3.846 \(\int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=172 \[ -\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(16*x)/(165*d^5*(d^2 - e^2*x^2)^(5/2)) - 1/(11*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(
5/2)) - 8/(99*d^2*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(99*d^3*e*(d + e*x)*(
d^2 - e^2*x^2)^(5/2)) + (64*x)/(495*d^7*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(495*d^
9*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.189518, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(16*x)/(165*d^5*(d^2 - e^2*x^2)^(5/2)) - 1/(11*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(
5/2)) - 8/(99*d^2*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(99*d^3*e*(d + e*x)*(
d^2 - e^2*x^2)^(5/2)) + (64*x)/(495*d^7*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(495*d^
9*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 21.6216, size = 148, normalized size = 0.86 \[ - \frac{1}{11 d e \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{8}{99 d^{2} e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{8}{99 d^{3} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{16 x}{165 d^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{64 x}{495 d^{7} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{128 x}{495 d^{9} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-1/(11*d*e*(d + e*x)**3*(d**2 - e**2*x**2)**(5/2)) - 8/(99*d**2*e*(d + e*x)**2*(
d**2 - e**2*x**2)**(5/2)) - 8/(99*d**3*e*(d + e*x)*(d**2 - e**2*x**2)**(5/2)) +
16*x/(165*d**5*(d**2 - e**2*x**2)**(5/2)) + 64*x/(495*d**7*(d**2 - e**2*x**2)**(
3/2)) + 128*x/(495*d**9*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.108955, size = 126, normalized size = 0.73 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-125 d^8+120 d^7 e x+680 d^6 e^2 x^2+400 d^5 e^3 x^3-720 d^4 e^4 x^4-832 d^3 e^5 x^5+64 d^2 e^6 x^6+384 d e^7 x^7+128 e^8 x^8\right )}{495 d^9 e (d-e x)^3 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-125*d^8 + 120*d^7*e*x + 680*d^6*e^2*x^2 + 400*d^5*e^3*x^3
- 720*d^4*e^4*x^4 - 832*d^3*e^5*x^5 + 64*d^2*e^6*x^6 + 384*d*e^7*x^7 + 128*e^8*
x^8))/(495*d^9*e*(d - e*x)^3*(d + e*x)^6)

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Maple [A]  time = 0.015, size = 121, normalized size = 0.7 \[ -{\frac{ \left ( -ex+d \right ) \left ( -128\,{e}^{8}{x}^{8}-384\,{e}^{7}{x}^{7}d-64\,{e}^{6}{x}^{6}{d}^{2}+832\,{e}^{5}{x}^{5}{d}^{3}+720\,{e}^{4}{x}^{4}{d}^{4}-400\,{e}^{3}{x}^{3}{d}^{5}-680\,{e}^{2}{x}^{2}{d}^{6}-120\,x{d}^{7}e+125\,{d}^{8} \right ) }{495\,e{d}^{9} \left ( ex+d \right ) ^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/495*(-e*x+d)*(-128*e^8*x^8-384*d*e^7*x^7-64*d^2*e^6*x^6+832*d^3*e^5*x^5+720*d
^4*e^4*x^4-400*d^5*e^3*x^3-680*d^6*e^2*x^2-120*d^7*e*x+125*d^8)/(e*x+d)^2/d^9/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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.419334, size = 929, normalized size = 5.4 \[ -\frac{128 \, e^{15} x^{16} + 1384 \, d e^{14} x^{15} - 1032 \, d^{2} e^{13} x^{14} - 23120 \, d^{3} e^{12} x^{13} - 20288 \, d^{4} e^{11} x^{12} + 106464 \, d^{5} e^{10} x^{11} + 159192 \, d^{6} e^{9} x^{10} - 192104 \, d^{7} e^{8} x^{9} - 437085 \, d^{8} e^{7} x^{8} + 95304 \, d^{9} e^{6} x^{7} + 576312 \, d^{10} e^{5} x^{6} + 128304 \, d^{11} e^{4} x^{5} - 372240 \, d^{12} e^{3} x^{4} - 179520 \, d^{13} e^{2} x^{3} + 95040 \, d^{14} e x^{2} + 63360 \, d^{15} x -{\left (125 \, e^{14} x^{15} - 649 \, d e^{13} x^{14} - 6947 \, d^{2} e^{12} x^{13} - 2897 \, d^{3} e^{11} x^{12} + 52751 \, d^{4} e^{10} x^{11} + 66429 \, d^{5} e^{9} x^{10} - 135113 \, d^{6} e^{8} x^{9} - 258819 \, d^{7} e^{7} x^{8} + 111936 \, d^{8} e^{6} x^{7} + 425832 \, d^{9} e^{5} x^{6} + 62304 \, d^{10} e^{4} x^{5} - 324720 \, d^{11} e^{3} x^{4} - 147840 \, d^{12} e^{2} x^{3} + 95040 \, d^{13} e x^{2} + 63360 \, d^{14} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{495 \,{\left (8 \, d^{10} e^{15} x^{15} + 24 \, d^{11} e^{14} x^{14} - 80 \, d^{12} e^{13} x^{13} - 304 \, d^{13} e^{12} x^{12} + 144 \, d^{14} e^{11} x^{11} + 1264 \, d^{15} e^{10} x^{10} + 416 \, d^{16} e^{9} x^{9} - 2400 \, d^{17} e^{8} x^{8} - 1816 \, d^{18} e^{7} x^{7} + 2168 \, d^{19} e^{6} x^{6} + 2544 \, d^{20} e^{5} x^{5} - 688 \, d^{21} e^{4} x^{4} - 1600 \, d^{22} e^{3} x^{3} - 192 \, d^{23} e^{2} x^{2} + 384 \, d^{24} e x + 128 \, d^{25} -{\left (d^{9} e^{15} x^{15} + 3 \, d^{10} e^{14} x^{14} - 31 \, d^{11} e^{13} x^{13} - 101 \, d^{12} e^{12} x^{12} + 123 \, d^{13} e^{11} x^{11} + 641 \, d^{14} e^{10} x^{10} + 67 \, d^{15} e^{9} x^{9} - 1599 \, d^{16} e^{8} x^{8} - 1024 \, d^{17} e^{7} x^{7} + 1792 \, d^{18} e^{6} x^{6} + 1888 \, d^{19} e^{5} x^{5} - 736 \, d^{20} e^{4} x^{4} - 1408 \, d^{21} e^{3} x^{3} - 128 \, d^{22} e^{2} x^{2} + 384 \, d^{23} e x + 128 \, d^{24}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-1/495*(128*e^15*x^16 + 1384*d*e^14*x^15 - 1032*d^2*e^13*x^14 - 23120*d^3*e^12*x
^13 - 20288*d^4*e^11*x^12 + 106464*d^5*e^10*x^11 + 159192*d^6*e^9*x^10 - 192104*
d^7*e^8*x^9 - 437085*d^8*e^7*x^8 + 95304*d^9*e^6*x^7 + 576312*d^10*e^5*x^6 + 128
304*d^11*e^4*x^5 - 372240*d^12*e^3*x^4 - 179520*d^13*e^2*x^3 + 95040*d^14*e*x^2
+ 63360*d^15*x - (125*e^14*x^15 - 649*d*e^13*x^14 - 6947*d^2*e^12*x^13 - 2897*d^
3*e^11*x^12 + 52751*d^4*e^10*x^11 + 66429*d^5*e^9*x^10 - 135113*d^6*e^8*x^9 - 25
8819*d^7*e^7*x^8 + 111936*d^8*e^6*x^7 + 425832*d^9*e^5*x^6 + 62304*d^10*e^4*x^5
- 324720*d^11*e^3*x^4 - 147840*d^12*e^2*x^3 + 95040*d^13*e*x^2 + 63360*d^14*x)*s
qrt(-e^2*x^2 + d^2))/(8*d^10*e^15*x^15 + 24*d^11*e^14*x^14 - 80*d^12*e^13*x^13 -
304*d^13*e^12*x^12 + 144*d^14*e^11*x^11 + 1264*d^15*e^10*x^10 + 416*d^16*e^9*x^
9 - 2400*d^17*e^8*x^8 - 1816*d^18*e^7*x^7 + 2168*d^19*e^6*x^6 + 2544*d^20*e^5*x^
5 - 688*d^21*e^4*x^4 - 1600*d^22*e^3*x^3 - 192*d^23*e^2*x^2 + 384*d^24*e*x + 128
*d^25 - (d^9*e^15*x^15 + 3*d^10*e^14*x^14 - 31*d^11*e^13*x^13 - 101*d^12*e^12*x^
12 + 123*d^13*e^11*x^11 + 641*d^14*e^10*x^10 + 67*d^15*e^9*x^9 - 1599*d^16*e^8*x
^8 - 1024*d^17*e^7*x^7 + 1792*d^18*e^6*x^6 + 1888*d^19*e^5*x^5 - 736*d^20*e^4*x^
4 - 1408*d^21*e^3*x^3 - 128*d^22*e^2*x^2 + 384*d^23*e*x + 128*d^24)*sqrt(-e^2*x^
2 + d^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]