3.53 \(\int \frac{(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=209 \[ -\frac{14 e^2 \sqrt{d^2-e^2 x^2}}{3 d^8 x}+\frac{2 e^3 (45 d+53 e x)}{15 d^8 \sqrt{d^2-e^2 x^2}}-\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^8}-\frac{e \sqrt{d^2-e^2 x^2}}{d^7 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^6 x^3}+\frac{e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(2*e^3*(d + e*x))/(5*d^4*(d^2 - e^2*x^2)^(5/2)) + (e^3*(20*d + 23*e*x))/(15*d^6*
(d^2 - e^2*x^2)^(3/2)) + (2*e^3*(45*d + 53*e*x))/(15*d^8*Sqrt[d^2 - e^2*x^2]) -
Sqrt[d^2 - e^2*x^2]/(3*d^6*x^3) - (e*Sqrt[d^2 - e^2*x^2])/(d^7*x^2) - (14*e^2*Sq
rt[d^2 - e^2*x^2])/(3*d^8*x) - (7*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^8

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Rubi [A]  time = 0.722024, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{14 e^2 \sqrt{d^2-e^2 x^2}}{3 d^8 x}+\frac{2 e^3 (45 d+53 e x)}{15 d^8 \sqrt{d^2-e^2 x^2}}-\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^8}-\frac{e \sqrt{d^2-e^2 x^2}}{d^7 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^6 x^3}+\frac{e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(2*e^3*(d + e*x))/(5*d^4*(d^2 - e^2*x^2)^(5/2)) + (e^3*(20*d + 23*e*x))/(15*d^6*
(d^2 - e^2*x^2)^(3/2)) + (2*e^3*(45*d + 53*e*x))/(15*d^8*Sqrt[d^2 - e^2*x^2]) -
Sqrt[d^2 - e^2*x^2]/(3*d^6*x^3) - (e*Sqrt[d^2 - e^2*x^2])/(d^7*x^2) - (14*e^2*Sq
rt[d^2 - e^2*x^2])/(3*d^8*x) - (7*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^8

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Rubi in Sympy [A]  time = 84.2737, size = 226, normalized size = 1.08 \[ - \frac{1}{3 d^{4} x^{3} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{e^{3}}{5 d^{5} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 e}{d^{5} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{23 e^{3}}{15 d^{6} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} - \frac{13 e^{2}}{3 d^{6} x \sqrt{d^{2} - e^{2} x^{2}}} + \frac{4 e^{3}}{d^{7} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{3 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{7} x^{2}} + \frac{176 e^{4} x}{15 d^{8} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{7 e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-1/(3*d**4*x**3*sqrt(d**2 - e**2*x**2)) + e**3/(5*d**5*(d - e*x)**2*sqrt(d**2 -
e**2*x**2)) + 2*e/(d**5*x**2*sqrt(d**2 - e**2*x**2)) + 23*e**3/(15*d**6*(d - e*x
)*sqrt(d**2 - e**2*x**2)) - 13*e**2/(3*d**6*x*sqrt(d**2 - e**2*x**2)) + 4*e**3/(
d**7*sqrt(d**2 - e**2*x**2)) - 3*e*sqrt(d**2 - e**2*x**2)/(d**7*x**2) + 176*e**4
*x/(15*d**8*sqrt(d**2 - e**2*x**2)) - 7*e**3*atanh(sqrt(d**2 - e**2*x**2)/d)/d**
8

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Mathematica [A]  time = 0.126912, size = 138, normalized size = 0.66 \[ \frac{-105 e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (5 d^6+5 d^5 e x+40 d^4 e^2 x^2-246 d^3 e^3 x^3+122 d^2 e^4 x^4+247 d e^5 x^5-176 e^6 x^6\right )}{x^3 (e x-d)^3 (d+e x)}+105 e^3 \log (x)}{15 d^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(5*d^6 + 5*d^5*e*x + 40*d^4*e^2*x^2 - 246*d^3*e^3*x^3 + 12
2*d^2*e^4*x^4 + 247*d*e^5*x^5 - 176*e^6*x^6))/(x^3*(-d + e*x)^3*(d + e*x)) + 105
*e^3*Log[x] - 105*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^8)

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Maple [A]  time = 0.025, size = 249, normalized size = 1.2 \[ -{\frac{1}{3\,{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{11\,{e}^{2}}{3\,{d}^{2}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{22\,{e}^{4}x}{5\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{88\,{e}^{4}x}{15\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{176\,{e}^{4}x}{15\,{d}^{8}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{e}{d{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{3}}{5\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{3}}{3\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+7\,{\frac{{e}^{3}}{{d}^{7}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-7\,{\frac{{e}^{3}}{{d}^{7}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/3/x^3/(-e^2*x^2+d^2)^(5/2)-11/3/d^2*e^2/x/(-e^2*x^2+d^2)^(5/2)+22/5/d^4*e^4*x
/(-e^2*x^2+d^2)^(5/2)+88/15/d^6*e^4*x/(-e^2*x^2+d^2)^(3/2)+176/15/d^8*e^4*x/(-e^
2*x^2+d^2)^(1/2)-1/d*e/x^2/(-e^2*x^2+d^2)^(5/2)+7/5/d^3*e^3/(-e^2*x^2+d^2)^(5/2)
+7/3/d^5*e^3/(-e^2*x^2+d^2)^(3/2)+7/d^7*e^3/(-e^2*x^2+d^2)^(1/2)-7/d^7*e^3/(d^2)
^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/((-e^2*x^2 + d^2)^(7/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284712, size = 953, normalized size = 4.56 \[ -\frac{176 \, e^{12} x^{12} - 943 \, d e^{11} x^{11} - 1898 \, d^{2} e^{10} x^{10} + 8404 \, d^{3} e^{9} x^{9} + 1788 \, d^{4} e^{8} x^{8} - 19305 \, d^{5} e^{7} x^{7} + 4075 \, d^{6} e^{6} x^{6} + 16090 \, d^{7} e^{5} x^{5} - 5350 \, d^{8} e^{4} x^{4} - 4400 \, d^{9} e^{3} x^{3} + 1040 \, d^{10} e^{2} x^{2} + 160 \, d^{11} e x + 160 \, d^{12} - 105 \,{\left (6 \, d e^{11} x^{11} - 12 \, d^{2} e^{10} x^{10} - 32 \, d^{3} e^{9} x^{9} + 76 \, d^{4} e^{8} x^{8} + 26 \, d^{5} e^{7} x^{7} - 128 \, d^{6} e^{6} x^{6} + 32 \, d^{7} e^{5} x^{5} + 64 \, d^{8} e^{4} x^{4} - 32 \, d^{9} e^{3} x^{3} -{\left (e^{11} x^{11} - 2 \, d e^{10} x^{10} - 17 \, d^{2} e^{9} x^{9} + 36 \, d^{3} e^{8} x^{8} + 30 \, d^{4} e^{7} x^{7} - 96 \, d^{5} e^{6} x^{6} + 16 \, d^{6} e^{5} x^{5} + 64 \, d^{7} e^{4} x^{4} - 32 \, d^{8} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 2 \,{\left (58 \, e^{11} x^{11} + 412 \, d e^{10} x^{10} - 1727 \, d^{2} e^{9} x^{9} - 1094 \, d^{3} e^{8} x^{8} + 6430 \, d^{4} e^{7} x^{7} - 920 \, d^{5} e^{6} x^{6} - 6975 \, d^{6} e^{5} x^{5} + 2385 \, d^{7} e^{4} x^{4} + 2160 \, d^{8} e^{3} x^{3} - 560 \, d^{9} e^{2} x^{2} - 80 \, d^{10} e x - 80 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (6 \, d^{9} e^{8} x^{11} - 12 \, d^{10} e^{7} x^{10} - 32 \, d^{11} e^{6} x^{9} + 76 \, d^{12} e^{5} x^{8} + 26 \, d^{13} e^{4} x^{7} - 128 \, d^{14} e^{3} x^{6} + 32 \, d^{15} e^{2} x^{5} + 64 \, d^{16} e x^{4} - 32 \, d^{17} x^{3} -{\left (d^{8} e^{8} x^{11} - 2 \, d^{9} e^{7} x^{10} - 17 \, d^{10} e^{6} x^{9} + 36 \, d^{11} e^{5} x^{8} + 30 \, d^{12} e^{4} x^{7} - 96 \, d^{13} e^{3} x^{6} + 16 \, d^{14} e^{2} x^{5} + 64 \, d^{15} e x^{4} - 32 \, d^{16} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(176*e^12*x^12 - 943*d*e^11*x^11 - 1898*d^2*e^10*x^10 + 8404*d^3*e^9*x^9 +
1788*d^4*e^8*x^8 - 19305*d^5*e^7*x^7 + 4075*d^6*e^6*x^6 + 16090*d^7*e^5*x^5 - 5
350*d^8*e^4*x^4 - 4400*d^9*e^3*x^3 + 1040*d^10*e^2*x^2 + 160*d^11*e*x + 160*d^12
- 105*(6*d*e^11*x^11 - 12*d^2*e^10*x^10 - 32*d^3*e^9*x^9 + 76*d^4*e^8*x^8 + 26*
d^5*e^7*x^7 - 128*d^6*e^6*x^6 + 32*d^7*e^5*x^5 + 64*d^8*e^4*x^4 - 32*d^9*e^3*x^3
- (e^11*x^11 - 2*d*e^10*x^10 - 17*d^2*e^9*x^9 + 36*d^3*e^8*x^8 + 30*d^4*e^7*x^7
- 96*d^5*e^6*x^6 + 16*d^6*e^5*x^5 + 64*d^7*e^4*x^4 - 32*d^8*e^3*x^3)*sqrt(-e^2*
x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + 2*(58*e^11*x^11 + 412*d*e^10*x^
10 - 1727*d^2*e^9*x^9 - 1094*d^3*e^8*x^8 + 6430*d^4*e^7*x^7 - 920*d^5*e^6*x^6 -
6975*d^6*e^5*x^5 + 2385*d^7*e^4*x^4 + 2160*d^8*e^3*x^3 - 560*d^9*e^2*x^2 - 80*d^
10*e*x - 80*d^11)*sqrt(-e^2*x^2 + d^2))/(6*d^9*e^8*x^11 - 12*d^10*e^7*x^10 - 32*
d^11*e^6*x^9 + 76*d^12*e^5*x^8 + 26*d^13*e^4*x^7 - 128*d^14*e^3*x^6 + 32*d^15*e^
2*x^5 + 64*d^16*e*x^4 - 32*d^17*x^3 - (d^8*e^8*x^11 - 2*d^9*e^7*x^10 - 17*d^10*e
^6*x^9 + 36*d^11*e^5*x^8 + 30*d^12*e^4*x^7 - 96*d^13*e^3*x^6 + 16*d^14*e^2*x^5 +
64*d^15*e*x^4 - 32*d^16*x^3)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**2/(x**4*(-(-d + e*x)*(d + e*x))**(7/2)), x)

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GIAC/XCAS [A]  time = 0.303236, size = 439, normalized size = 2.1 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (2 \, x{\left (\frac{53 \, x e^{8}}{d^{8}} + \frac{45 \, e^{7}}{d^{7}}\right )} - \frac{235 \, e^{6}}{d^{6}}\right )} x - \frac{200 \, e^{5}}{d^{5}}\right )} x + \frac{135 \, e^{4}}{d^{4}}\right )} x + \frac{116 \, e^{3}}{d^{3}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} + \frac{x^{3}{\left (\frac{6 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} + \frac{57 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} + e^{8}\right )} e}{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{8}} - \frac{7 \, e^{3}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{8}} - \frac{{\left (\frac{57 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{16}}{x} + \frac{6 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{14}}{x^{2}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{12}}{x^{3}}\right )} e^{\left (-15\right )}}{24 \, d^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((((2*x*(53*x*e^8/d^8 + 45*e^7/d^7) - 235*e^6/d^6)*x
- 200*e^5/d^5)*x + 135*e^4/d^4)*x + 116*e^3/d^3)/(x^2*e^2 - d^2)^3 + 1/24*x^3*(6
*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^6/x + 57*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^4/
x^2 + e^8)*e/((d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^8) - 7*e^3*ln(1/2*abs(-2*d*e -
2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^8 - 1/24*(57*(d*e + sqrt(-x^2*e^2 + d
^2)*e)*d^16*e^16/x + 6*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^16*e^14/x^2 + (d*e + s
qrt(-x^2*e^2 + d^2)*e)^3*d^16*e^12/x^3)*e^(-15)/d^24