3.136 \(\int \frac{x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=162 \[ \frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(x^6*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d - 7*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (x^2*(24*d - 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d
- 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) + (7*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(2*e^8)

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Rubi [A]  time = 0.510835, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(x^6*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d - 7*e*x))/(15*e^4*(d^2
- e^2*x^2)^(3/2)) + (x^2*(24*d - 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d
- 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) + (7*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^
2]])/(2*e^8)

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Rubi in Sympy [A]  time = 98.3398, size = 260, normalized size = 1.6 \[ \frac{d^{6}}{5 e^{8} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{d^{5}}{e^{8} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{d^{4} x}{15 e^{7} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{3 d^{3}}{e^{8} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 d^{2} x^{3}}{3 e^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{13 d^{2} x}{15 e^{7} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{7 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{8}} + \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{e^{8}} + \frac{x^{5}}{3 e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{3}}{3 e^{5} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{5 x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**6/(5*e**8*(d + e*x)*(d**2 - e**2*x**2)**(3/2)) - d**5/(e**8*(d**2 - e**2*x**2
)**(3/2)) + d**4*x/(15*e**7*(d**2 - e**2*x**2)**(3/2)) + 3*d**3/(e**8*sqrt(d**2
- e**2*x**2)) + 2*d**2*x**3/(3*e**5*(d**2 - e**2*x**2)**(3/2)) - 13*d**2*x/(15*e
**7*sqrt(d**2 - e**2*x**2)) + 7*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e**8) +
d*sqrt(d**2 - e**2*x**2)/e**8 + x**5/(3*e**3*(d**2 - e**2*x**2)**(3/2)) - 5*x**
3/(3*e**5*sqrt(d**2 - e**2*x**2)) - 5*x*sqrt(d**2 - e**2*x**2)/(2*e**7)

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Mathematica [A]  time = 0.15006, size = 128, normalized size = 0.79 \[ \frac{105 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (96 d^6-9 d^5 e x-249 d^4 e^2 x^2-4 d^3 e^3 x^3+176 d^2 e^4 x^4+15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^2 (d+e x)^3}}{30 e^8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(96*d^6 - 9*d^5*e*x - 249*d^4*e^2*x^2 - 4*d^3*e^3*x^3 + 17
6*d^2*e^4*x^4 + 15*d*e^5*x^5 - 15*e^6*x^6))/((d - e*x)^2*(d + e*x)^3) + 105*d^2*
ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8)

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Maple [F]  time = 180., size = 0, normalized size = 0. \[ \text{hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

int(x^7/(e*x+d)/(-e^2*x^2+d^2)^(5/2),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(x^7/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.326503, size = 963, normalized size = 5.94 \[ -\frac{15 \, e^{12} x^{12} - 15 \, d e^{11} x^{11} - 446 \, d^{2} e^{10} x^{10} - 302 \, d^{3} e^{9} x^{9} + 3561 \, d^{4} e^{8} x^{8} + 3441 \, d^{5} e^{7} x^{7} - 9282 \, d^{6} e^{6} x^{6} - 9282 \, d^{7} e^{5} x^{5} + 9520 \, d^{8} e^{4} x^{4} + 9520 \, d^{9} e^{3} x^{3} - 3360 \, d^{10} e^{2} x^{2} - 3360 \, d^{11} e x + 210 \,{\left (6 \, d^{3} e^{9} x^{9} + 6 \, d^{4} e^{8} x^{8} - 44 \, d^{5} e^{7} x^{7} - 44 \, d^{6} e^{6} x^{6} + 102 \, d^{7} e^{5} x^{5} + 102 \, d^{8} e^{4} x^{4} - 96 \, d^{9} e^{3} x^{3} - 96 \, d^{10} e^{2} x^{2} + 32 \, d^{11} e x + 32 \, d^{12} -{\left (d^{2} e^{9} x^{9} + d^{3} e^{8} x^{8} - 19 \, d^{4} e^{7} x^{7} - 19 \, d^{5} e^{6} x^{6} + 66 \, d^{6} e^{5} x^{5} + 66 \, d^{7} e^{4} x^{4} - 80 \, d^{8} e^{3} x^{3} - 80 \, d^{9} e^{2} x^{2} + 32 \, d^{10} e x + 32 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 2 \,{\left (45 \, d e^{10} x^{10} + 3 \, d^{2} e^{9} x^{9} - 720 \, d^{3} e^{8} x^{8} - 660 \, d^{4} e^{7} x^{7} + 2891 \, d^{5} e^{6} x^{6} + 2891 \, d^{6} e^{5} x^{5} - 3920 \, d^{7} e^{4} x^{4} - 3920 \, d^{8} e^{3} x^{3} + 1680 \, d^{9} e^{2} x^{2} + 1680 \, d^{10} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (6 \, d e^{17} x^{9} + 6 \, d^{2} e^{16} x^{8} - 44 \, d^{3} e^{15} x^{7} - 44 \, d^{4} e^{14} x^{6} + 102 \, d^{5} e^{13} x^{5} + 102 \, d^{6} e^{12} x^{4} - 96 \, d^{7} e^{11} x^{3} - 96 \, d^{8} e^{10} x^{2} + 32 \, d^{9} e^{9} x + 32 \, d^{10} e^{8} -{\left (e^{17} x^{9} + d e^{16} x^{8} - 19 \, d^{2} e^{15} x^{7} - 19 \, d^{3} e^{14} x^{6} + 66 \, d^{4} e^{13} x^{5} + 66 \, d^{5} e^{12} x^{4} - 80 \, d^{6} e^{11} x^{3} - 80 \, d^{7} e^{10} x^{2} + 32 \, d^{8} e^{9} x + 32 \, d^{9} e^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

-1/30*(15*e^12*x^12 - 15*d*e^11*x^11 - 446*d^2*e^10*x^10 - 302*d^3*e^9*x^9 + 356
1*d^4*e^8*x^8 + 3441*d^5*e^7*x^7 - 9282*d^6*e^6*x^6 - 9282*d^7*e^5*x^5 + 9520*d^
8*e^4*x^4 + 9520*d^9*e^3*x^3 - 3360*d^10*e^2*x^2 - 3360*d^11*e*x + 210*(6*d^3*e^
9*x^9 + 6*d^4*e^8*x^8 - 44*d^5*e^7*x^7 - 44*d^6*e^6*x^6 + 102*d^7*e^5*x^5 + 102*
d^8*e^4*x^4 - 96*d^9*e^3*x^3 - 96*d^10*e^2*x^2 + 32*d^11*e*x + 32*d^12 - (d^2*e^
9*x^9 + d^3*e^8*x^8 - 19*d^4*e^7*x^7 - 19*d^5*e^6*x^6 + 66*d^6*e^5*x^5 + 66*d^7*
e^4*x^4 - 80*d^8*e^3*x^3 - 80*d^9*e^2*x^2 + 32*d^10*e*x + 32*d^11)*sqrt(-e^2*x^2
+ d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 2*(45*d*e^10*x^10 + 3*d^2*e
^9*x^9 - 720*d^3*e^8*x^8 - 660*d^4*e^7*x^7 + 2891*d^5*e^6*x^6 + 2891*d^6*e^5*x^5
- 3920*d^7*e^4*x^4 - 3920*d^8*e^3*x^3 + 1680*d^9*e^2*x^2 + 1680*d^10*e*x)*sqrt(
-e^2*x^2 + d^2))/(6*d*e^17*x^9 + 6*d^2*e^16*x^8 - 44*d^3*e^15*x^7 - 44*d^4*e^14*
x^6 + 102*d^5*e^13*x^5 + 102*d^6*e^12*x^4 - 96*d^7*e^11*x^3 - 96*d^8*e^10*x^2 +
32*d^9*e^9*x + 32*d^10*e^8 - (e^17*x^9 + d*e^16*x^8 - 19*d^2*e^15*x^7 - 19*d^3*e
^14*x^6 + 66*d^4*e^13*x^5 + 66*d^5*e^12*x^4 - 80*d^6*e^11*x^3 - 80*d^7*e^10*x^2
+ 32*d^8*e^9*x + 32*d^9*e^8)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**7/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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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(x^7/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

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