3.20 \(\int \frac{1}{(3 i x+4 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{4096 (8 x+3 i)}{10935 \sqrt{4 x^2+3 i x}}+\frac{128 (8 x+3 i)}{1215 \left (4 x^2+3 i x\right )^{3/2}}+\frac{2 (8 x+3 i)}{45 \left (4 x^2+3 i x\right )^{5/2}} \]

[Out]

(2*(3*I + 8*x))/(45*((3*I)*x + 4*x^2)^(5/2)) + (128*(3*I + 8*x))/(1215*((3*I)*x + 4*x^2)^(3/2)) + (4096*(3*I +
8*x))/(10935*Sqrt[(3*I)*x + 4*x^2])

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Rubi [A]  time = 0.0124143, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {614, 613} \[ \frac{4096 (8 x+3 i)}{10935 \sqrt{4 x^2+3 i x}}+\frac{128 (8 x+3 i)}{1215 \left (4 x^2+3 i x\right )^{3/2}}+\frac{2 (8 x+3 i)}{45 \left (4 x^2+3 i x\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((3*I)*x + 4*x^2)^(-7/2),x]

[Out]

(2*(3*I + 8*x))/(45*((3*I)*x + 4*x^2)^(5/2)) + (128*(3*I + 8*x))/(1215*((3*I)*x + 4*x^2)^(3/2)) + (4096*(3*I +
8*x))/(10935*Sqrt[(3*I)*x + 4*x^2])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx &=\frac{2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac{64}{45} \int \frac{1}{\left (3 i x+4 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac{128 (3 i+8 x)}{1215 \left (3 i x+4 x^2\right )^{3/2}}+\frac{2048 \int \frac{1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx}{1215}\\ &=\frac{2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac{128 (3 i+8 x)}{1215 \left (3 i x+4 x^2\right )^{3/2}}+\frac{4096 (3 i+8 x)}{10935 \sqrt{3 i x+4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0148155, size = 48, normalized size = 0.61 \[ \frac{524288 x^5+983040 i x^4-552960 x^3-69120 i x^2-6480 x+1458 i}{10935 (x (4 x+3 i))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((3*I)*x + 4*x^2)^(-7/2),x]

[Out]

(1458*I - 6480*x - (69120*I)*x^2 - 552960*x^3 + (983040*I)*x^4 + 524288*x^5)/(10935*(x*(3*I + 4*x))^(5/2))

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Maple [A]  time = 0.108, size = 62, normalized size = 0.8 \begin{align*}{\frac{6\,i+16\,x}{45} \left ( 3\,ix+4\,{x}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{384\,i+1024\,x}{1215} \left ( 3\,ix+4\,{x}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{12288\,i+32768\,x}{10935}{\frac{1}{\sqrt{3\,ix+4\,{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*I*x+4*x^2)^(7/2),x)

[Out]

2/45*(3*I+8*x)/(3*I*x+4*x^2)^(5/2)+128/1215*(3*I+8*x)/(3*I*x+4*x^2)^(3/2)+4096/10935*(3*I+8*x)/(3*I*x+4*x^2)^(
1/2)

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Maxima [A]  time = 1.37852, size = 111, normalized size = 1.41 \begin{align*} \frac{32768 \, x}{10935 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{4096 i}{3645 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{1024 \, x}{1215 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}} + \frac{128 i}{405 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}} + \frac{16 \, x}{45 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}}} + \frac{2 i}{15 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(7/2),x, algorithm="maxima")

[Out]

32768/10935*x/sqrt(4*x^2 + 3*I*x) + 4096/3645*I/sqrt(4*x^2 + 3*I*x) + 1024/1215*x/(4*x^2 + 3*I*x)^(3/2) + 128/
405*I/(4*x^2 + 3*I*x)^(3/2) + 16/45*x/(4*x^2 + 3*I*x)^(5/2) + 2/15*I/(4*x^2 + 3*I*x)^(5/2)

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Fricas [A]  time = 2.41308, size = 288, normalized size = 3.65 \begin{align*} \frac{1048576 \, x^{6} + 2359296 i \, x^{5} - 1769472 \, x^{4} - 442368 i \, x^{3} +{\left (524288 \, x^{5} + 983040 i \, x^{4} - 552960 \, x^{3} - 69120 i \, x^{2} - 6480 \, x + 1458 i\right )} \sqrt{4 \, x^{2} + 3 i \, x}}{699840 \, x^{6} + 1574640 i \, x^{5} - 1180980 \, x^{4} - 295245 i \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(7/2),x, algorithm="fricas")

[Out]

(1048576*x^6 + 2359296*I*x^5 - 1769472*x^4 - 442368*I*x^3 + (524288*x^5 + 983040*I*x^4 - 552960*x^3 - 69120*I*
x^2 - 6480*x + 1458*I)*sqrt(4*x^2 + 3*I*x))/(699840*x^6 + 1574640*I*x^5 - 1180980*x^4 - 295245*I*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError