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

Optimal. Leaf size=60 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m-3;m-\frac{1}{2};\frac{d+e x}{2 d}\right )}{d e (3-2 m) \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

-(((d + e*x)^m*Hypergeometric2F1[1, -3 + m, -1/2 + m, (d + e*x)/(2*d)])/(d*e*(3
- 2*m)*(d^2 - e^2*x^2)^(3/2)))

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Rubi [A]  time = 0.159863, antiderivative size = 83, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2^{m-\frac{3}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{3}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{d-e x}{2 d}\right )}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-3/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(3/2 - m)*Hypergeometric2F1[-3/2, 5/2 -
m, -1/2, (d - e*x)/(2*d)])/(3*d*e*(d^2 - e^2*x^2)^(3/2))

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Rubi in Sympy [A]  time = 24.357, size = 83, normalized size = 1.38 \[ \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m + \frac{5}{2}, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{12 d^{3} e \left (d - e x\right )^{2} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((d/2 + e*x/2)/d)**(-m - 1/2)*(d + e*x)**(m + 1/2)*sqrt(d**2 - e**2*x**2)*hyper(
(-m + 5/2, -3/2), (-1/2,), (d/2 - e*x/2)/d)/(12*d**3*e*(d - e*x)**2*sqrt(d + e*x
))

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Mathematica [A]  time = 0.133069, size = 92, normalized size = 1.53 \[ \frac{2^{m-\frac{3}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{d-e x}{2 d}\right )}{\left (3 d^3 e-3 d^2 e^2 x\right ) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-3/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(1/2 - m)*Hypergeometric2F1[-3/2, 5/2 -
m, -1/2, (d - e*x)/(2*d)])/((3*d^3*e - 3*d^2*e^2*x)*Sqrt[d^2 - e^2*x^2])

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^m/(-e^2*x^2+d^2)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{{\left (e^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^m/((e^4*x^4 - 2*d^2*e^2*x^2 + d^4)*sqrt(-e^2*x^2 + d^2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(5/2), x)