right )+144 a \sqrt{d} \sqrt{c^2 x^2+1} (2 c f-g) (2 c f+g) \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )-144 b c^2 f^2 \sqrt{c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-256 b c f g \sqrt{c^2 d x^2+d} \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )-9 b g^2 \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{1152 c^3 \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.355, size = 791, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c^{2} d x^{2} + d}{\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c^{2} d x^{2} + d}{\left (g x + f\right )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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