Optimal. Leaf size=69 \[ \frac{1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}+\frac{27 (8 x+3 i) \sqrt{4 x^2+3 i x}}{1024}+\frac{243 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{4096} \]
[Out]
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Rubi [A] time = 0.0368329, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}+\frac{27 (8 x+3 i) \sqrt{4 x^2+3 i x}}{1024}+\frac{243 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{4096} \]
Antiderivative was successfully verified.
[In] Int[((3*I)*x + 4*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.43116, size = 56, normalized size = 0.81 \[ \frac{\left (8 x + 3 i\right ) \left (4 x^{2} + 3 i x\right )^{\frac{3}{2}}}{32} + \frac{27 \left (8 x + 3 i\right ) \sqrt{4 x^{2} + 3 i x}}{1024} + \frac{243 \operatorname{asinh}{\left (\frac{8 x}{3} + i \right )}}{4096} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3*I*x+4*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0730044, size = 83, normalized size = 1.2 \[ \frac{2 x \left (4096 x^4+7680 i x^3-3744 x^2+108 i x-243\right )+243 \sqrt{x} \sqrt{4 x+3 i} \log \left (2 \sqrt{x}+\sqrt{4 x+3 i}\right )}{2048 \sqrt{x (4 x+3 i)}} \]
Antiderivative was successfully verified.
[In] Integrate[((3*I)*x + 4*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 51, normalized size = 0.7 \[{\frac{3\,i+8\,x}{32} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{81\,i+216\,x}{1024}\sqrt{3\,ix+4\,{x}^{2}}}+{\frac{243}{4096}{\it Arcsinh} \left ({\frac{8\,x}{3}}+i \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3*I*x+4*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.793266, size = 103, normalized size = 1.49 \[ \frac{1}{4} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}} x + \frac{3}{32} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}} + \frac{27}{128} \, \sqrt{4 \, x^{2} + 3 i \, x} x + \frac{81}{1024} i \, \sqrt{4 \, x^{2} + 3 i \, x} + \frac{243}{4096} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 3 i \, x} + 3 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 + 3*I*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219453, size = 278, normalized size = 4.03 \[ -\frac{2147483648 \, x^{8} + 6442450944 i \, x^{7} - 7247757312 \, x^{6} - 3623878656 i \, x^{5} + 623738880 \, x^{4} - 83607552 i \, x^{3} + 33219072 \, x^{2} +{\left (63700992 \, x^{4} + 95551488 i \, x^{3} - 44789760 \, x^{2} -{\left (31850496 \, x^{3} + 35831808 i \, x^{2} - 11197440 \, x - 839808 i\right )} \sqrt{4 \, x^{2} + 3 i \, x} - 6718464 i \, x + 157464\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 3 i \, x} - \frac{3}{4} i\right ) -{\left (1073741824 \, x^{7} + 2818572288 i \, x^{6} - 2642411520 \, x^{5} - 990904320 i \, x^{4} + 65028096 \, x^{3} - 38320128 i \, x^{2} + 5505408 \, x - 34992 i\right )} \sqrt{4 \, x^{2} + 3 i \, x} + 1399680 i \, x + 45927}{1073741824 \, x^{4} + 1610612736 i \, x^{3} - 754974720 \, x^{2} -{\left (536870912 \, x^{3} + 603979776 i \, x^{2} - 188743680 \, x - 14155776 i\right )} \sqrt{4 \, x^{2} + 3 i \, x} - 113246208 i \, x + 2654208} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 + 3*I*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (4 x^{2} + 3 i x\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*I*x+4*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 + 3*I*x)^(3/2),x, algorithm="giac")
[Out]