Optimal. Leaf size=187 \[ \frac {i f^3 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b f^3 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f^3 \left (c^2 x^2+1\right )^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5712, 651, 5819, 12, 627, 43} \[ \frac {i f^3 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b f^3 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f^3 \left (c^2 x^2+1\right )^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 627
Rule 651
Rule 5712
Rule 5819
Rubi steps
\begin {align*} \int \frac {\sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{5/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(f-i c f x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {i f^3 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {i f^3 (1-i c x)^3}{3 c \left (1+c^2 x^2\right )^2} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {i f^3 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(1-i c x)^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {i f^3 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1-i c x}{(1+i c x)^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {i f^3 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {2}{(-i+c x)^2}+\frac {i}{-i+c x}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i b f^3 \left (1+c^2 x^2\right )^{5/2}}{3 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f^3 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f^3 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 141, normalized size = 0.75 \[ \frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left (-(c x+i) \left (a \sqrt {c^2 x^2+1}+b c x-i b\right )-b (c x+i) \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+b (c x-i)^2 \log (d+i c d x)\right )}{3 c d^3 (c x-i)^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 565, normalized size = 3.02 \[ -\frac {24 \, \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b c x + 3 \, {\left (4 \, b c^{2} x^{2} + 8 i \, b c x - 4 \, b\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (3 \, c^{4} d^{3} x^{3} - 3 i \, c^{3} d^{3} x^{2} + 3 \, c^{2} d^{3} x - 3 i \, c d^{3}\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}} \log \left (\frac {3 \, {\left (-2 i \, b c^{6} x^{2} - 4 \, b c^{5} x + 4 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} + 2 \, {\left (3 i \, c^{9} d^{3} x^{4} + 6 \, c^{8} d^{3} x^{3} + 3 i \, c^{7} d^{3} x^{2} + 6 \, c^{6} d^{3} x\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}}}{3 \, {\left (16 \, b c^{3} x^{3} - 16 i \, b c^{2} x^{2} + 16 \, b c x - 16 i \, b\right )}}\right ) - 2 \, {\left (3 \, c^{4} d^{3} x^{3} - 3 i \, c^{3} d^{3} x^{2} + 3 \, c^{2} d^{3} x - 3 i \, c d^{3}\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}} \log \left (\frac {3 \, {\left (-2 i \, b c^{6} x^{2} - 4 \, b c^{5} x + 4 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} + 2 \, {\left (-3 i \, c^{9} d^{3} x^{4} - 6 \, c^{8} d^{3} x^{3} - 3 i \, c^{7} d^{3} x^{2} - 6 \, c^{6} d^{3} x\right )} \sqrt {\frac {b^{2} f}{c^{2} d^{5}}}}{3 \, {\left (16 \, b c^{3} x^{3} - 16 i \, b c^{2} x^{2} + 16 \, b c x - 16 i \, b\right )}}\right ) + 3 \, {\left (4 \, a c^{2} x^{2} + 8 i \, a c x - 4 \, a\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{3 \, {\left (12 \, c^{4} d^{3} x^{3} - 12 i \, c^{3} d^{3} x^{2} + 12 \, c^{2} d^{3} x - 12 i \, c d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (c x \right )\right ) \sqrt {-i c f x +f}}{\left (i c d x +d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 219, normalized size = 1.17 \[ \frac {1}{3} \, b c {\left (\frac {6 \, \sqrt {f}}{3 i \, c^{3} d^{\frac {5}{2}} x + 3 \, c^{2} d^{\frac {5}{2}}} + \frac {\sqrt {f} \log \left (c x - i\right )}{c^{2} d^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {2 i \, \sqrt {c^{2} d f x^{2} + d f}}{c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}} + \frac {3 i \, \sqrt {c^{2} d f x^{2} + d f}}{3 i \, c^{2} d^{3} x + 3 \, c d^{3}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 i \, \sqrt {c^{2} d f x^{2} + d f}}{c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}} + \frac {3 i \, \sqrt {c^{2} d f x^{2} + d f}}{3 i \, c^{2} d^{3} x + 3 \, c d^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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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