3.2 \(\int (3 i x+4 x^2)^{7/2} \, dx\)

Optimal. Leaf size=121 \[ \frac{1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}+\frac{21 (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}}{2048}+\frac{945 (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}}{131072}+\frac{25515 (8 x+3 i) \sqrt{4 x^2+3 i x}}{4194304}+\frac{229635 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{16777216} \]

[Out]

(25515*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/4194304 + (945*(3*I + 8*x)*((3*I)*x + 4*x^2)^(3/2))/131072 + (21*(3*
I + 8*x)*((3*I)*x + 4*x^2)^(5/2))/2048 + ((3*I + 8*x)*((3*I)*x + 4*x^2)^(7/2))/64 + ((229635*I)/16777216)*ArcS
in[1 - ((8*I)/3)*x]

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Rubi [A]  time = 0.0308269, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {612, 619, 215} \[ \frac{1}{64} (8 x+3 i) \left (4 x^2+3 i x\right )^{7/2}+\frac{21 (8 x+3 i) \left (4 x^2+3 i x\right )^{5/2}}{2048}+\frac{945 (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}}{131072}+\frac{25515 (8 x+3 i) \sqrt{4 x^2+3 i x}}{4194304}+\frac{229635 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{16777216} \]

Antiderivative was successfully verified.

[In]

Int[((3*I)*x + 4*x^2)^(7/2),x]

[Out]

(25515*(3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/4194304 + (945*(3*I + 8*x)*((3*I)*x + 4*x^2)^(3/2))/131072 + (21*(3*
I + 8*x)*((3*I)*x + 4*x^2)^(5/2))/2048 + ((3*I + 8*x)*((3*I)*x + 4*x^2)^(7/2))/64 + ((229635*I)/16777216)*ArcS
in[1 - ((8*I)/3)*x]

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]

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]

Rubi steps

\begin{align*} \int \left (3 i x+4 x^2\right )^{7/2} \, dx &=\frac{1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac{63}{128} \int \left (3 i x+4 x^2\right )^{5/2} \, dx\\ &=\frac{21 (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}}{2048}+\frac{1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac{945 \int \left (3 i x+4 x^2\right )^{3/2} \, dx}{4096}\\ &=\frac{945 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{131072}+\frac{21 (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}}{2048}+\frac{1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac{25515 \int \sqrt{3 i x+4 x^2} \, dx}{262144}\\ &=\frac{25515 (3 i+8 x) \sqrt{3 i x+4 x^2}}{4194304}+\frac{945 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{131072}+\frac{21 (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}}{2048}+\frac{1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac{229635 \int \frac{1}{\sqrt{3 i x+4 x^2}} \, dx}{8388608}\\ &=\frac{25515 (3 i+8 x) \sqrt{3 i x+4 x^2}}{4194304}+\frac{945 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{131072}+\frac{21 (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}}{2048}+\frac{1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac{76545 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{9}}} \, dx,x,3 i+8 x\right )}{16777216}\\ &=\frac{25515 (3 i+8 x) \sqrt{3 i x+4 x^2}}{4194304}+\frac{945 (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}}{131072}+\frac{21 (3 i+8 x) \left (3 i x+4 x^2\right )^{5/2}}{2048}+\frac{1}{64} (3 i+8 x) \left (3 i x+4 x^2\right )^{7/2}+\frac{229635 i \sin ^{-1}\left (1-\frac{8 i x}{3}\right )}{16777216}\\ \end{align*}

Mathematica [A]  time = 0.0894461, size = 119, normalized size = 0.98 \[ \frac{\sqrt{x (4 x+3 i)} \left (2 \sqrt{3-4 i x} \sqrt{x} \left (33554432 x^7+88080384 i x^6-79429632 x^5-25067520 i x^4+82944 x^3-72576 i x^2-68040 x+76545 i\right )-229635 \sqrt [4]{-1} \sin ^{-1}\left ((1+i) \sqrt{\frac{2}{3}} \sqrt{x}\right )\right )}{8388608 \sqrt{3-4 i x} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((3*I)*x + 4*x^2)^(7/2),x]

[Out]

(Sqrt[x*(3*I + 4*x)]*(2*Sqrt[3 - (4*I)*x]*Sqrt[x]*(76545*I - 68040*x - (72576*I)*x^2 + 82944*x^3 - (25067520*I
)*x^4 - 79429632*x^5 + (88080384*I)*x^6 + 33554432*x^7) - 229635*(-1)^(1/4)*ArcSin[(1 + I)*Sqrt[2/3]*Sqrt[x]])
)/(8388608*Sqrt[3 - (4*I)*x]*Sqrt[x])

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Maple [A]  time = 0.104, size = 91, normalized size = 0.8 \begin{align*}{\frac{3\,i+8\,x}{64} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{63\,i+168\,x}{2048} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{2835\,i+7560\,x}{131072} \left ( 3\,ix+4\,{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{76545\,i+204120\,x}{4194304}\sqrt{3\,ix+4\,{x}^{2}}}+{\frac{229635}{16777216}{\it Arcsinh} \left ({\frac{8\,x}{3}}+i \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*I*x+4*x^2)^(7/2),x)

[Out]

1/64*(3*I+8*x)*(3*I*x+4*x^2)^(7/2)+21/2048*(3*I+8*x)*(3*I*x+4*x^2)^(5/2)+945/131072*(3*I+8*x)*(3*I*x+4*x^2)^(3
/2)+25515/4194304*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)+229635/16777216*arcsinh(8/3*x+I)

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Maxima [A]  time = 1.89731, size = 176, normalized size = 1.45 \begin{align*} \frac{1}{8} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{7}{2}} x + \frac{3}{64} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{7}{2}} + \frac{21}{256} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}} x + \frac{63}{2048} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}} + \frac{945}{16384} \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}} x + \frac{2835}{131072} i \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}} + \frac{25515}{524288} \, \sqrt{4 \, x^{2} + 3 i \, x} x + \frac{76545}{4194304} i \, \sqrt{4 \, x^{2} + 3 i \, x} + \frac{229635}{16777216} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 3 i \, x} + 3 i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(7/2),x, algorithm="maxima")

[Out]

1/8*(4*x^2 + 3*I*x)^(7/2)*x + 3/64*I*(4*x^2 + 3*I*x)^(7/2) + 21/256*(4*x^2 + 3*I*x)^(5/2)*x + 63/2048*I*(4*x^2
+ 3*I*x)^(5/2) + 945/16384*(4*x^2 + 3*I*x)^(3/2)*x + 2835/131072*I*(4*x^2 + 3*I*x)^(3/2) + 25515/524288*sqrt(
4*x^2 + 3*I*x)*x + 76545/4194304*I*sqrt(4*x^2 + 3*I*x) + 229635/16777216*log(8*x + 4*sqrt(4*x^2 + 3*I*x) + 3*I
)

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Fricas [A]  time = 2.28309, size = 320, normalized size = 2.64 \begin{align*} \frac{1}{268435456} \,{\left (2147483648 \, x^{7} + 5637144576 i \, x^{6} - 5083496448 \, x^{5} - 1604321280 i \, x^{4} + 5308416 \, x^{3} - 4644864 i \, x^{2} - 4354560 \, x + 4898880 i\right )} \sqrt{4 \, x^{2} + 3 i \, x} - \frac{229635}{16777216} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 3 i \, x} - \frac{3}{4} i\right ) - \frac{1165671}{268435456} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(7/2),x, algorithm="fricas")

[Out]

1/268435456*(2147483648*x^7 + 5637144576*I*x^6 - 5083496448*x^5 - 1604321280*I*x^4 + 5308416*x^3 - 4644864*I*x
^2 - 4354560*x + 4898880*I)*sqrt(4*x^2 + 3*I*x) - 229635/16777216*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - 11
65671/268435456

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (4 x^{2} + 3 i x\right )^{\frac{7}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x**2)**(7/2),x)

[Out]

Integral((4*x**2 + 3*I*x)**(7/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*I*x+4*x^2)^(7/2),x, algorithm="giac")

[Out]

integrate((4*x^2 + 3*I*x)^(7/2), x)