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

Optimal. Leaf size=124 \[ \frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{5 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e}+\frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2} \]

[Out]

(5*d^5*x*Sqrt[d^2 - e^2*x^2])/16 + (5*d^3*x*(d^2 - e^2*x^2)^(3/2))/24 + (d*x*(d^
2 - e^2*x^2)^(5/2))/6 + (d^2 - e^2*x^2)^(7/2)/(7*e) + (5*d^7*ArcTan[(e*x)/Sqrt[d
^2 - e^2*x^2]])/(16*e)

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Rubi [A]  time = 0.11197, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{5 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e}+\frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(5*d^5*x*Sqrt[d^2 - e^2*x^2])/16 + (5*d^3*x*(d^2 - e^2*x^2)^(3/2))/24 + (d*x*(d^
2 - e^2*x^2)^(5/2))/6 + (d^2 - e^2*x^2)^(7/2)/(7*e) + (5*d^7*ArcTan[(e*x)/Sqrt[d
^2 - e^2*x^2]])/(16*e)

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Rubi in Sympy [A]  time = 19.1468, size = 105, normalized size = 0.85 \[ \frac{5 d^{7} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{16 e} + \frac{5 d^{5} x \sqrt{d^{2} - e^{2} x^{2}}}{16} + \frac{5 d^{3} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24} + \frac{d x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{7 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*d**7*atan(e*x/sqrt(d**2 - e**2*x**2))/(16*e) + 5*d**5*x*sqrt(d**2 - e**2*x**2)
/16 + 5*d**3*x*(d**2 - e**2*x**2)**(3/2)/24 + d*x*(d**2 - e**2*x**2)**(5/2)/6 +
(d**2 - e**2*x**2)**(7/2)/(7*e)

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Mathematica [A]  time = 0.114896, size = 113, normalized size = 0.91 \[ \frac{105 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right )}{336 e} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(48*d^6 + 231*d^5*e*x - 144*d^4*e^2*x^2 - 182*d^3*e^3*x^3 +
144*d^2*e^4*x^4 + 56*d*e^5*x^5 - 48*e^6*x^6) + 105*d^7*ArcTan[(e*x)/Sqrt[d^2 -
e^2*x^2]])/(336*e)

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Maple [A]  time = 0.023, size = 181, normalized size = 1.5 \[{\frac{1}{7\,e} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{dx}{6} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{3}x}{24} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{5}x}{16}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{5\,{d}^{7}}{16}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7/e*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+1/6*d*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^
(5/2)*x+5/24*d^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+5/16*d^5*(-(d/e+x)^2*e^2
+2*d*e*(d/e+x))^(1/2)*x+5/16*d^7/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(d/e+x)^2*e^
2+2*d*e*(d/e+x))^(1/2))

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Maxima [A]  time = 0.813554, size = 174, normalized size = 1.4 \[ -\frac{5 i \, d^{7} \arcsin \left (\frac{e x}{d} + 2\right )}{16 \, e} + \frac{5}{16} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x + \frac{5 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{8 \, e} + \frac{5}{24} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3} x + \frac{1}{6} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}{7 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/16*I*d^7*arcsin(e*x/d + 2)/e + 5/16*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^5*x + 5
/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^6/e + 5/24*(-e^2*x^2 + d^2)^(3/2)*d^3*x + 1
/6*(-e^2*x^2 + d^2)^(5/2)*d*x + 1/7*(-e^2*x^2 + d^2)^(7/2)/e

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Fricas [A]  time = 0.230232, size = 682, normalized size = 5.5 \[ -\frac{48 \, e^{14} x^{14} - 56 \, d e^{13} x^{13} - 1344 \, d^{2} e^{12} x^{12} + 1582 \, d^{3} e^{11} x^{11} + 8736 \, d^{4} e^{10} x^{10} - 10605 \, d^{5} e^{9} x^{9} - 25536 \, d^{6} e^{8} x^{8} + 32767 \, d^{7} e^{7} x^{7} + 39648 \, d^{8} e^{6} x^{6} - 53816 \, d^{9} e^{5} x^{5} - 32256 \, d^{10} e^{4} x^{4} + 44912 \, d^{11} e^{3} x^{3} + 10752 \, d^{12} e^{2} x^{2} - 14784 \, d^{13} e x + 210 \,{\left (7 \, d^{8} e^{6} x^{6} - 56 \, d^{10} e^{4} x^{4} + 112 \, d^{12} e^{2} x^{2} - 64 \, d^{14} -{\left (d^{7} e^{6} x^{6} - 24 \, d^{9} e^{4} x^{4} + 80 \, d^{11} e^{2} x^{2} - 64 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 7 \,{\left (48 \, d e^{12} x^{12} - 56 \, d^{2} e^{11} x^{11} - 528 \, d^{3} e^{10} x^{10} + 630 \, d^{4} e^{9} x^{9} + 2064 \, d^{5} e^{8} x^{8} - 2583 \, d^{6} e^{7} x^{7} - 3936 \, d^{7} e^{6} x^{6} + 5272 \, d^{8} e^{5} x^{5} + 3840 \, d^{9} e^{4} x^{4} - 5360 \, d^{10} e^{3} x^{3} - 1536 \, d^{11} e^{2} x^{2} + 2112 \, d^{12} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{336 \,{\left (7 \, d e^{7} x^{6} - 56 \, d^{3} e^{5} x^{4} + 112 \, d^{5} e^{3} x^{2} - 64 \, d^{7} e -{\left (e^{7} x^{6} - 24 \, d^{2} e^{5} x^{4} + 80 \, d^{4} e^{3} x^{2} - 64 \, d^{6} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/336*(48*e^14*x^14 - 56*d*e^13*x^13 - 1344*d^2*e^12*x^12 + 1582*d^3*e^11*x^11
+ 8736*d^4*e^10*x^10 - 10605*d^5*e^9*x^9 - 25536*d^6*e^8*x^8 + 32767*d^7*e^7*x^7
+ 39648*d^8*e^6*x^6 - 53816*d^9*e^5*x^5 - 32256*d^10*e^4*x^4 + 44912*d^11*e^3*x
^3 + 10752*d^12*e^2*x^2 - 14784*d^13*e*x + 210*(7*d^8*e^6*x^6 - 56*d^10*e^4*x^4
+ 112*d^12*e^2*x^2 - 64*d^14 - (d^7*e^6*x^6 - 24*d^9*e^4*x^4 + 80*d^11*e^2*x^2 -
64*d^13)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 7*(4
8*d*e^12*x^12 - 56*d^2*e^11*x^11 - 528*d^3*e^10*x^10 + 630*d^4*e^9*x^9 + 2064*d^
5*e^8*x^8 - 2583*d^6*e^7*x^7 - 3936*d^7*e^6*x^6 + 5272*d^8*e^5*x^5 + 3840*d^9*e^
4*x^4 - 5360*d^10*e^3*x^3 - 1536*d^11*e^2*x^2 + 2112*d^12*e*x)*sqrt(-e^2*x^2 + d
^2))/(7*d*e^7*x^6 - 56*d^3*e^5*x^4 + 112*d^5*e^3*x^2 - 64*d^7*e - (e^7*x^6 - 24*
d^2*e^5*x^4 + 80*d^4*e^3*x^2 - 64*d^6*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 48.1925, size = 813, normalized size = 6.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**5*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2))
+ I*e**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**2*
asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) - d**4*e*Piecewise((x
**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - 2
*d**3*e**2*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 +
e**2*x**2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*
sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d)/(8*e**3)
- d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**
2)) - e**2*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) + 2*d**2*e**3*Piecewise((
-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*
e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) + d
*e**4*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e*
*2*x**2/d**2)) - I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(2
4*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**2*x**2/d**2)), Abs(
e**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**
2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt
(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2)), True)) - e**5*
Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**2 - e
**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d*
*2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError