3.4 \(\int \left (3 i x+4 x^2\right )^{3/2} \, dx\)

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]

(27*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/1024 + ((3*I + 8*x)*((3*I)*x + 4*x^2)^(3/
2))/32 + ((243*I)/4096)*ArcSin[1 - ((8*I)/3)*x]

_______________________________________________________________________________________

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]

(27*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/1024 + ((3*I + 8*x)*((3*I)*x + 4*x^2)^(3/
2))/32 + ((243*I)/4096)*ArcSin[1 - ((8*I)/3)*x]

_______________________________________________________________________________________

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]

(8*x + 3*I)*(4*x**2 + 3*I*x)**(3/2)/32 + 27*(8*x + 3*I)*sqrt(4*x**2 + 3*I*x)/102
4 + 243*asinh(8*x/3 + I)/4096

_______________________________________________________________________________________

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]

(2*x*(-243 + (108*I)*x - 3744*x^2 + (7680*I)*x^3 + 4096*x^4) + 243*Sqrt[x]*Sqrt[
3*I + 4*x]*Log[2*Sqrt[x] + Sqrt[3*I + 4*x]])/(2048*Sqrt[x*(3*I + 4*x)])

_______________________________________________________________________________________

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]

1/32*(3*I+8*x)*(3*I*x+4*x^2)^(3/2)+27/1024*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)+243/409
6*arcsinh(8/3*x+I)

_______________________________________________________________________________________

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]

1/4*(4*x^2 + 3*I*x)^(3/2)*x + 3/32*I*(4*x^2 + 3*I*x)^(3/2) + 27/128*sqrt(4*x^2 +
3*I*x)*x + 81/1024*I*sqrt(4*x^2 + 3*I*x) + 243/4096*log(8*x + 4*sqrt(4*x^2 + 3*
I*x) + 3*I)

_______________________________________________________________________________________

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]

-(2147483648*x^8 + 6442450944*I*x^7 - 7247757312*x^6 - 3623878656*I*x^5 + 623738
880*x^4 - 83607552*I*x^3 + 33219072*x^2 + (63700992*x^4 + 95551488*I*x^3 - 44789
760*x^2 - (31850496*x^3 + 35831808*I*x^2 - 11197440*x - 839808*I)*sqrt(4*x^2 + 3
*I*x) - 6718464*I*x + 157464)*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - (1073741
824*x^7 + 2818572288*I*x^6 - 2642411520*x^5 - 990904320*I*x^4 + 65028096*x^3 - 3
8320128*I*x^2 + 5505408*x - 34992*I)*sqrt(4*x^2 + 3*I*x) + 1399680*I*x + 45927)/
(1073741824*x^4 + 1610612736*I*x^3 - 754974720*x^2 - (536870912*x^3 + 603979776*
I*x^2 - 188743680*x - 14155776*I)*sqrt(4*x^2 + 3*I*x) - 113246208*I*x + 2654208)

_______________________________________________________________________________________

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]

Integral((4*x**2 + 3*I*x)**(3/2), x)

_______________________________________________________________________________________

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]

integrate((4*x^2 + 3*I*x)^(3/2), x)