Optimal. Leaf size=615 \[ \frac{5 f^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{22 i b f^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 f^3 \left (c^2 x^2+1\right )^2}{27 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 b^2 f^3 x \left (c^2 x^2+1\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{68 i b^2 f^3 \left (c^2 x^2+1\right )}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b^2 f^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}} \]
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Rubi [A] time = 0.780721, antiderivative size = 615, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used = {5712, 5831, 3317, 3296, 2638, 3311, 32, 2635, 8, 2633} \[ \frac{5 f^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{22 i b f^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 f^3 \left (c^2 x^2+1\right )^2}{27 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 b^2 f^3 x \left (c^2 x^2+1\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{68 i b^2 f^3 \left (c^2 x^2+1\right )}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b^2 f^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5831
Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 32
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \frac{(f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+i c d x}} \, dx &=\frac{\sqrt{1+c^2 x^2} \int \frac{(f-i c f x)^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 (c f-i c f \sinh (x))^3 \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int \left (c^3 f^3 (a+b x)^2-3 i c^3 f^3 (a+b x)^2 \sinh (x)-3 c^3 f^3 (a+b x)^2 \sinh ^2(x)+i c^3 f^3 (a+b x)^2 \sinh ^3(x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{f^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (i f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh ^3(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (3 i f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (3 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{3 b c f^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 i f^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{f^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (2 i f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (3 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (6 i b f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (2 i b^2 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh ^3(x) \, dx,x,\sinh ^{-1}(c x)\right )}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (3 b^2 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{6 i b f^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{5 f^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (4 i b f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (2 i b^2 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt{1+c^2 x^2}\right )}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (6 i b^2 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (3 b^2 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\sinh ^{-1}(c x)\right )}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{56 i b^2 f^3 \left (1+c^2 x^2\right )}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 f^3 \left (1+c^2 x^2\right )^2}{27 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b^2 f^3 \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{22 i b f^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{5 f^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (4 i b^2 f^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{68 i b^2 f^3 \left (1+c^2 x^2\right )}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 b^2 f^3 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 f^3 \left (1+c^2 x^2\right )^2}{27 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b^2 f^3 \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{22 i b f^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{5 f^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ \end{align*}
Mathematica [A] time = 3.60554, size = 723, normalized size = 1.18 \[ \frac{-792 i a^2 f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+72 i a^2 c^2 f^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-324 a^2 c f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+540 a^2 \sqrt{d} f^{5/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )+18 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (30 a-45 i b \sqrt{c^2 x^2+1}-9 b \sinh \left (2 \sinh ^{-1}(c x)\right )+i b \cosh \left (3 \sinh ^{-1}(c x)\right )\right )+6 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (27 b \cosh \left (2 \sinh ^{-1}(c x)\right )+2 i \left (27 a \sqrt{c^2 x^2+1} (-5+2 i c x)+3 a \cosh \left (3 \sinh ^{-1}(c x)\right )-4 b c x \left (c^2 x^2-33\right )\right )\right )+1620 i a b c f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}-12 i a b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )+162 a b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-1620 i b^2 f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+180 b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-81 b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )+4 i b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (3 \sinh ^{-1}(c x)\right )}{216 c d \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.323, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ( f-icfx \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{d+icdx}}}}\, dx \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 (\frac{{\left (i \, b^{2} c^{2} f^{2} x^{2} - 2 \, b^{2} c f^{2} x - i \, b^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} +{\left (2 i \, a b c^{2} f^{2} x^{2} - 4 \, a b c f^{2} x - 2 i \, a b f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (i \, a^{2} c^{2} f^{2} x^{2} - 2 \, a^{2} c f^{2} x - i \, a^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{c d x - i \, d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (-i \, c f x + f\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt{i \, c d x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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