3.845 \(\int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 x}{45 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(2*x)/(15*d^4*(d^2 - e^2*x^2)^(5/2)) - 1/(9*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2
)) - 1/(9*d^2*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (8*x)/(45*d^6*(d^2 - e^2*x^2)
^(3/2)) + (16*x)/(45*d^8*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.134694, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 x}{45 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(2*x)/(15*d^4*(d^2 - e^2*x^2)^(5/2)) - 1/(9*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2
)) - 1/(9*d^2*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (8*x)/(45*d^6*(d^2 - e^2*x^2)
^(3/2)) + (16*x)/(45*d^8*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 15.4269, size = 119, normalized size = 0.86 \[ - \frac{1}{9 d e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{1}{9 d^{2} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{2 x}{15 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{8 x}{45 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{16 x}{45 d^{8} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-1/(9*d*e*(d + e*x)**2*(d**2 - e**2*x**2)**(5/2)) - 1/(9*d**2*e*(d + e*x)*(d**2
- e**2*x**2)**(5/2)) + 2*x/(15*d**4*(d**2 - e**2*x**2)**(5/2)) + 8*x/(45*d**6*(d
**2 - e**2*x**2)**(3/2)) + 16*x/(45*d**8*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.095849, size = 115, normalized size = 0.83 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-10 d^7+25 d^6 e x+60 d^5 e^2 x^2-10 d^4 e^3 x^3-80 d^3 e^4 x^4-24 d^2 e^5 x^5+32 d e^6 x^6+16 e^7 x^7\right )}{45 d^8 e (d-e x)^3 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-10*d^7 + 25*d^6*e*x + 60*d^5*e^2*x^2 - 10*d^4*e^3*x^3 - 8
0*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 32*d*e^6*x^6 + 16*e^7*x^7))/(45*d^8*e*(d - e*x)
^3*(d + e*x)^5)

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Maple [A]  time = 0.014, size = 110, normalized size = 0.8 \[ -{\frac{ \left ( -ex+d \right ) \left ( -16\,{e}^{7}{x}^{7}-32\,{e}^{6}{x}^{6}d+24\,{e}^{5}{x}^{5}{d}^{2}+80\,{e}^{4}{x}^{4}{d}^{3}+10\,{e}^{3}{x}^{3}{d}^{4}-60\,{e}^{2}{x}^{2}{d}^{5}-25\,x{d}^{6}e+10\,{d}^{7} \right ) }{ \left ( 45\,ex+45\,d \right ){d}^{8}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/45*(-e*x+d)*(-16*e^7*x^7-32*d*e^6*x^6+24*d^2*e^5*x^5+80*d^3*e^4*x^4+10*d^4*e^
3*x^3-60*d^5*e^2*x^2-25*d^6*e*x+10*d^7)/(e*x+d)/d^8/e/(-e^2*x^2+d^2)^(7/2)

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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(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305126, size = 807, normalized size = 5.81 \[ -\frac{10 \, e^{13} x^{14} - 92 \, d e^{12} x^{13} - 484 \, d^{2} e^{11} x^{12} + 524 \, d^{3} e^{10} x^{11} + 3632 \, d^{4} e^{9} x^{10} + 34 \, d^{5} e^{8} x^{9} - 10704 \, d^{6} e^{7} x^{8} - 4563 \, d^{7} e^{6} x^{7} + 15228 \, d^{8} e^{5} x^{6} + 10104 \, d^{9} e^{4} x^{5} - 10560 \, d^{10} e^{3} x^{4} - 8880 \, d^{11} e^{2} x^{3} + 2880 \, d^{12} e x^{2} + 2880 \, d^{13} x +{\left (16 \, e^{12} x^{13} + 102 \, d e^{11} x^{12} - 268 \, d^{2} e^{10} x^{11} - 1478 \, d^{3} e^{9} x^{10} + 446 \, d^{4} e^{8} x^{9} + 6150 \, d^{5} e^{7} x^{8} + 1941 \, d^{6} e^{6} x^{7} - 11028 \, d^{7} e^{5} x^{6} - 6744 \, d^{8} e^{4} x^{5} + 9120 \, d^{9} e^{3} x^{4} + 7440 \, d^{10} e^{2} x^{3} - 2880 \, d^{11} e x^{2} - 2880 \, d^{12} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{45 \,{\left (d^{8} e^{14} x^{14} + 2 \, d^{9} e^{13} x^{13} - 26 \, d^{10} e^{12} x^{12} - 54 \, d^{11} e^{11} x^{11} + 128 \, d^{12} e^{10} x^{10} + 310 \, d^{13} e^{9} x^{9} - 222 \, d^{14} e^{8} x^{8} - 754 \, d^{15} e^{7} x^{7} + 79 \, d^{16} e^{6} x^{6} + 912 \, d^{17} e^{5} x^{5} + 184 \, d^{18} e^{4} x^{4} - 544 \, d^{19} e^{3} x^{3} - 208 \, d^{20} e^{2} x^{2} + 128 \, d^{21} e x + 64 \, d^{22} +{\left (7 \, d^{9} e^{12} x^{12} + 14 \, d^{10} e^{11} x^{11} - 63 \, d^{11} e^{10} x^{10} - 140 \, d^{12} e^{9} x^{9} + 161 \, d^{13} e^{8} x^{8} + 462 \, d^{14} e^{7} x^{7} - 113 \, d^{15} e^{6} x^{6} - 688 \, d^{16} e^{5} x^{5} - 104 \, d^{17} e^{4} x^{4} + 480 \, d^{18} e^{3} x^{3} + 176 \, d^{19} e^{2} x^{2} - 128 \, d^{20} e x - 64 \, d^{21}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/45*(10*e^13*x^14 - 92*d*e^12*x^13 - 484*d^2*e^11*x^12 + 524*d^3*e^10*x^11 + 3
632*d^4*e^9*x^10 + 34*d^5*e^8*x^9 - 10704*d^6*e^7*x^8 - 4563*d^7*e^6*x^7 + 15228
*d^8*e^5*x^6 + 10104*d^9*e^4*x^5 - 10560*d^10*e^3*x^4 - 8880*d^11*e^2*x^3 + 2880
*d^12*e*x^2 + 2880*d^13*x + (16*e^12*x^13 + 102*d*e^11*x^12 - 268*d^2*e^10*x^11
- 1478*d^3*e^9*x^10 + 446*d^4*e^8*x^9 + 6150*d^5*e^7*x^8 + 1941*d^6*e^6*x^7 - 11
028*d^7*e^5*x^6 - 6744*d^8*e^4*x^5 + 9120*d^9*e^3*x^4 + 7440*d^10*e^2*x^3 - 2880
*d^11*e*x^2 - 2880*d^12*x)*sqrt(-e^2*x^2 + d^2))/(d^8*e^14*x^14 + 2*d^9*e^13*x^1
3 - 26*d^10*e^12*x^12 - 54*d^11*e^11*x^11 + 128*d^12*e^10*x^10 + 310*d^13*e^9*x^
9 - 222*d^14*e^8*x^8 - 754*d^15*e^7*x^7 + 79*d^16*e^6*x^6 + 912*d^17*e^5*x^5 + 1
84*d^18*e^4*x^4 - 544*d^19*e^3*x^3 - 208*d^20*e^2*x^2 + 128*d^21*e*x + 64*d^22 +
(7*d^9*e^12*x^12 + 14*d^10*e^11*x^11 - 63*d^11*e^10*x^10 - 140*d^12*e^9*x^9 + 1
61*d^13*e^8*x^8 + 462*d^14*e^7*x^7 - 113*d^15*e^6*x^6 - 688*d^16*e^5*x^5 - 104*d
^17*e^4*x^4 + 480*d^18*e^3*x^3 + 176*d^19*e^2*x^2 - 128*d^20*e*x - 64*d^21)*sqrt
(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**2), x)

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GIAC/XCAS [A]  time = 0.684718, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

sage0*x