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

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 = 0.0353751, 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 \[ \frac{4096 (8 x+3 i)}{10935 \sqrt{4 x^2+3 i x}}+\frac{128 (8 x+3 i)}{1215 \left (4 x^2+3 i x\right )^{3/2}}+\frac{2 (8 x+3 i)}{45 \left (4 x^2+3 i x\right )^{5/2}} \]

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])

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Rubi in Sympy [A]  time = 2.58007, size = 65, normalized size = 0.82 \[ \frac{128 \left (8 x + 3 i\right )}{1215 \left (4 x^{2} + 3 i x\right )^{\frac{3}{2}}} + \frac{2 \left (8 x + 3 i\right )}{45 \left (4 x^{2} + 3 i x\right )^{\frac{5}{2}}} + \frac{2048 \left (16 x + 6 i\right )}{10935 \sqrt{4 x^{2} + 3 i x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*I*x+4*x**2)**(7/2),x)

[Out]

128*(8*x + 3*I)/(1215*(4*x**2 + 3*I*x)**(3/2)) + 2*(8*x + 3*I)/(45*(4*x**2 + 3*I
*x)**(5/2)) + 2048*(16*x + 6*I)/(10935*sqrt(4*x**2 + 3*I*x))

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Mathematica [A]  time = 0.0322578, 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)/(10
935*(x*(3*I + 4*x))^(5/2))

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Maple [A]  time = 0.012, size = 62, normalized size = 0.8 \[{\frac{6\,i+16\,x}{45} \left ( 3\,ix+4\,{x}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{384\,i+1024\,x}{1215} \left ( 3\,ix+4\,{x}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{12288\,i+32768\,x}{10935}{\frac{1}{\sqrt{3\,ix+4\,{x}^{2}}}}} \]

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)+4096/1
0935*(3*I+8*x)/(3*I*x+4*x^2)^(1/2)

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Maxima [A]  time = 0.69541, size = 111, normalized size = 1.41 \[ \frac{32768 \, x}{10935 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{4096 i}{3645 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{1024 \, x}{1215 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}} + \frac{128 i}{405 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}} + \frac{16 \, x}{45 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}}} + \frac{2 i}{15 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 3*I*x)^(-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 = 0.225158, size = 197, normalized size = 2.49 \[ \frac{327680 \, x^{4} + 491520 i \, x^{3} - 224640 \, x^{2} -{\left (163840 \, x^{3} + 184320 i \, x^{2} - 54720 \, x - 3240 i\right )} \sqrt{4 \, x^{2} + 3 i \, x} - 30240 i \, x + 486}{251658240 \, x^{10} + 943718400 i \, x^{9} - 1459814400 \, x^{8} - 1194393600 i \, x^{7} + 548985600 \, x^{6} + 137868480 i \, x^{5} - 16621200 \, x^{4} - 656100 i \, x^{3} -{\left (125829120 \, x^{9} + 424673280 i \, x^{8} - 579502080 \, x^{7} - 406425600 i \, x^{6} + 153187200 \, x^{5} + 29218320 i \, x^{4} - 2274480 \, x^{3} - 32805 i \, x^{2}\right )} \sqrt{4 \, x^{2} + 3 i \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(327680*x^4 + 491520*I*x^3 - 224640*x^2 - (163840*x^3 + 184320*I*x^2 - 54720*x -
3240*I)*sqrt(4*x^2 + 3*I*x) - 30240*I*x + 486)/(251658240*x^10 + 943718400*I*x^
9 - 1459814400*x^8 - 1194393600*I*x^7 + 548985600*x^6 + 137868480*I*x^5 - 166212
00*x^4 - 656100*I*x^3 - (125829120*x^9 + 424673280*I*x^8 - 579502080*x^7 - 40642
5600*I*x^6 + 153187200*x^5 + 29218320*I*x^4 - 2274480*x^3 - 32805*I*x^2)*sqrt(4*
x^2 + 3*I*x))

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

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError