Optimal. Leaf size=416 \[ -\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{c^2 x^2+1}}+\frac{2 i d^2 \left (c^2 x^2+1\right ) \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{3}{8} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{b c^3 d^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}-\frac{2 i b c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{c^2 x^2+1}}-\frac{3 b c d^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}-\frac{2 i b d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.59734, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {5712, 5821, 5682, 5675, 30, 5717, 5742, 5758} \[ -\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{c^2 x^2+1}}+\frac{2 i d^2 \left (c^2 x^2+1\right ) \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{3}{8} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{b c^3 d^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}-\frac{2 i b c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{c^2 x^2+1}}-\frac{3 b c d^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}-\frac{2 i b d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5821
Rule 5682
Rule 5675
Rule 30
Rule 5717
Rule 5742
Rule 5758
Rubi steps
\begin{align*} \int (d+i c d x)^{5/2} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left (\sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int (d+i c d x)^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (\sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \left (d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+2 i c d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-c^2 d^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 (d^2 \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 ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (2 i c d^2 \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 ) \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (c^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\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} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 i d^2 \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 )}{3 c}+\frac{\left (d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\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 (2 i b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 \sqrt{1+c^2 x^2}}-\frac{\left (b c d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int x \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (c^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\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^3 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int x^3 \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 i b d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{1+c^2 x^2}}-\frac{b c d^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{4 \sqrt{1+c^2 x^2}}-\frac{2 i b c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{1+c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 i d^2 \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 )}{3 c}+\frac{d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt{1+c^2 x^2}}+\frac{\left (d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\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^2 \sqrt{d+i c d x} \sqrt{f-i c f x}\right ) \int x \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 i b d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{1+c^2 x^2}}-\frac{3 b c d^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{1+c^2 x^2}}-\frac{2 i b c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{1+c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 i d^2 \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 )}{3 c}+\frac{5 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.42174, size = 361, normalized size = 0.87 \[ \frac{48 a d^2 \sqrt{c^2 x^2+1} \left (-6 c^3 x^3+16 i c^2 x^2+9 c x+16 i\right ) \sqrt{d+i c d x} \sqrt{f-i c f x}+720 a d^{5/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )-64 i b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (-3 \sinh ^{-1}(c x) \left (3 \sqrt{c^2 x^2+1}+\cosh \left (3 \sinh ^{-1}(c x)\right )\right )+9 c x+\sinh \left (3 \sinh ^{-1}(c x)\right )\right )+144 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (2 \sinh ^{-1}(c x)^2+2 \sinh \left (2 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)-\cosh \left (2 \sinh ^{-1}(c x)\right )\right )+9 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{1152 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.379, size = 0, normalized size = 0. \begin{align*} \int \left ( d+icdx \right ) ^{{\frac{5}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) \sqrt{f-icfx}\, 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 (b c^{2} d^{2} x^{2} - 2 i \, b c d^{2} x - b d^{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 (a c^{2} d^{2} x^{2} - 2 i \, a c d^{2} x - a d^{2}\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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