Optimal. Leaf size=508 \[ -\frac {b c f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}+\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {c^2 x^2+1}}-\frac {i f \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {c^2 x^2+1}}+\frac {1}{2} f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 i b^2 f \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x}}{27 c}-\frac {b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {4 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c} \]
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Rubi [A] time = 0.65, antiderivative size = 508, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {5712, 5821, 5682, 5675, 5661, 321, 215, 5717, 5679, 444, 43} \[ \frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {c^2 x^2+1}}-\frac {b c f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}+\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {c^2 x^2+1}}-\frac {i f \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c}+\frac {1}{2} f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 i b^2 f \left (c^2 x^2+1\right ) \sqrt {d+i c d x} \sqrt {f-i c f x}}{27 c}-\frac {b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {4 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 215
Rule 321
Rule 444
Rule 5661
Rule 5675
Rule 5679
Rule 5682
Rule 5712
Rule 5717
Rule 5821
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 )^2 \, 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 )^2 \, 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 )^2-i c f x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2\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 )^2 \, 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 )^2 \, 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 )^2-\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 )^2}{3 c}+\frac {\left (f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (2 i b f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\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 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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 )^2-\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 )^2}{3 c}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {1+c^2 x^2}}-\frac {\left (2 i b^2 c f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x \left (1+\frac {c^2 x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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 )^2-\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 )^2}{3 c}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {1+c^2 x^2}}-\frac {\left (b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (i b^2 c f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt {1+c^2 x^2}}+\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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 )^2-\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 )^2}{3 c}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {1+c^2 x^2}}-\frac {\left (i b^2 c f \sqrt {d+i c d x} \sqrt {f-i c f x}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+c^2 x}}+\frac {1}{3} \sqrt {1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {4 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {1}{4} b^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {2 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt {1+c^2 x^2}}+\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )}{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 )^2-\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 )^2}{3 c}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.83, size = 705, normalized size = 1.39 \[ \frac {108 a^2 \sqrt {d} f^{3/2} \sqrt {c^2 x^2+1} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-72 i a^2 c^2 f x^2 \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}-72 i a^2 f \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+108 a^2 c f x \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+18 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^2 \left (6 a-3 i b \sqrt {c^2 x^2+1}+3 b \sinh \left (2 \sinh ^{-1}(c x)\right )-i b \cosh \left (3 \sinh ^{-1}(c x)\right )\right )+6 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (-9 b \cosh \left (2 \sinh ^{-1}(c x)\right )+2 \left (-9 i a \sqrt {c^2 x^2+1}+9 a \sinh \left (2 \sinh ^{-1}(c x)\right )-3 i a \cosh \left (3 \sinh ^{-1}(c x)\right )+9 i b c x+i b \sinh \left (3 \sinh ^{-1}(c x)\right )\right )\right )+108 i a b c f x \sqrt {d+i c d x} \sqrt {f-i c f x}+12 i a b f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )-54 a b f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-108 i b^2 f \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+36 b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^3+27 b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )-4 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (3 \sinh ^{-1}(c x)\right )}{216 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-i \, b^{2} c f x + b^{2} f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (-2 i \, a b c f x + 2 \, a b f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (-i \, a^{2} c f x + a^{2} f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}, x\right ) \]
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 = 0.30, size = 0, normalized size = 0.00 \[ \int \left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2} \sqrt {i c d x +d}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {i d \left (c x - i\right )} \left (- i f \left (c x + i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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