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

Optimal. Leaf size=103 \[ \frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)} \]

[Out]

Sqrt[d^2 - e^2*x^2]/(5*d*e*(d - e*x)^3) + (2*Sqrt[d^2 - e^2*x^2])/(15*d^2*e*(d -
e*x)^2) + (2*Sqrt[d^2 - e^2*x^2])/(15*d^3*e*(d - e*x))

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Rubi [A]  time = 0.136913, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d^2 - e^2*x^2]/(5*d*e*(d - e*x)^3) + (2*Sqrt[d^2 - e^2*x^2])/(15*d^2*e*(d -
e*x)^2) + (2*Sqrt[d^2 - e^2*x^2])/(15*d^3*e*(d - e*x))

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Rubi in Sympy [A]  time = 21.2142, size = 80, normalized size = 0.78 \[ \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d e \left (d - e x\right )^{3}} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{2} e \left (d - e x\right )^{2}} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{3} e \left (d - e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(d**2 - e**2*x**2)/(5*d*e*(d - e*x)**3) + 2*sqrt(d**2 - e**2*x**2)/(15*d**2*
e*(d - e*x)**2) + 2*sqrt(d**2 - e**2*x**2)/(15*d**3*e*(d - e*x))

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Mathematica [A]  time = 0.0425405, size = 53, normalized size = 0.51 \[ \frac{\sqrt{d^2-e^2 x^2} \left (7 d^2-6 d e x+2 e^2 x^2\right )}{15 d^3 e (d-e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(7*d^2 - 6*d*e*x + 2*e^2*x^2))/(15*d^3*e*(d - e*x)^3)

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Maple [A]  time = 0., size = 55, normalized size = 0.5 \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{4} \left ( 2\,{e}^{2}{x}^{2}-6\,dex+7\,{d}^{2} \right ) }{15\,{d}^{3}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(e*x+d)^4*(2*e^2*x^2-6*d*e*x+7*d^2)/d^3/e/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.694023, size = 136, normalized size = 1.32 \[ \frac{e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{2 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e*x^2/(-e^2*x^2 + d^2)^(5/2) + 4/5*d*x/(-e^2*x^2 + d^2)^(5/2) + 7/15*d^2/((-
e^2*x^2 + d^2)^(5/2)*e) + 1/15*x/((-e^2*x^2 + d^2)^(3/2)*d) + 2/15*x/(sqrt(-e^2*
x^2 + d^2)*d^3)

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Fricas [A]  time = 0.28108, size = 271, normalized size = 2.63 \[ \frac{9 \, e^{4} x^{5} - 35 \, d e^{3} x^{4} + 20 \, d^{2} e^{2} x^{3} + 60 \, d^{3} e x^{2} - 60 \, d^{4} x + 5 \,{\left (e^{3} x^{4} + 2 \, d e^{2} x^{3} - 12 \, d^{2} e x^{2} + 12 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{5} x^{5} - 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} + 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x + 4 \, d^{8} +{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} + 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(9*e^4*x^5 - 35*d*e^3*x^4 + 20*d^2*e^2*x^3 + 60*d^3*e*x^2 - 60*d^4*x + 5*(e
^3*x^4 + 2*d*e^2*x^3 - 12*d^2*e*x^2 + 12*d^3*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^5*x
^5 - 5*d^4*e^4*x^4 + 5*d^5*e^3*x^3 + 5*d^6*e^2*x^2 - 10*d^7*e*x + 4*d^8 + (d^3*e
^4*x^4 - 7*d^5*e^2*x^2 + 10*d^6*e*x - 4*d^7)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.298221, size = 95, normalized size = 0.92 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (7 \, d^{2} e^{\left (-1\right )} +{\left ({\left (x{\left (\frac{2 \, x^{2} e^{4}}{d^{3}} - \frac{5 \, e^{2}}{d}\right )} + 5 \, e\right )} x + 15 \, d\right )} x\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*(7*d^2*e^(-1) + ((x*(2*x^2*e^4/d^3 - 5*e^2/d) + 5*e)*
x + 15*d)*x)/(x^2*e^2 - d^2)^3