Optimal. Leaf size=282 \[ \frac {i b d \left (1+c^2 x^2\right )^{5/2}}{6 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (i-c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (1+c^2 x^2\right )^{5/2} \text {ArcTan}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {5796, 653, 197,
5837, 641, 46, 209, 266} \begin {gather*} \frac {2 d x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (-c x+i) \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 {i b d \left (c^2 x^2+1\right )^{5/2} \text {ArcTan}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (c^2 x^2+1\right )^{5/2}}{6 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (c^2 x^2+1\right )^{5/2} \log \left (c^2 x^2+1\right )}{3 c (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 46
Rule 197
Rule 209
Rule 266
Rule 641
Rule 653
Rule 5796
Rule 5837
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(d+i c d x) \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 {d (i-c x) \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 {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (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 \left (-\frac {d (i-c x)}{3 c \left (1+c^2 x^2\right )^2}+\frac {2 d x}{3 \left (1+c^2 x^2\right )}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac {d (i-c x) \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 {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {i-c x}{\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 {\left (2 b c d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac {d (i-c x) \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 {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{(-i-c x)^2 (i-c x)} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac {d (i-c x) \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 {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {i}{2 (i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {i b d \left (1+c^2 x^2\right )^{5/2}}{6 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (i-c x) \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 {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (i b d \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {i b d \left (1+c^2 x^2\right )^{5/2}}{6 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {d (i-c x) \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 {2 d x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i b d \left (1+c^2 x^2\right )^{5/2} \tan ^{-1}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b d \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{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 = 0.36, size = 202, normalized size = 0.72 \begin {gather*} \frac {\sqrt {f-i c f x} \left (4 i a-8 a c x+8 i a c^2 x^2-2 b \sqrt {1+c^2 x^2}+4 i b \left (1+2 i c x+2 c^2 x^2\right ) \sinh ^{-1}(c x)+5 b (1-i c x) \sqrt {1+c^2 x^2} \log (d (-1+i c x))+3 b \sqrt {1+c^2 x^2} \log (d+i c d x)-3 i b c x \sqrt {1+c^2 x^2} \log (d+i c d x)\right )}{12 c d f^3 (i+c x)^2 \sqrt {d+i c d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {a +b \arcsinh \left (c x \right )}{\left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 237, normalized size = 0.84 \begin {gather*} \frac {1}{12} \, b c {\left (\frac {2 i \, \sqrt {d} \sqrt {f}}{c^{3} d^{2} f^{3} x + i \, c^{2} d^{2} f^{3}} - \frac {5 \, \log \left (c x + i\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}} - \frac {3 \, \log \left (c x - i\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {3 i}{-3 i \, \sqrt {c^{2} d f x^{2} + d f} c^{2} d f^{2} x + 3 \, \sqrt {c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {c^{2} d f x^{2} + d f} d f^{2}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {3 i}{-3 i \, \sqrt {c^{2} d f x^{2} + d f} c^{2} d f^{2} x + 3 \, \sqrt {c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {c^{2} d f x^{2} + d f} d f^{2}}\right )} \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 {a+b\,\mathrm {asinh}\left (c\,x\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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