3.5 \(\int \sqrt {3 i x+4 x^2} \, dx\)

Optimal. Leaf size=43 \[ \frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)+\frac {9}{64} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right ) \]

[Out]

-9/64*I*arcsin(-1+8/3*I*x)+1/16*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {612, 619, 215} \[ \frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)+\frac {9}{64} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(3*I)*x + 4*x^2],x]

[Out]

((3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/16 + ((9*I)/64)*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 \sqrt {3 i x+4 x^2} \, dx &=\frac {1}{16} (3 i+8 x) \sqrt {3 i x+4 x^2}+\frac {9}{32} \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx\\ &=\frac {1}{16} (3 i+8 x) \sqrt {3 i x+4 x^2}+\frac {3}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )\\ &=\frac {1}{16} (3 i+8 x) \sqrt {3 i x+4 x^2}+\frac {9}{64} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 64, normalized size = 1.49 \[ \frac {1}{32} \sqrt {x (4 x+3 i)} \left (16 x-\frac {9 \sqrt [4]{-1} \sin ^{-1}\left ((1+i) \sqrt {\frac {2}{3}} \sqrt {x}\right )}{\sqrt {3-4 i x} \sqrt {x}}+6 i\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(3*I)*x + 4*x^2],x]

[Out]

(Sqrt[x*(3*I + 4*x)]*(6*I + 16*x - (9*(-1)^(1/4)*ArcSin[(1 + I)*Sqrt[2/3]*Sqrt[x]])/(Sqrt[3 - (4*I)*x]*Sqrt[x]
)))/32

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fricas [A]  time = 0.80, size = 39, normalized size = 0.91 \[ \frac {1}{256} \, \sqrt {4 \, x^{2} + 3 i \, x} {\left (128 \, x + 48 i\right )} - \frac {9}{64} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) - \frac {9}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/256*sqrt(4*x^2 + 3*I*x)*(128*x + 48*I) - 9/64*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - 9/256

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giac [A]  time = 0.63, size = 1, normalized size = 0.02 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(1/2),x, algorithm="giac")

[Out]

0

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maple [A]  time = 0.10, size = 31, normalized size = 0.72 \[ \frac {9 \arcsinh \left (\frac {8 x}{3}+i\right )}{64}+\frac {\left (8 x +3 i\right ) \sqrt {4 x^{2}+3 i x}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+3*I*x)^(1/2),x)

[Out]

1/16*(8*x+3*I)*(4*x^2+3*I*x)^(1/2)+9/64*arcsinh(8/3*x+I)

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maxima [A]  time = 2.96, size = 49, normalized size = 1.14 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} + 3 i \, x} x + \frac {3}{16} i \, \sqrt {4 \, x^{2} + 3 i \, x} + \frac {9}{64} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(4*x^2 + 3*I*x)*x + 3/16*I*sqrt(4*x^2 + 3*I*x) + 9/64*log(8*x + 4*sqrt(4*x^2 + 3*I*x) + 3*I)

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mupad [B]  time = 0.09, size = 39, normalized size = 0.91 \[ \frac {9\,\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{64}+\left (\frac {x}{2}+\frac {3}{16}{}\mathrm {i}\right )\,\sqrt {4\,x^2+x\,3{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*3i + 4*x^2)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 x^{2} + 3 i x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x**2)**(1/2),x)

[Out]

Integral(sqrt(4*x**2 + 3*I*x), x)

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