Optimal. Leaf size=548 \[ \frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac {245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (1+c^2 x^2\right )^2}+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac {115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt {1+c^2 x^2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}} \]
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Rubi [A]
time = 0.39, 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 = {5796, 5786,
5785, 5783, 5776, 327, 221, 5798, 201} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5796
Rule 5798
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 = 1.46, size = 735, normalized size = 1.34 \begin {gather*} \frac {9504 a^2 c d^2 f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+7488 a^2 c^3 d^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2304 a^2 c^5 d^2 f^2 x^5 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1440 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^3-3240 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 )+4320 a^2 d^{5/2} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+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 )-12 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (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 )-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 )\right )+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 )}{13824 c \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, 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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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