Optimal. Leaf size=459 \[ \frac {3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )}+\frac {3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \left (c^2 x^2+1\right )^{3/2}}-\frac {i f \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac {1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}+\frac {i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (c^2 x^2+1\right )^{3/2}}+\frac {2 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (c^2 x^2+1\right )^{3/2}}+\frac {i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (c^2 x^2+1\right )^{3/2}}-\frac {b c^3 f x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}} \]
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Rubi [A] time = 0.43, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5712, 5821, 5684, 5682, 5675, 30, 14, 5717, 194} \[ \frac {3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )}+\frac {3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \left (c^2 x^2+1\right )^{3/2}}-\frac {i f \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac {1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (c^2 x^2+1\right )^{3/2}}-\frac {b c^3 f x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}+\frac {2 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (c^2 x^2+1\right )^{3/2}}-\frac {5 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}+\frac {i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 194
Rule 5675
Rule 5682
Rule 5684
Rule 5712
Rule 5717
Rule 5821
Rubi steps
\begin {align*} \int (d+i c d x)^{3/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int (f-i c f x) \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac {\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (f \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-i c f x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac {\left (f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}-\frac {\left (i c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac {\left (3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (i b f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (b c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )}-\frac {i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac {\left (3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (i b f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (b c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (3 b c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac {i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c^3 f x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}+\frac {i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (1+c^2 x^2\right )^{3/2}}+\frac {1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )}-\frac {i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac {3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \left (1+c^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.67, size = 683, normalized size = 1.49 \[ \frac {3600 a d^{3/2} f^{5/2} \sqrt {c^2 x^2+1} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-3840 i a c^2 d f^2 x^2 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+6000 a c d f^2 x \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}-1920 i a d f^2 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}-1920 i a c^4 d f^2 x^4 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+2400 a c^3 d f^2 x^3 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+60 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (5 \left (-4 i \sqrt {c^2 x^2+1}+8 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )-10 i \cosh \left (3 \sinh ^{-1}(c x)\right )-2 i \cosh \left (5 \sinh ^{-1}(c x)\right )\right )+1200 i b c d f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}+1800 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^2+200 i b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )+24 i b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (5 \sinh ^{-1}(c x)\right )-1200 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-75 b d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (4 \sinh ^{-1}(c x)\right )}{9600 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-i \, b c^{3} d f^{2} x^{3} + b c^{2} d f^{2} x^{2} - i \, b c d f^{2} x + b d f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (-i \, a c^{3} d f^{2} x^{3} + a c^{2} d f^{2} x^{2} - i \, a c d f^{2} x + a d f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}, x\right ) \]
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 \[ \int \left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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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