Optimal. Leaf size=304 \[ \frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {c^2 x^2+1}}+\frac {i d \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {c^2 x^2+1}}-\frac {i b d x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {c^2 x^2+1}}-\frac {i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.34, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5712, 5821, 5682, 5675, 30, 5717} \[ \frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {c^2 x^2+1}}+\frac {i d \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac {i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {c^2 x^2+1}}-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {c^2 x^2+1}}-\frac {i b d x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5675
Rule 5682
Rule 5712
Rule 5717
Rule 5821
Rubi steps
\begin {align*} \int (d+i c d x)^{3/2} \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) \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 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+i c d x \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 \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 (i c d \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 {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac {i d \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 \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 (i b d \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 \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {i b d x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {1+c^2 x^2}}-\frac {i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )+\frac {i d \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 \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}}\\ \end {align*}
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Mathematica [A] time = 1.58, size = 273, normalized size = 0.90 \[ \frac {12 a d \left (2 i c^2 x^2+3 c x+2 i\right ) \sqrt {d+i c d x} \sqrt {f-i c f x}+36 a d^{3/2} \sqrt {f} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+\frac {9 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 \sinh ^{-1}(c x)^2+2 \sinh \left (2 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)-\cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {2 i b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-3 \sinh ^{-1}(c x) \left (3 \sqrt {c^2 x^2+1}+\cosh \left (3 \sinh ^{-1}(c x)\right )\right )+9 c x+\sinh \left (3 \sinh ^{-1}(c x)\right )\right )}{\sqrt {c^2 x^2+1}}}{72 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (i \, b c d x + b d\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (i \, a c d x + a d\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}, x\right ) \]
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 = 0.28, size = 0, normalized size = 0.00 \[ \int \left (i c d x +d \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right ) \sqrt {-i c f x +f}\, dx \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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