Optimal. Leaf size=381 \[ \frac {11 i b f^3 x \sqrt {1+c^2 x^2}}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 b c f^3 x^2 \sqrt {1+c^2 x^2}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i b c^2 f^3 x^3 \sqrt {1+c^2 x^2}}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {11 i f^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 f^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i c f^3 x^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {5 f^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rubi [A]
time = 0.43, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5796, 5838,
5783, 5798, 8, 5812, 30} \begin {gather*} \frac {5 f^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 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 )}{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 \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 )}{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}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {11 i b f^3 x \sqrt {c^2 x^2+1}}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i b c^2 f^3 x^3 \sqrt {c^2 x^2+1}}{9 \sqrt {d+i c d x} \sqrt {f-i c f x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5783
Rule 5796
Rule 5798
Rule 5812
Rule 5838
Rubi steps
\begin {align*} \int \frac {(f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{\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 )}{\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} \int \left (\frac {f^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {3 i c f^3 x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {3 c^2 f^3 x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {i c^3 f^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {\left (f^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (3 i c f^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (3 c^2 f^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (i c^3 f^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\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 )}{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 \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 )}{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 )^2}{2 b 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 ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (3 i b f^3 \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i c f^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (3 b c f^3 \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (i b c^2 f^3 \sqrt {1+c^2 x^2}\right ) \int x^2 \, dx}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {3 i b f^3 x \sqrt {1+c^2 x^2}}{\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}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i b c^2 f^3 x^3 \sqrt {1+c^2 x^2}}{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 )}{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 \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 )}{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 )^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (2 i b f^3 \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {11 i b f^3 x \sqrt {1+c^2 x^2}}{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}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i b c^2 f^3 x^3 \sqrt {1+c^2 x^2}}{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 )}{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 \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 )}{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 )^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 465, normalized size = 1.22 \begin {gather*} \frac {264 i b c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-8 i b c^3 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}-264 i a f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-108 a c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+24 i a c^2 f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+90 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^2+27 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-6 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (9 (5 i+2 c x) \sqrt {1+c^2 x^2}-i \cosh \left (3 \sinh ^{-1}(c x)\right )\right )+180 a \sqrt {d} 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 )}{72 c d \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 \[\int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )}{\sqrt {i c d x +d}}\, dx\]
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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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