3.162 \(\int \frac{\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\left (67-32 x^2\right ) \sqrt{x^4+5 x^2+3}}{12 x^2}+\frac{49}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{527 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{24 \sqrt{3}}-\frac{\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{6 x^6} \]

[Out]

-((67 - 32*x^2)*Sqrt[3 + 5*x^2 + x^4])/(12*x^2) - ((2 + 7*x^2)*(3 + 5*x^2 + x^4)
^(3/2))/(6*x^6) + (49*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/4 - (527*A
rcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(24*Sqrt[3])

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Rubi [A]  time = 0.263227, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{\left (67-32 x^2\right ) \sqrt{x^4+5 x^2+3}}{12 x^2}+\frac{49}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{527 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{24 \sqrt{3}}-\frac{\left (7 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^7,x]

[Out]

-((67 - 32*x^2)*Sqrt[3 + 5*x^2 + x^4])/(12*x^2) - ((2 + 7*x^2)*(3 + 5*x^2 + x^4)
^(3/2))/(6*x^6) + (49*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/4 - (527*A
rcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(24*Sqrt[3])

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Rubi in Sympy [A]  time = 26.6391, size = 114, normalized size = 0.9 \[ \frac{49 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{4} - \frac{527 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{72} - \frac{\left (- 64 x^{2} + 134\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{24 x^{2}} - \frac{\left (42 x^{2} + 12\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{36 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)*(x**4+5*x**2+3)**(3/2)/x**7,x)

[Out]

49*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x**2 + 3)))/4 - 527*sqrt(3)*atanh(sqrt(3)
*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/72 - (-64*x**2 + 134)*sqrt(x**4 + 5*x
**2 + 3)/(24*x**2) - (42*x**2 + 12)*(x**4 + 5*x**2 + 3)**(3/2)/(36*x**6)

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Mathematica [A]  time = 0.192967, size = 112, normalized size = 0.88 \[ \frac{49}{4} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+\frac{527 \left (2 \log (x)-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )}{24 \sqrt{3}}+\sqrt{x^4+5 x^2+3} \left (-\frac{1}{x^6}-\frac{31}{6 x^4}-\frac{47}{4 x^2}+\frac{3}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^7,x]

[Out]

(3/2 - x^(-6) - 31/(6*x^4) - 47/(4*x^2))*Sqrt[3 + 5*x^2 + x^4] + (49*Log[5 + 2*x
^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/4 + (527*(2*Log[x] - Log[6 + 5*x^2 + 2*Sqrt[3]*Sq
rt[3 + 5*x^2 + x^4]]))/(24*Sqrt[3])

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Maple [A]  time = 0.026, size = 117, normalized size = 0.9 \[{\frac{49}{4}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-{\frac{1}{{x}^{6}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{31}{6\,{x}^{4}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{47}{4\,{x}^{2}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{527\,\sqrt{3}}{72}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }+{\frac{3}{2}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^7,x)

[Out]

49/4*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))-(x^4+5*x^2+3)^(1/2)/x^6-31/6*(x^4+5*x^2+3)^
(1/2)/x^4-47/4*(x^4+5*x^2+3)^(1/2)/x^2-527/72*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4
+5*x^2+3)^(1/2))*3^(1/2)+3/2*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.826634, size = 208, normalized size = 1.64 \[ \frac{67}{36} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{11}{54} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - \frac{527}{72} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{431}{36} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{79 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{108 \, x^{2}} - \frac{11 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}}}{54 \, x^{4}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}}}{9 \, x^{6}} + \frac{49}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 5*x^2 + 3)^(3/2)*(3*x^2 + 2)/x^7,x, algorithm="maxima")

[Out]

67/36*sqrt(x^4 + 5*x^2 + 3)*x^2 + 11/54*(x^4 + 5*x^2 + 3)^(3/2) - 527/72*sqrt(3)
*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 431/36*sqrt(x^4 + 5*x^2
+ 3) - 79/108*(x^4 + 5*x^2 + 3)^(3/2)/x^2 - 11/54*(x^4 + 5*x^2 + 3)^(5/2)/x^4 -
1/9*(x^4 + 5*x^2 + 3)^(5/2)/x^6 + 49/4*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.273047, size = 554, normalized size = 4.36 \[ -\frac{2 \, \sqrt{3}{\left (2304 \, x^{14} + 20160 \, x^{12} + 4256 \, x^{10} - 332728 \, x^{8} - 900266 \, x^{6} - 781877 \, x^{4} - 230318 \, x^{2} - 30828\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 294 \,{\left (8 \, \sqrt{3}{\left (16 \, x^{12} + 120 \, x^{10} + 274 \, x^{8} + 185 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (128 \, x^{14} + 1280 \, x^{12} + 4384 \, x^{10} + 5920 \, x^{8} + 2569 \, x^{6}\right )}\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 527 \,{\left (128 \, x^{14} + 1280 \, x^{12} + 4384 \, x^{10} + 5920 \, x^{8} + 2569 \, x^{6} - 8 \,{\left (16 \, x^{12} + 120 \, x^{10} + 274 \, x^{8} + 185 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - \sqrt{3}{\left (4608 \, x^{16} + 51840 \, x^{14} + 101824 \, x^{12} - 690976 \, x^{10} - 3367088 \, x^{8} - 5249243 \, x^{6} - 3352784 \, x^{4} - 885984 \, x^{2} - 106560\right )}}{24 \,{\left (8 \, \sqrt{3}{\left (16 \, x^{12} + 120 \, x^{10} + 274 \, x^{8} + 185 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (128 \, x^{14} + 1280 \, x^{12} + 4384 \, x^{10} + 5920 \, x^{8} + 2569 \, x^{6}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 5*x^2 + 3)^(3/2)*(3*x^2 + 2)/x^7,x, algorithm="fricas")

[Out]

-1/24*(2*sqrt(3)*(2304*x^14 + 20160*x^12 + 4256*x^10 - 332728*x^8 - 900266*x^6 -
781877*x^4 - 230318*x^2 - 30828)*sqrt(x^4 + 5*x^2 + 3) + 294*(8*sqrt(3)*(16*x^1
2 + 120*x^10 + 274*x^8 + 185*x^6)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(128*x^14 + 12
80*x^12 + 4384*x^10 + 5920*x^8 + 2569*x^6))*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3)
- 5) + 527*(128*x^14 + 1280*x^12 + 4384*x^10 + 5920*x^8 + 2569*x^6 - 8*(16*x^12
+ 120*x^10 + 274*x^8 + 185*x^6)*sqrt(x^4 + 5*x^2 + 3))*log((6*x^2 + sqrt(3)*(2*
x^4 + 5*x^2 + 6) - 2*sqrt(x^4 + 5*x^2 + 3)*(sqrt(3)*x^2 + 3))/(2*x^4 - 2*sqrt(x^
4 + 5*x^2 + 3)*x^2 + 5*x^2)) - sqrt(3)*(4608*x^16 + 51840*x^14 + 101824*x^12 - 6
90976*x^10 - 3367088*x^8 - 5249243*x^6 - 3352784*x^4 - 885984*x^2 - 106560))/(8*
sqrt(3)*(16*x^12 + 120*x^10 + 274*x^8 + 185*x^6)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)
*(128*x^14 + 1280*x^12 + 4384*x^10 + 5920*x^8 + 2569*x^6))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)*(x**4+5*x**2+3)**(3/2)/x**7,x)

[Out]

Integral((3*x**2 + 2)*(x**4 + 5*x**2 + 3)**(3/2)/x**7, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 5*x^2 + 3)^(3/2)*(3*x^2 + 2)/x^7,x, algorithm="giac")

[Out]

integrate((x^4 + 5*x^2 + 3)^(3/2)*(3*x^2 + 2)/x^7, x)