3.1.20 \(\int \frac {1}{(3 i x+4 x^2)^{7/2}} \, dx\) [20]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 = {628, 627} \begin {gather*} \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}} \end {gather*}

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 627

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]

Rule 628

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]

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*}

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Mathematica [A]
time = 0.09, size = 48, normalized size = 0.61 \begin {gather*} \frac {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}} \end {gather*}

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 = 0.42, size = 62, normalized size = 0.78

method	result	size
risch	\(\frac {\frac {524288}{10935} x^{5}+\frac {65536}{729} i x^{4}-\frac {4096}{81} x^{3}-\frac {512}{81} i x^{2}-\frac {16}{27} x +\frac {2}{15} i}{x^{2} \left (3 i+4 x \right )^{2} \sqrt {x \left (3 i+4 x \right )}}\)	\(52\)
default	\(\frac {\frac {2 i}{15}+\frac {16 x}{45}}{\left (4 x^{2}+3 i x \right )^{\frac {5}{2}}}+\frac {\frac {128 i}{405}+\frac {1024 x}{1215}}{\left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}+\frac {\frac {4096 i}{3645}+\frac {32768 x}{10935}}{\sqrt {4 x^{2}+3 i x}}\)	\(62\)
trager	\(\frac {\left (\frac {1054}{4271484375}+\frac {224 i}{1423828125}\right ) \left (12288000000 i x^{5}[phone] x^{5}[phone] i x^{4}-23040000000 x^{4}-12960000000 i x^{3}-17280000000 x^{3}-2160000000 i x^{2}[phone] x^{2}-151875000 i x -202500000 x [phone] i-34171875\right ) \sqrt {4 x^{2}+3 i x}}{\left (12 i x -16 x -12 i-9\right )^{3} x^{3}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 82, normalized size = 1.04 \begin {gather*} \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 {gather*}

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 = 1.65, size = 83, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (524288 \, x^{6} + 1179648 i \, x^{5} - 884736 \, x^{4} - 221184 i \, x^{3} + {\left (262144 \, x^{5} + 491520 i \, x^{4} - 276480 \, x^{3} - 34560 i \, x^{2} - 3240 \, x + 729 i\right )} \sqrt {4 \, x^{2} + 3 i \, x}\right )}}{10935 \, {\left (64 \, x^{6} + 144 i \, x^{5} - 108 \, x^{4} - 27 i \, x^{3}\right )}} \end {gather*}

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]

2/10935*(524288*x^6 + 1179648*I*x^5 - 884736*x^4 - 221184*I*x^3 + (262144*x^5 + 491520*I*x^4 - 276480*x^3 - 34
560*I*x^2 - 3240*x + 729*I)*sqrt(4*x^2 + 3*I*x))/(64*x^6 + 144*I*x^5 - 108*x^4 - 27*I*x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (4 x^{2} + 3 i x\right )^{\frac {7}{2}}}\, dx \end {gather*}

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 [A]
time = 1.78, size = 84, normalized size = 1.06 \begin {gather*} \frac {{\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x + 15 i\right )} x - 135\right )} x - 135 i\right )} x - 405\right )} x + 729 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 )}}{10935 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{3}} \end {gather*}

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]

1/10935*(8*(32*(8*(16*(8*x + 15*I)*x - 135)*x - 135*I)*x - 405)*x + 729*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)/(4*x^2 + 3*I*x)^3

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Mupad [B]
time = 0.29, size = 40, normalized size = 0.51 \begin {gather*} -\frac {-524288\,x^5-x^4\,983040{}\mathrm {i}+552960\,x^3+x^2\,69120{}\mathrm {i}+6480\,x-1458{}\mathrm {i}}{10935\,{\left (x\,\left (4\,x+3{}\mathrm {i}\right )\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*3i + 4*x^2)^(7/2),x)

[Out]

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

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