3.942 \(\int (d+e x)^m \left (d^2-e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=59 \[ \frac{\left (d^2-e^2 x^2\right )^{5/2} (d+e x)^m \, _2F_1\left (1,m+5;m+\frac{7}{2};\frac{d+e x}{2 d}\right )}{d e (2 m+5)} \]

[Out]

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

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Rubi [A]  time = 0.147306, antiderivative size = 83, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2^{m+\frac{5}{2}} \left (d^2-e^2 x^2\right )^{5/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{5}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{d-e x}{2 d}\right )}{5 d e} \]

Antiderivative was successfully verified.

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

[Out]

-(2^(5/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-5/2 - m)*(d^2 - e^2*x^2)^(5/2)*Hyperge
ometric2F1[5/2, -3/2 - m, 7/2, (d - e*x)/(2*d)])/(5*d*e)

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Rubi in Sympy [A]  time = 25.7722, size = 83, normalized size = 1.41 \[ - \frac{4 d \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d - e x\right )^{2} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{3}{2}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{5 e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*d*((d/2 + e*x/2)/d)**(-m - 1/2)*(d - e*x)**2*(d + e*x)**(m + 1/2)*sqrt(d**2 -
e**2*x**2)*hyper((-m - 3/2, 5/2), (7/2,), (d/2 - e*x/2)/d)/(5*e*sqrt(d + e*x))

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Mathematica [C]  time = 0.496324, size = 244, normalized size = 4.14 \[ \frac{2 d \sqrt{d-e x} (d+e x)^m \left (d \left (-2^{m+\frac{1}{2}}\right ) \sqrt{d-e x} \sqrt{d^2-e^2 x^2} \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )-\frac{4 e^3 x^3 \sqrt{d+e x} F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )}{8 d F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )+e x \left ((2 m+1) F_1\left (4;-\frac{1}{2},\frac{1}{2}-m;5;\frac{e x}{d},-\frac{e x}{d}\right )-F_1\left (4;\frac{1}{2},-m-\frac{1}{2};5;\frac{e x}{d},-\frac{e x}{d}\right )\right )}\right )}{3 e} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*d*Sqrt[d - e*x]*(d + e*x)^m*((-4*e^3*x^3*Sqrt[d + e*x]*AppellF1[3, -1/2, -1/2
- m, 4, (e*x)/d, -((e*x)/d)])/(8*d*AppellF1[3, -1/2, -1/2 - m, 4, (e*x)/d, -((e
*x)/d)] + e*x*((1 + 2*m)*AppellF1[4, -1/2, 1/2 - m, 5, (e*x)/d, -((e*x)/d)] - Ap
pellF1[4, 1/2, -1/2 - m, 5, (e*x)/d, -((e*x)/d)])) - 2^(1/2 + m)*d*Sqrt[d - e*x]
*(1 + (e*x)/d)^(-1/2 - m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5
/2, (d - e*x)/(2*d)]))/(3*e)

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^m*(-e^2*x^2+d^2)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^m, x)