Optimal. Leaf size=508 \[ \frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{c^2 x^2+1}}+\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{c^2 x^2+1}}-\frac{i f \left (c^2 x^2+1\right ) \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 i b^2 f \left (c^2 x^2+1\right ) \sqrt{d+i c d x} \sqrt{f-i c f x}}{27 c}-\frac{b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{4 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c} \]
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Rubi [A] time = 0.649009, antiderivative size = 508, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.297, Rules used = {5712, 5821, 5682, 5675, 5661, 321, 215, 5717, 5679, 444, 43} \[ \frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{c^2 x^2+1}}+\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{c^2 x^2+1}}-\frac{i f \left (c^2 x^2+1\right ) \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 i b^2 f \left (c^2 x^2+1\right ) \sqrt{d+i c d x} \sqrt{f-i c f x}}{27 c}-\frac{b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{4 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5821
Rule 5682
Rule 5675
Rule 5661
Rule 321
Rule 215
Rule 5717
Rule 5679
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+i c d x} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{\left (\sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int (f-i c f x) \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (\sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \left (f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-i c f x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (i c f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{i f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{\left (f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}+\frac{\left (2 i b f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt{1+c^2 x^2}}-\frac{\left (b c f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{i f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{1+c^2 x^2}}-\frac{\left (2 i b^2 c f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \frac{x \left (1+\frac{c^2 x^2}{3}\right )}{\sqrt{1+c^2 x^2}} \, dx}{3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{4} b^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}+\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{i f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{1+c^2 x^2}}-\frac{\left (b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{4 \sqrt{1+c^2 x^2}}-\frac{\left (i b^2 c f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{c^2 x}{3}}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{4} b^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{1+c^2 x^2}}+\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{i f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{1+c^2 x^2}}-\frac{\left (i b^2 c f \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1+c^2 x}}+\frac{1}{3} \sqrt{1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{4 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c}+\frac{1}{4} b^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{2 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac{b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{1+c^2 x^2}}+\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{i f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.81958, size = 705, normalized size = 1.39 \[ \frac{108 a^2 \sqrt{d} f^{3/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )-72 i a^2 c^2 f x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-72 i a^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+108 a^2 c f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+18 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (6 a-3 i b \sqrt{c^2 x^2+1}+3 b \sinh \left (2 \sinh ^{-1}(c x)\right )-i b \cosh \left (3 \sinh ^{-1}(c x)\right )\right )+6 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (-9 b \cosh \left (2 \sinh ^{-1}(c x)\right )+2 \left (-9 i a \sqrt{c^2 x^2+1}+9 a \sinh \left (2 \sinh ^{-1}(c x)\right )-3 i a \cosh \left (3 \sinh ^{-1}(c x)\right )+9 i b c x+i b \sinh \left (3 \sinh ^{-1}(c x)\right )\right )\right )+108 i a b c f x \sqrt{d+i c d x} \sqrt{f-i c f x}+12 i a b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )-54 a b f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-108 i b^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+36 b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+27 b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )-4 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (3 \sinh ^{-1}(c x)\right )}{216 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int \left ( f-icfx \right ) ^{{\frac{3}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}\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 ({\left (-i \, b^{2} c f x + b^{2} f\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 f x + 2 \, a b f\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 f x + a^{2} f\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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