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

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]

1/32*(3*I+8*x)*(3*I*x+4*x^2)^(3/2)-243/4096*I*arcsin(-1+8/3*I*x)+27/1024*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {612, 619, 215} \[ \frac {1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}+\frac {27 (8 x+3 i) \sqrt {4 x^2+3 i x}}{1024}+\frac {243 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{4096} \]

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]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \left (3 i x+4 x^2\right )^{3/2} \, dx &=\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {27}{64} \int \sqrt {3 i x+4 x^2} \, dx\\ &=\frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {243 \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx}{2048}\\ &=\frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {81 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )}{4096}\\ &=\frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {243 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{4096}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 76, normalized size = 1.10 \[ \frac {\sqrt {x (4 x+3 i)} \left (2048 x^3+2304 i x^2-144 x-\frac {243 \sqrt [4]{-1} \sin ^{-1}\left ((1+i) \sqrt {\frac {2}{3}} \sqrt {x}\right )}{\sqrt {3-4 i x} \sqrt {x}}+162 i\right )}{2048} \]

Antiderivative was successfully verified.

[In]

Integrate[((3*I)*x + 4*x^2)^(3/2),x]

[Out]

(Sqrt[x*(3*I + 4*x)]*(162*I - 144*x + (2304*I)*x^2 + 2048*x^3 - (243*(-1)^(1/4)*ArcSin[(1 + I)*Sqrt[2/3]*Sqrt[
x]])/(Sqrt[3 - (4*I)*x]*Sqrt[x])))/2048

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fricas [A]  time = 1.17, size = 49, normalized size = 0.71 \[ \frac {1}{32768} \, {\left (32768 \, x^{3} + 36864 i \, x^{2} - 2304 \, x + 2592 i\right )} \sqrt {4 \, x^{2} + 3 i \, x} - \frac {243}{4096} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) - \frac {567}{32768} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(3/2),x, algorithm="fricas")

[Out]

1/32768*(32768*x^3 + 36864*I*x^2 - 2304*x + 2592*I)*sqrt(4*x^2 + 3*I*x) - 243/4096*log(-2*x + sqrt(4*x^2 + 3*I
*x) - 3/4*I) - 567/32768

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giac [A]  time = 0.61, size = 1, normalized size = 0.01 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(3/2),x, algorithm="giac")

[Out]

0

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maple [A]  time = 0.12, size = 51, normalized size = 0.74 \[ \frac {243 \arcsinh \left (\frac {8 x}{3}+i\right )}{4096}+\frac {\left (8 x +3 i\right ) \left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}{32}+\frac {27 \left (8 x +3 i\right ) \sqrt {4 x^{2}+3 i x}}{1024} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+3*I*x)^(3/2),x)

[Out]

1/32*(8*x+3*I)*(4*x^2+3*I*x)^(3/2)+27/1024*(8*x+3*I)*(4*x^2+3*I*x)^(1/2)+243/4096*arcsinh(8/3*x+I)

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maxima [A]  time = 2.98, size = 76, normalized size = 1.10 \[ \frac {1}{4} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} x + \frac {3}{32} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} + \frac {27}{128} \, \sqrt {4 \, x^{2} + 3 i \, x} x + \frac {81}{1024} i \, \sqrt {4 \, x^{2} + 3 i \, x} + \frac {243}{4096} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(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)

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mupad [B]  time = 0.16, size = 60, normalized size = 0.87 \[ \frac {243\,\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{4096}+\frac {\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{3/2}}{16}+\frac {27\,\left (\frac {x}{2}+\frac {3}{16}{}\mathrm {i}\right )\,\sqrt {4\,x^2+x\,3{}\mathrm {i}}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*3i + 4*x^2)^(3/2),x)

[Out]

(243*log(x + (x*(4*x + 3i))^(1/2)/2 + 3i/8))/4096 + ((4*x + 3i/2)*(x*3i + 4*x^2)^(3/2))/16 + (27*(x/2 + 3i/16)
*(x*3i + 4*x^2)^(1/2))/64

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (4 x^{2} + 3 i x\right )^{\frac {3}{2}}\, dx \]

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)

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