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

Optimal. Leaf size=206 \[ -\frac{\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}+\frac{d^6 (2 m+9) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{2 d^5 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

[Out]

-(((g*x)^(1 + m)*(d^2 - e^2*x^2)^(7/2))/(g*(8 + m))) + (d^6*(9 + 2*m)*(g*x)^(1 +
m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/
d^2])/(g*(1 + m)*(8 + m)*Sqrt[1 - (e^2*x^2)/d^2]) + (2*d^5*e*(g*x)^(2 + m)*Sqrt[
d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(g^
2*(2 + m)*Sqrt[1 - (e^2*x^2)/d^2])

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Rubi [A]  time = 0.423761, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}+\frac{d^6 (2 m+9) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{2 d^5 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-(((g*x)^(1 + m)*(d^2 - e^2*x^2)^(7/2))/(g*(8 + m))) + (d^6*(9 + 2*m)*(g*x)^(1 +
m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/
d^2])/(g*(1 + m)*(8 + m)*Sqrt[1 - (e^2*x^2)/d^2]) + (2*d^5*e*(g*x)^(2 + m)*Sqrt[
d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(g^
2*(2 + m)*Sqrt[1 - (e^2*x^2)/d^2])

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Rubi in Sympy [A]  time = 52.6554, size = 214, normalized size = 1.04 \[ \frac{d^{6} \left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} + \frac{2 d^{5} e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} + \frac{d^{4} e^{2} \left (g x\right )^{m + 3} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{3} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**6*(g*x)**(m + 1)*sqrt(d**2 - e**2*x**2)*hyper((-5/2, m/2 + 1/2), (m/2 + 3/2,)
, e**2*x**2/d**2)/(g*sqrt(1 - e**2*x**2/d**2)*(m + 1)) + 2*d**5*e*(g*x)**(m + 2)
*sqrt(d**2 - e**2*x**2)*hyper((-5/2, m/2 + 1), (m/2 + 2,), e**2*x**2/d**2)/(g**2
*sqrt(1 - e**2*x**2/d**2)*(m + 2)) + d**4*e**2*(g*x)**(m + 3)*sqrt(d**2 - e**2*x
**2)*hyper((-5/2, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2/d**2)/(g**3*sqrt(1 - e**2*
x**2/d**2)*(m + 3))

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Mathematica [A]  time = 0.555276, size = 335, normalized size = 1.63 \[ \frac{d^2 x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (\frac{e^6 x^6 \, _2F_1\left (-\frac{1}{2},\frac{m+7}{2};\frac{m+9}{2};\frac{e^2 x^2}{d^2}\right )}{m+7}+\frac{2 d e^5 x^5 \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+3;\frac{m}{2}+4;\frac{e^2 x^2}{d^2}\right )}{m+6}-\frac{d^2 e^4 x^4 \, _2F_1\left (-\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\frac{e^2 x^2}{d^2}\right )}{m+5}+\frac{d^6 \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}+\frac{2 d^5 e x \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )}{m+2}-\frac{d^4 e^2 x^2 \, _2F_1\left (-\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}-\frac{4 d^3 e^3 x^3 \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+2;\frac{m}{2}+3;\frac{e^2 x^2}{d^2}\right )}{m+4}\right )}{\sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*((2*d^5*e*x*Hypergeometric2F1[-1/2, 1 + m
/2, 2 + m/2, (e^2*x^2)/d^2])/(2 + m) - (4*d^3*e^3*x^3*Hypergeometric2F1[-1/2, 2
+ m/2, 3 + m/2, (e^2*x^2)/d^2])/(4 + m) + (2*d*e^5*x^5*Hypergeometric2F1[-1/2, 3
+ m/2, 4 + m/2, (e^2*x^2)/d^2])/(6 + m) + (d^6*Hypergeometric2F1[-1/2, (1 + m)/
2, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) - (d^4*e^2*x^2*Hypergeometric2F1[-1/2, (3
+ m)/2, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) - (d^2*e^4*x^4*Hypergeometric2F1[-1/2
, (5 + m)/2, (7 + m)/2, (e^2*x^2)/d^2])/(5 + m) + (e^6*x^6*Hypergeometric2F1[-1/
2, (7 + m)/2, (9 + m)/2, (e^2*x^2)/d^2])/(7 + m)))/Sqrt[d^2 - e^2*x^2]

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} + 2 \, d^{5} e x + d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^6*x^6 + 2*d*e^5*x^5 - d^2*e^4*x^4 - 4*d^3*e^3*x^3 - d^4*e^2*x^2 + 2*
d^5*e*x + d^6)*sqrt(-e^2*x^2 + d^2)*(g*x)^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((g*x)**m*(e*x+d)**2*(-e**2*x**2+d**2)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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