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

Optimal. Leaf size=69 \[ \frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {243 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{4096} \]

[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 = {626, 633, 221} \begin {gather*} \frac {243 i \text {ArcSin}\left (1-\frac {8 i x}{3}\right )}{4096}+\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} \end {gather*}

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 221

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 626

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 633

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 \text {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 = 0.08, size = 83, normalized size = 1.20 \begin {gather*} \frac {2 x \left (-243+108 i x-3744 x^2+7680 i x^3+4096 x^4\right )-243 \sqrt {x} \sqrt {3 i+4 x} \log \left (-2 \sqrt {x}+\sqrt {3 i+4 x}\right )}{2048 \sqrt {x (3 i+4 x)}} \end {gather*}

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] + Sq
rt[3*I + 4*x]])/(2048*Sqrt[x*(3*I + 4*x)])

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Maple [A]
time = 0.40, size = 51, normalized size = 0.74

method	result	size
risch	\(\frac {\left (1024 x^{3}+1152 i x^{2}-72 x +81 i\right ) x \left (3 i+4 x \right )}{1024 \sqrt {x \left (3 i+4 x \right )}}+\frac {243 \arcsinh \left (i+\frac {8 x}{3}\right )}{4096}\)	\(47\)
default	\(\frac {\left (3 i+8 x \right ) \left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}{32}+\frac {27 \left (3 i+8 x \right ) \sqrt {4 x^{2}+3 i x}}{1024}+\frac {243 \arcsinh \left (i+\frac {8 x}{3}\right )}{4096}\)	\(51\)
trager	\(\left (\frac {9}{8} i x^{2}+x^{3}+\frac {81}{1024} i-\frac {9}{128} x \right ) \sqrt {4 x^{2}+3 i x}+\frac {243 \ln \left (440 x +144+165 i-192 i \sqrt {4 x^{2}+3 i x}-384 i x +220 \sqrt {4 x^{2}+3 i x}\right )}{4096}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)+243/4096*arcsinh(I+8/3*x)

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Maxima [A]
time = 0.54, size = 76, normalized size = 1.10 \begin {gather*} \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 ) \end {gather*}

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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Fricas [A]
time = 1.57, size = 49, normalized size = 0.71 \begin {gather*} \frac {1}{1024} \, {\left (1024 \, x^{3} + 1152 i \, x^{2} - 72 \, x + 81 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(1024*x^3 + 1152*I*x^2 - 72*x + 81*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (4 x^{2} + 3 i x\right )^{\frac {3}{2}}\, dx \end {gather*}

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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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (45) = 90\).
time = 1.94, size = 120, normalized size = 1.74 \begin {gather*} \frac {1}{2048} \, {\left (8 \, {\left (16 \, {\left (8 \, x + 9 i\right )} x - 9\right )} x + 81 i\right )} \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - \frac {243}{4096} \, \log \left (2 \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - 8 \, x - 3 i\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2048*(8*(16*(8*x + 9*I)*x - 9)*x + 81*I)*sqrt(8*x^2 + 2*sqrt(16*x^2 + 9)*x)*(3*I*x/(4*x^2 + sqrt(16*x^4 + 9*
x^2)) + 1) - 243/4096*log(2*sqrt(8*x^2 + 2*sqrt(16*x^2 + 9)*x)*(3*I*x/(4*x^2 + sqrt(16*x^4 + 9*x^2)) + 1) - 8*
x - 3*I)

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Mupad [B]
time = 0.16, size = 60, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

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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