Optimal. Leaf size=304 \[ \frac {i b f x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {b c f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {1+c^2 x^2}}+\frac {i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac {i f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {1+c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, 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 = {5796, 5838,
5785, 5783, 30, 5798} \begin {gather*} \frac {f \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 f \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} f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{4 \sqrt {c^2 x^2+1}}+\frac {i b f 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 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 5783
Rule 5785
Rule 5796
Rule 5798
Rule 5838
Rubi steps
\begin {align*} \int \sqrt {d+i c d x} (f-i c f x)^{3/2} \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 (f-i c f 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 (f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-i c f 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 (f \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 f \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} f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac {i f \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 (f \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 f \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 f \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 f x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {b c f 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 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac {i f \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 {f \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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.97, size = 273, normalized size = 0.90 \begin {gather*} \frac {12 a f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-2 i+3 c x-2 i c^2 x^2\right )+36 a \sqrt {d} f^{3/2} \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 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 \sinh ^{-1}(c x)^2-\cosh \left (2 \sinh ^{-1}(c x)\right )+2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {2 i b f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (9 c x-3 \sinh ^{-1}(c x) \left (3 \sqrt {1+c^2 x^2}+\cosh \left (3 \sinh ^{-1}(c x)\right )\right )+\sinh \left (3 \sinh ^{-1}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}}{72 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right ) \sqrt {i c d x +d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {i d \left (c x - i\right )} \left (- i f \left (c x + i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________