3.81 \(\int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx\)
Optimal. Leaf size=225 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac{33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}+\frac{33 e^{10} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^2}-\frac{33 e^8 \sqrt{d^2-e^2 x^2}}{256 d x^2}+\frac{11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac{11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac{5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7} \]
[Out]
(-33*e^8*Sqrt[d^2 - e^2*x^2])/(256*d*x^2) + (11*e^6*(d^2 - e^2*x^2)^(3/2))/(128*
d*x^4) - (11*e^4*(d^2 - e^2*x^2)^(5/2))/(160*d*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/
(10*x^10) - (e*(d^2 - e^2*x^2)^(7/2))/(3*x^9) - (33*e^2*(d^2 - e^2*x^2)^(7/2))/(
80*d*x^8) - (5*e^3*(d^2 - e^2*x^2)^(7/2))/(21*d^2*x^7) + (33*e^10*ArcTanh[Sqrt[d
^2 - e^2*x^2]/d])/(256*d^2)
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Rubi [A] time = 0.535826, antiderivative size = 225, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259
\[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac{33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}+\frac{33 e^{10} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^2}-\frac{33 e^8 \sqrt{d^2-e^2 x^2}}{256 d x^2}+\frac{11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac{11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac{5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]
[Out]
(-33*e^8*Sqrt[d^2 - e^2*x^2])/(256*d*x^2) + (11*e^6*(d^2 - e^2*x^2)^(3/2))/(128*
d*x^4) - (11*e^4*(d^2 - e^2*x^2)^(5/2))/(160*d*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/
(10*x^10) - (e*(d^2 - e^2*x^2)^(7/2))/(3*x^9) - (33*e^2*(d^2 - e^2*x^2)^(7/2))/(
80*d*x^8) - (5*e^3*(d^2 - e^2*x^2)^(7/2))/(21*d^2*x^7) + (33*e^10*ArcTanh[Sqrt[d
^2 - e^2*x^2]/d])/(256*d^2)
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Rubi in Sympy [A] time = 138.234, size = 280, normalized size = 1.24 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{10 x^{10}} - \frac{d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{3 x^{9}} - \frac{9 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{80 x^{8}} + \frac{16 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{21 x^{7}} + \frac{139 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{160 x^{6}} - \frac{2 d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{7 x^{5}} - \frac{117 d e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{128 x^{4}} - \frac{8 e^{7} \sqrt{d^{2} - e^{2} x^{2}}}{21 x^{3}} + \frac{33 e^{8} \sqrt{d^{2} - e^{2} x^{2}}}{256 d x^{2}} + \frac{33 e^{10} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{256 d^{2}} + \frac{5 e^{9} \sqrt{d^{2} - e^{2} x^{2}}}{21 d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**11,x)
[Out]
-d**7*sqrt(d**2 - e**2*x**2)/(10*x**10) - d**6*e*sqrt(d**2 - e**2*x**2)/(3*x**9)
- 9*d**5*e**2*sqrt(d**2 - e**2*x**2)/(80*x**8) + 16*d**4*e**3*sqrt(d**2 - e**2*
x**2)/(21*x**7) + 139*d**3*e**4*sqrt(d**2 - e**2*x**2)/(160*x**6) - 2*d**2*e**5*
sqrt(d**2 - e**2*x**2)/(7*x**5) - 117*d*e**6*sqrt(d**2 - e**2*x**2)/(128*x**4) -
8*e**7*sqrt(d**2 - e**2*x**2)/(21*x**3) + 33*e**8*sqrt(d**2 - e**2*x**2)/(256*d
*x**2) + 33*e**10*atanh(sqrt(d**2 - e**2*x**2)/d)/(256*d**2) + 5*e**9*sqrt(d**2
- e**2*x**2)/(21*d**2*x)
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Mathematica [A] time = 0.266482, size = 161, normalized size = 0.72 \[ -\frac{-3465 e^{10} x^{10} \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (2688 d^9+8960 d^8 e x+3024 d^7 e^2 x^2-20480 d^6 e^3 x^3-23352 d^5 e^4 x^4+7680 d^4 e^5 x^5+24570 d^3 e^6 x^6+10240 d^2 e^7 x^7-3465 d e^8 x^8-6400 e^9 x^9\right )+3465 e^{10} x^{10} \log (x)}{26880 d^2 x^{10}} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]
[Out]
-(Sqrt[d^2 - e^2*x^2]*(2688*d^9 + 8960*d^8*e*x + 3024*d^7*e^2*x^2 - 20480*d^6*e^
3*x^3 - 23352*d^5*e^4*x^4 + 7680*d^4*e^5*x^5 + 24570*d^3*e^6*x^6 + 10240*d^2*e^7
*x^7 - 3465*d*e^8*x^8 - 6400*e^9*x^9) + 3465*e^10*x^10*Log[x] - 3465*e^10*x^10*L
og[d + Sqrt[d^2 - e^2*x^2]])/(26880*d^2*x^10)
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Maple [A] time = 0.209, size = 278, normalized size = 1.2 \[ -{\frac{d}{10\,{x}^{10}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{33\,{e}^{2}}{80\,d{x}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{4}}{160\,{d}^{3}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{e}^{6}}{640\,{d}^{5}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{33\,{e}^{8}}{1280\,{d}^{7}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{33\,{e}^{10}}{1280\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{11\,{e}^{10}}{256\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{33\,{e}^{10}}{256\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{33\,{e}^{10}}{256\,d}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{5\,{e}^{3}}{21\,{d}^{2}{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{e}{3\,{x}^{9}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^11,x)
[Out]
-1/10*d*(-e^2*x^2+d^2)^(7/2)/x^10-33/80*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^8-11/160/d^
3*e^4/x^6*(-e^2*x^2+d^2)^(7/2)+11/640/d^5*e^6/x^4*(-e^2*x^2+d^2)^(7/2)-33/1280/d
^7*e^8/x^2*(-e^2*x^2+d^2)^(7/2)-33/1280/d^7*e^10*(-e^2*x^2+d^2)^(5/2)-11/256/d^5
*e^10*(-e^2*x^2+d^2)^(3/2)-33/256/d^3*e^10*(-e^2*x^2+d^2)^(1/2)+33/256/d*e^10/(d
^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-5/21*e^3*(-e^2*x^2+d^
2)^(7/2)/d^2/x^7-1/3*e*(-e^2*x^2+d^2)^(7/2)/x^9
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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^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^11,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.983389, size = 980, normalized size = 4.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^11,x, algorithm="fricas")
[Out]
-1/26880*(64000*d*e^19*x^19 + 34650*d^2*e^18*x^18 - 1190400*d^3*e^17*x^17 - 8347
50*d^4*e^16*x^16 + 6988800*d^5*e^15*x^15 + 7293300*d^6*e^14*x^14 - 17863680*d^7*
e^13*x^13 - 30318960*d^8*e^12*x^12 + 17236480*d^9*e^11*x^11 + 66909024*d^10*e^10
*x^10 + 12582400*d^11*e^9*x^9 - 81177600*d^12*e^8*x^8 - 48742400*d^13*e^7*x^7 +
51340800*d^14*e^6*x^6 + 50585600*d^15*e^5*x^5 - 12042240*d^16*e^4*x^4 - 24248320
*d^17*e^3*x^3 - 2580480*d^18*e^2*x^2 + 4587520*d^19*e*x + 1376256*d^20 + 3465*(e
^20*x^20 - 50*d^2*e^18*x^18 + 400*d^4*e^16*x^16 - 1120*d^6*e^14*x^14 + 1280*d^8*
e^12*x^12 - 512*d^10*e^10*x^10 + 2*(5*d*e^18*x^18 - 80*d^3*e^16*x^16 + 336*d^5*e
^14*x^14 - 512*d^7*e^12*x^12 + 256*d^9*e^10*x^10)*sqrt(-e^2*x^2 + d^2))*log(-(d
- sqrt(-e^2*x^2 + d^2))/x) - (6400*e^19*x^19 + 3465*d*e^18*x^18 - 330240*d^2*e^1
7*x^17 - 197820*d^3*e^16*x^16 + 3064320*d^4*e^15*x^15 + 2637852*d^5*e^14*x^14 -
10859520*d^6*e^13*x^13 - 14879424*d^7*e^12*x^12 + 15555840*d^8*e^11*x^11 + 41442
912*d^9*e^10*x^10 + 857600*d^10*e^9*x^9 - 60453120*d^11*e^8*x^8 - 31109120*d^12*
e^7*x^7 + 44782080*d^13*e^6*x^6 + 40181760*d^14*e^5*x^5 - 12816384*d^15*e^4*x^4
- 21954560*d^16*e^3*x^3 - 1892352*d^17*e^2*x^2 + 4587520*d^18*e*x + 1376256*d^19
)*sqrt(-e^2*x^2 + d^2))/(d^2*e^10*x^20 - 50*d^4*e^8*x^18 + 400*d^6*e^6*x^16 - 11
20*d^8*e^4*x^14 + 1280*d^10*e^2*x^12 - 512*d^12*x^10 + 2*(5*d^3*e^8*x^18 - 80*d^
5*e^6*x^16 + 336*d^7*e^4*x^14 - 512*d^9*e^2*x^12 + 256*d^11*x^10)*sqrt(-e^2*x^2
+ d^2))
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Sympy [A] time = 137.099, size = 2159, normalized size = 9.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**11,x)
[Out]
d**7*Piecewise((-d**2/(10*e*x**11*sqrt(d**2/(e**2*x**2) - 1)) + 9*e/(80*x**9*sqr
t(d**2/(e**2*x**2) - 1)) + e**3/(480*d**2*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 7*e
**5/(1920*d**4*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**7/(768*d**6*x**3*sqrt(d**
2/(e**2*x**2) - 1)) - 7*e**9/(256*d**8*x*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**10*a
cosh(d/(e*x))/(256*d**9), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(10*e*x**11*sqrt(-
d**2/(e**2*x**2) + 1)) - 9*I*e/(80*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(4
80*d**2*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**5/(1920*d**4*x**5*sqrt(-d**2/
(e**2*x**2) + 1)) - 7*I*e**7/(768*d**6*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 7*I*e
**9/(256*d**8*x*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**10*asin(d/(e*x))/(256*d**9
), True)) + 3*d**6*e*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(9*x**8) + e**3*sq
rt(d**2/(e**2*x**2) - 1)/(63*d**2*x**6) + 2*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105
*d**4*x**4) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(315*d**6*x**2) + 16*e**9*sqrt(d
**2/(e**2*x**2) - 1)/(315*d**8), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e
**2*x**2) + 1)/(9*x**8) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(63*d**2*x**6) + 2*
I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**4) + 8*I*e**7*sqrt(-d**2/(e**2*x
**2) + 1)/(315*d**6*x**2) + 16*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(315*d**8), Tr
ue)) + d**5*e**2*Piecewise((-d**2/(8*e*x**9*sqrt(d**2/(e**2*x**2) - 1)) + 7*e/(4
8*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(192*d**2*x**5*sqrt(d**2/(e**2*x**2) -
1)) + 5*e**5/(384*d**4*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 5*e**7/(128*d**6*x*sq
rt(d**2/(e**2*x**2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d**2/(e**2*x**
2)) > 1), (I*d**2/(8*e*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-
d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 5*
I*e**5/(384*d**4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*x*sqrt(-
d**2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) - 5*d**4*e**3
*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2)
- 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7
*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-
d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**
4) + 4*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/
(e**2*x**2) + 1)/(105*d**6), True)) - 5*d**3*e**4*Piecewise((-d**2/(6*e*x**7*sqr
t(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d
**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1
)) + e**6*acosh(d/(e*x))/(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**
7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - I
*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/
(e**2*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + d**2*e**5*Piecewise
((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2
*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6*x**6*s
qrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(
-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2/d**2) > 1),
(3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*d*e**2*x**2
*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1 -
e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x*
*2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), True)) + 3*d*e**6*Piecewise((-d**2/(4
*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d**2/(e**2*x**2) - 1)) -
e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d
**2/(e**2*x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8
*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1
)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + e**7*Piecewise((-e*sqrt(d**2/(e**2*
x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x
**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2
*x**2) + 1)/(3*d**2), True))
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GIAC/XCAS [A] time = 0.31593, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^11,x, algorithm="giac")
[Out]
Done