Optimal. Leaf size=548 \[ \frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (c^2 x^2+1\right )^{5/2}}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (c^2 x^2+1\right )}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (c^2 x^2+1\right )^2}-\frac {b \sqrt {c^2 x^2+1} (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {c^2 x^2+1}}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (c^2 x^2+1\right )^{5/2}}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (c^2 x^2+1\right )}+\frac {245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (c^2 x^2+1\right )^2}-\frac {115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (c^2 x^2+1\right )^{5/2}}+\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \]
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Rubi [A] time = 0.61, antiderivative size = 548, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {5712, 5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (c^2 x^2+1\right )^{5/2}}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (c^2 x^2+1\right )}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (c^2 x^2+1\right )^2}-\frac {b \sqrt {c^2 x^2+1} (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {c^2 x^2+1}}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (c^2 x^2+1\right )^{5/2}}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (c^2 x^2+1\right )}+\frac {245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (c^2 x^2+1\right )^2}-\frac {115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (c^2 x^2+1\right )^{5/2}}+\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5682
Rule 5684
Rule 5712
Rule 5717
Rubi steps
\begin {align*} \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {\left ((d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\left (1+c^2 x^2\right )^{5/2}}\\ &=\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (5 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{6 \left (1+c^2 x^2\right )^{5/2}}-\frac {\left (b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \left (1+c^2 x^2\right )^{5/2}}\\ &=-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac {\left (5 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{8 \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{18 \left (1+c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{12 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {1+c^2 x^2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac {\left (5 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{108 \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{48 \left (1+c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {1+c^2 x^2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{144 \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{64 \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 c^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac {245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (1+c^2 x^2\right )^2}+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {1+c^2 x^2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{288 \left (1+c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{128 \left (1+c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac {245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (1+c^2 x^2\right )^2}+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac {115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {1+c^2 x^2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 2.34, size = 735, normalized size = 1.34 \[ \frac {4320 a^2 d^{5/2} f^{5/2} \sqrt {c^2 x^2+1} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+9504 a^2 c d^2 f^2 x \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+2304 a^2 c^5 d^2 f^2 x^5 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+7488 a^2 c^3 d^2 f^2 x^3 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+72 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^2 \left (60 a+45 b \sinh \left (2 \sinh ^{-1}(c x)\right )+9 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )-3240 a b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-324 a b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (6 \sinh ^{-1}(c x)\right )-12 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (-540 a \sinh \left (2 \sinh ^{-1}(c x)\right )-108 a \sinh \left (4 \sinh ^{-1}(c x)\right )-12 a \sinh \left (6 \sinh ^{-1}(c x)\right )+270 b \cosh \left (2 \sinh ^{-1}(c x)\right )+27 b \cosh \left (4 \sinh ^{-1}(c x)\right )+2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )+1440 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^3+1620 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )+81 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (6 \sinh ^{-1}(c x)\right )}{13824 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} c^{4} d^{2} f^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} f^{2} x^{2} + b^{2} d^{2} f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (a b c^{4} d^{2} f^{2} x^{4} + 2 \, a b c^{2} d^{2} f^{2} x^{2} + a b d^{2} f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a^{2} c^{4} d^{2} f^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} f^{2} x^{2} + a^{2} d^{2} f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}, x\right ) \]
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 = 0.30, size = 0, normalized size = 0.00 \[ \int \left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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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