3.3 \(\int (3 i x+4 x^2)^{5/2} \, dx\)

Optimal. Leaf size=95 \[ \frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}+\frac {15 (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}}{1024}+\frac {405 (8 x+3 i) \sqrt {4 x^2+3 i x}}{32768}+\frac {3645 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{131072} \]

[Out]

15/1024*(3*I+8*x)*(3*I*x+4*x^2)^(3/2)+1/48*(3*I+8*x)*(3*I*x+4*x^2)^(5/2)-3645/131072*I*arcsin(-1+8/3*I*x)+405/
32768*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {612, 619, 215} \[ \frac {1}{48} (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}+\frac {15 (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}}{1024}+\frac {405 (8 x+3 i) \sqrt {4 x^2+3 i x}}{32768}+\frac {3645 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{131072} \]

Antiderivative was successfully verified.

[In]

Int[((3*I)*x + 4*x^2)^(5/2),x]

[Out]

(405*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/32768 + (15*(3*I + 8*x)*((3*I)*x + 4*x^2)^(3/2))/1024 + ((3*I + 8*x)*(
(3*I)*x + 4*x^2)^(5/2))/48 + ((3645*I)/131072)*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 )^{5/2} \, dx &=\frac {1}{48} (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}+\frac {15}{32} \int \left (3 i x+4 x^2\right )^{3/2} \, dx\\ &=\frac {15 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{1024}+\frac {1}{48} (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}+\frac {405 \int \sqrt {3 i x+4 x^2} \, dx}{2048}\\ &=\frac {405 (3 i+8 x) \sqrt {3 i x+4 x^2}}{32768}+\frac {15 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{1024}+\frac {1}{48} (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}+\frac {3645 \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx}{65536}\\ &=\frac {405 (3 i+8 x) \sqrt {3 i x+4 x^2}}{32768}+\frac {15 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{1024}+\frac {1}{48} (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}+\frac {1215 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )}{131072}\\ &=\frac {405 (3 i+8 x) \sqrt {3 i x+4 x^2}}{32768}+\frac {15 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{1024}+\frac {1}{48} (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}+\frac {3645 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{131072}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 88, normalized size = 0.93 \[ \frac {\sqrt {x (4 x+3 i)} \left (524288 x^5+983040 i x^4-497664 x^3-6912 i x^2-6480 x-\frac {10935 \sqrt [4]{-1} \sin ^{-1}\left ((1+i) \sqrt {\frac {2}{3}} \sqrt {x}\right )}{\sqrt {3-4 i x} \sqrt {x}}+7290 i\right )}{196608} \]

Antiderivative was successfully verified.

[In]

Integrate[((3*I)*x + 4*x^2)^(5/2),x]

[Out]

(Sqrt[x*(3*I + 4*x)]*(7290*I - 6480*x - (6912*I)*x^2 - 497664*x^3 + (983040*I)*x^4 + 524288*x^5 - (10935*(-1)^
(1/4)*ArcSin[(1 + I)*Sqrt[2/3]*Sqrt[x]])/(Sqrt[3 - (4*I)*x]*Sqrt[x])))/196608

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fricas [A]  time = 0.97, size = 59, normalized size = 0.62 \[ \frac {1}{3145728} \, {\left (8388608 \, x^{5} + 15728640 i \, x^{4} - 7962624 \, x^{3} - 110592 i \, x^{2} - 103680 \, x + 116640 i\right )} \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3645}{131072} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) - \frac {8991}{1048576} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(5/2),x, algorithm="fricas")

[Out]

1/3145728*(8388608*x^5 + 15728640*I*x^4 - 7962624*x^3 - 110592*I*x^2 - 103680*x + 116640*I)*sqrt(4*x^2 + 3*I*x
) - 3645/131072*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - 8991/1048576

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giac [A]  time = 0.46, 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)^(5/2),x, algorithm="giac")

[Out]

0

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maple [A]  time = 0.09, size = 71, normalized size = 0.75 \[ \frac {3645 \arcsinh \left (\frac {8 x}{3}+i\right )}{131072}+\frac {\left (8 x +3 i\right ) \left (4 x^{2}+3 i x \right )^{\frac {5}{2}}}{48}+\frac {15 \left (8 x +3 i\right ) \left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}{1024}+\frac {405 \left (8 x +3 i\right ) \sqrt {4 x^{2}+3 i x}}{32768} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+3*I*x)^(5/2),x)

[Out]

1/48*(8*x+3*I)*(4*x^2+3*I*x)^(5/2)+15/1024*(8*x+3*I)*(4*x^2+3*I*x)^(3/2)+405/32768*(8*x+3*I)*(4*x^2+3*I*x)^(1/
2)+3645/131072*arcsinh(8/3*x+I)

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maxima [A]  time = 3.01, size = 103, normalized size = 1.08 \[ \frac {1}{6} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {5}{2}} x + \frac {1}{16} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {5}{2}} + \frac {15}{128} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} x + \frac {45}{1024} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} + \frac {405}{4096} \, \sqrt {4 \, x^{2} + 3 i \, x} x + \frac {1215}{32768} i \, \sqrt {4 \, x^{2} + 3 i \, x} + \frac {3645}{131072} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(4*x^2 + 3*I*x)^(5/2)*x + 1/16*I*(4*x^2 + 3*I*x)^(5/2) + 15/128*(4*x^2 + 3*I*x)^(3/2)*x + 45/1024*I*(4*x^2
+ 3*I*x)^(3/2) + 405/4096*sqrt(4*x^2 + 3*I*x)*x + 1215/32768*I*sqrt(4*x^2 + 3*I*x) + 3645/131072*log(8*x + 4*
sqrt(4*x^2 + 3*I*x) + 3*I)

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mupad [B]  time = 0.30, size = 80, normalized size = 0.84 \[ \frac {3645\,\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{131072}+\frac {15\,\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{3/2}}{512}+\frac {\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{5/2}}{24}+\frac {405\,\left (\frac {x}{2}+\frac {3}{16}{}\mathrm {i}\right )\,\sqrt {4\,x^2+x\,3{}\mathrm {i}}}{2048} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*3i + 4*x^2)^(5/2),x)

[Out]

(3645*log(x + (x*(4*x + 3i))^(1/2)/2 + 3i/8))/131072 + (15*(4*x + 3i/2)*(x*3i + 4*x^2)^(3/2))/512 + ((4*x + 3i
/2)*(x*3i + 4*x^2)^(5/2))/24 + (405*(x/2 + 3i/16)*(x*3i + 4*x^2)^(1/2))/2048

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (4 x^{2} + 3 i x\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x**2)**(5/2),x)

[Out]

Integral((4*x**2 + 3*I*x)**(5/2), x)

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