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

Optimal. Leaf size=59 \[ \frac{\left (d^2-e^2 x^2\right )^{9/2} (d+e x)^m \, _2F_1\left (1,m+9;m+\frac{11}{2};\frac{d+e x}{2 d}\right )}{d e (2 m+9)} \]

[Out]

((d + e*x)^m*(d^2 - e^2*x^2)^(9/2)*Hypergeometric2F1[1, 9 + m, 11/2 + m, (d + e*
x)/(2*d)])/(d*e*(9 + 2*m))

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Rubi [A]  time = 0.146824, 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 \[ -\frac{2^{m+\frac{9}{2}} \left (d^2-e^2 x^2\right )^{9/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{9}{2}} \, _2F_1\left (\frac{9}{2},-m-\frac{7}{2};\frac{11}{2};\frac{d-e x}{2 d}\right )}{9 d e} \]

Antiderivative was successfully verified.

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

[Out]

-(2^(9/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-9/2 - m)*(d^2 - e^2*x^2)^(9/2)*Hyperge
ometric2F1[9/2, -7/2 - m, 11/2, (d - e*x)/(2*d)])/(9*d*e)

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Rubi in Sympy [A]  time = 25.1323, size = 85, normalized size = 1.44 \[ - \frac{16 d^{3} \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d - e x\right )^{4} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{7}{2}, \frac{9}{2} \\ \frac{11}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{9 e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*d**3*((d/2 + e*x/2)/d)**(-m - 1/2)*(d - e*x)**4*(d + e*x)**(m + 1/2)*sqrt(d*
*2 - e**2*x**2)*hyper((-m - 7/2, 9/2), (11/2,), (d/2 - e*x/2)/d)/(9*e*sqrt(d + e
*x))

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Mathematica [C]  time = 4.94665, size = 531, normalized size = 9. \[ \frac{2 d \sqrt{d-e x} (d+e x)^m \left (-\frac{84 d^4 e^3 x^3 \sqrt{d+e x} F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )}{8 d F_1\left (3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right )+e x \left ((2 m+1) F_1\left (4;-\frac{1}{2},\frac{1}{2}-m;5;\frac{e x}{d},-\frac{e x}{d}\right )-F_1\left (4;\frac{1}{2},-m-\frac{1}{2};5;\frac{e x}{d},-\frac{e x}{d}\right )\right )}+\frac{378 d^2 e^5 x^5 \sqrt{d+e x} F_1\left (5;-\frac{1}{2},-m-\frac{1}{2};6;\frac{e x}{d},-\frac{e x}{d}\right )}{60 d F_1\left (5;-\frac{1}{2},-m-\frac{1}{2};6;\frac{e x}{d},-\frac{e x}{d}\right )+5 e x \left ((2 m+1) F_1\left (6;-\frac{1}{2},\frac{1}{2}-m;7;\frac{e x}{d},-\frac{e x}{d}\right )-F_1\left (6;\frac{1}{2},-m-\frac{1}{2};7;\frac{e x}{d},-\frac{e x}{d}\right )\right )}-\frac{24 e^7 x^7 \sqrt{d+e x} F_1\left (7;-\frac{1}{2},-m-\frac{1}{2};8;\frac{e x}{d},-\frac{e x}{d}\right )}{16 d F_1\left (7;-\frac{1}{2},-m-\frac{1}{2};8;\frac{e x}{d},-\frac{e x}{d}\right )+e x \left ((2 m+1) F_1\left (8;-\frac{1}{2},\frac{1}{2}-m;9;\frac{e x}{d},-\frac{e x}{d}\right )-F_1\left (8;\frac{1}{2},-m-\frac{1}{2};9;\frac{e x}{d},-\frac{e x}{d}\right )\right )}-7 d^5 2^{m+\frac{1}{2}} \sqrt{d-e x} \sqrt{d^2-e^2 x^2} \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )\right )}{21 e} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*d*Sqrt[d - e*x]*(d + e*x)^m*((-84*d^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)]
- AppellF1[4, 1/2, -1/2 - m, 5, (e*x)/d, -((e*x)/d)])) + (378*d^2*e^5*x^5*Sqrt[
d + e*x]*AppellF1[5, -1/2, -1/2 - m, 6, (e*x)/d, -((e*x)/d)])/(60*d*AppellF1[5,
-1/2, -1/2 - m, 6, (e*x)/d, -((e*x)/d)] + 5*e*x*((1 + 2*m)*AppellF1[6, -1/2, 1/2
- m, 7, (e*x)/d, -((e*x)/d)] - AppellF1[6, 1/2, -1/2 - m, 7, (e*x)/d, -((e*x)/d
)])) - (24*e^7*x^7*Sqrt[d + e*x]*AppellF1[7, -1/2, -1/2 - m, 8, (e*x)/d, -((e*x)
/d)])/(16*d*AppellF1[7, -1/2, -1/2 - m, 8, (e*x)/d, -((e*x)/d)] + e*x*((1 + 2*m)
*AppellF1[8, -1/2, 1/2 - m, 9, (e*x)/d, -((e*x)/d)] - AppellF1[8, 1/2, -1/2 - m,
9, (e*x)/d, -((e*x)/d)])) - 7*2^(1/2 + m)*d^5*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)]
))/(21*e)

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Maple [F]  time = 0.056, 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. \[ \int{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(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)*(
e*x + d)^m, 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. \[ \int{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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