3.840 \(\int \frac{(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0818264, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.4687, size = 51, normalized size = 0.76 \[ \frac{\left (d + e x\right )^{4}}{3 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{\left (d + e x\right )^{5}}{15 d^{2} e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

(d + e*x)**4/(3*d*e*(d**2 - e**2*x**2)**(5/2)) - (d + e*x)**5/(15*d**2*e*(d**2 -
e**2*x**2)**(5/2))

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Mathematica [A]  time = 0.0337796, size = 53, normalized size = 0.79 \[ \frac{\sqrt{d^2-e^2 x^2} \left (4 d^2+3 d e x-e^2 x^2\right )}{15 d^2 e (d-e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(4*d^2 + 3*d*e*x - e^2*x^2))/(15*d^2*e*(d - e*x)^3)

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Maple [A]  time = 0.01, size = 44, normalized size = 0.7 \[{\frac{ \left ( ex+d \right ) ^{5} \left ( -ex+d \right ) \left ( -ex+4\,d \right ) }{15\,{d}^{2}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

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

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Maxima [A]  time = 0.730808, size = 166, normalized size = 2.48 \[ \frac{e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{11 \, d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} - \frac{x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

1/2*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) + 4/3*d*e*x^2/(-e^2*x^2 + d^2)^(5/2) + 11/10*
d^2*x/(-e^2*x^2 + d^2)^(5/2) + 4/15*d^3/((-e^2*x^2 + d^2)^(5/2)*e) - 1/30*x/(-e^
2*x^2 + d^2)^(3/2) - 1/15*x/(sqrt(-e^2*x^2 + d^2)*d^2)

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Fricas [A]  time = 0.226753, size = 271, normalized size = 4.04 \[ \frac{3 \, e^{4} x^{5} - 20 \, d e^{3} x^{4} + 35 \, d^{2} e^{2} x^{3} + 30 \, d^{3} e x^{2} - 60 \, d^{4} x + 5 \,{\left (e^{3} x^{4} - d e^{2} x^{3} - 6 \, d^{2} e x^{2} + 12 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{5} x^{5} - 5 \, d^{3} e^{4} x^{4} + 5 \, d^{4} e^{3} x^{3} + 5 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x + 4 \, d^{7} +{\left (d^{2} e^{4} x^{4} - 7 \, d^{4} e^{2} x^{2} + 10 \, d^{5} e x - 4 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(3*e^4*x^5 - 20*d*e^3*x^4 + 35*d^2*e^2*x^3 + 30*d^3*e*x^2 - 60*d^4*x + 5*(e
^3*x^4 - d*e^2*x^3 - 6*d^2*e*x^2 + 12*d^3*x)*sqrt(-e^2*x^2 + d^2))/(d^2*e^5*x^5
- 5*d^3*e^4*x^4 + 5*d^4*e^3*x^3 + 5*d^5*e^2*x^2 - 10*d^6*e*x + 4*d^7 + (d^2*e^4*
x^4 - 7*d^4*e^2*x^2 + 10*d^5*e*x - 4*d^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**4/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.230313, size = 95, normalized size = 1.42 \[ -\frac{{\left (4 \, d^{3} e^{\left (-1\right )} +{\left (15 \, d^{2} -{\left (x{\left (\frac{x^{2} e^{4}}{d^{2}} - 10 \, e^{2}\right )} - 20 \, d e\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-1/15*(4*d^3*e^(-1) + (15*d^2 - (x*(x^2*e^4/d^2 - 10*e^2) - 20*d*e)*x)*x)*sqrt(-
x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3