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

Optimal. Leaf size=68 \[ -\frac{(d-e x) (d+e x)^{m+1} \, _2F_1\left (1,m-5;m-\frac{3}{2};\frac{d+e x}{2 d}\right )}{d e (5-2 m) \left (d^2-e^2 x^2\right )^{7/2}} \]

[Out]

-(((d - e*x)*(d + e*x)^(1 + m)*Hypergeometric2F1[1, -5 + m, -3/2 + m, (d + e*x)/
(2*d)])/(d*e*(5 - 2*m)*(d^2 - e^2*x^2)^(7/2)))

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Rubi [A]  time = 0.162846, antiderivative size = 83, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2^{m-\frac{5}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{5}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-5/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(5/2 - m)*Hypergeometric2F1[-5/2, 7/2 -
m, -3/2, (d - e*x)/(2*d)])/(5*d*e*(d^2 - e^2*x^2)^(5/2))

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Rubi in Sympy [A]  time = 24.3192, size = 83, normalized size = 1.22 \[ \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m + \frac{7}{2}, - \frac{5}{2} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{40 d^{4} e \left (d - e x\right )^{3} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(-e**2*x**2+d**2)**(7/2),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 + 7/2, -5/2), (-3/2,), (d/2 - e*x/2)/d)/(40*d**4*e*(d - e*x)**3*sqrt(d + e*x
))

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Mathematica [A]  time = 0.106014, size = 91, normalized size = 1.34 \[ \frac{2^{m-\frac{5}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-5/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(1/2 - m)*Hypergeometric2F1[-5/2, 7/2 -
m, -3/2, (d - e*x)/(2*d)])/(5*d^3*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^m/(-e^2*x^2+d^2)^(7/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (e x + d\right )}^{m}}{{\left (e^{6} x^{6} - 3 \, d^{2} e^{4} x^{4} + 3 \, d^{4} e^{2} x^{2} - d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(e*x + d)^m/((e^6*x^6 - 3*d^2*e^4*x^4 + 3*d^4*e^2*x^2 - d^6)*sqrt(-e^2
*x^2 + d^2)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(7/2), x)