Optimal. Leaf size=615 \[ \frac {5 f^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {22 i b f^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b c^2 f^3 x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 f^3 \left (c^2 x^2+1\right )^2}{27 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 b^2 f^3 x \left (c^2 x^2+1\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {68 i b^2 f^3 \left (c^2 x^2+1\right )}{9 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b^2 f^3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rubi [A] time = 0.78, antiderivative size = 615, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, 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 8
Rule 32
Rule 2633
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3317
Rule 5712
Rule 5831
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*}
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Mathematica [A] time = 3.62, size = 723, normalized size = 1.18 \[ \frac {540 a^2 \sqrt {d} f^{5/2} \sqrt {c^2 x^2+1} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-792 i a^2 f^2 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+72 i a^2 c^2 f^2 x^2 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}-324 a^2 c f^2 x \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+18 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^2 \left (30 a-45 i b \sqrt {c^2 x^2+1}-9 b \sinh \left (2 \sinh ^{-1}(c x)\right )+i b \cosh \left (3 \sinh ^{-1}(c x)\right )\right )+6 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (27 b \cosh \left (2 \sinh ^{-1}(c x)\right )+2 i \left (27 a \sqrt {c^2 x^2+1} (-5+2 i c x)+3 a \cosh \left (3 \sinh ^{-1}(c x)\right )-4 b c x \left (c^2 x^2-33\right )\right )\right )+1620 i a b c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-12 i a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )+162 a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-1620 i b^2 f^2 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+180 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^3-81 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )+4 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (3 \sinh ^{-1}(c x)\right )}{216 c d \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (i \, b^{2} c^{2} f^{2} x^{2} - 2 \, b^{2} c f^{2} x - i \, b^{2} f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (2 i \, a b c^{2} f^{2} x^{2} - 4 \, a b c f^{2} x - 2 i \, a b f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (i \, a^{2} c^{2} f^{2} x^{2} - 2 \, a^{2} c f^{2} x - i \, a^{2} f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{c d x - i \, d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}{\sqrt {i c d x +d}}\, dx \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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