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

Optimal. Leaf size=162 \[ \frac{d^5 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^4 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

[Out]

(d^5*(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)*Sqrt[1 - (e^2*x^2)/d^2]) + (d^4*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 = 0.224537, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{d^5 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^4 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(d^5*(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)*Sqrt[1 - (e^2*x^2)/d^2]) + (d^4*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 = 26.5831, size = 138, normalized size = 0.85 \[ \frac{d^{5} \left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} + \frac{d^{4} e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**5*(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)) + d**4*e*(g*x)**(m + 2)*s
qrt(d**2 - e**2*x**2)*hyper((-5/2, m/2 + 1), (m/2 + 2,), e**2*x**2/d**2)/(g**2*s
qrt(1 - e**2*x**2/d**2)*(m + 2))

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Mathematica [A]  time = 0.396327, size = 289, normalized size = 1.78 \[ \frac{d^2 x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (\frac{e^5 x^5 \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+3;\frac{m}{2}+4;\frac{e^2 x^2}{d^2}\right )}{m+6}+\frac{d e^4 x^4 \, _2F_1\left (-\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\frac{e^2 x^2}{d^2}\right )}{m+5}-\frac{2 d^2 e^3 x^3 \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+2;\frac{m}{2}+3;\frac{e^2 x^2}{d^2}\right )}{m+4}+\frac{d^5 \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}+\frac{d^4 e x \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )}{m+2}-\frac{2 d^3 e^2 x^2 \, _2F_1\left (-\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}\right )}{\sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^5*x^5 + d*e^4*x^4 - 2*d^2*e^3*x^3 - 2*d^3*e^2*x^2 + d^4*e*x + d^5)*s
qrt(-e^2*x^2 + d^2)*(g*x)^m, x)

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Sympy [A]  time = 176.499, size = 374, normalized size = 2.31 \[ \frac{d^{6} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{d^{5} e g^{m} x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} - \frac{d^{4} e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} - \frac{d^{3} e^{3} g^{m} x^{4} x^{m} \Gamma \left (\frac{m}{2} + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac{m}{2} + 3\right )} + \frac{d^{2} e^{4} g^{m} x^{5} x^{m} \Gamma \left (\frac{m}{2} + \frac{5}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{5}{2} \\ \frac{m}{2} + \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{d e^{5} g^{m} x^{6} x^{m} \Gamma \left (\frac{m}{2} + 3\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 3 \\ \frac{m}{2} + 4 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**6*g**m*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), e**2*x*
*2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 3/2)) + d**5*e*g**m*x**2*x**m*gamma(m/
2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*g
amma(m/2 + 2)) - d**4*e**2*g**m*x**3*x**m*gamma(m/2 + 3/2)*hyper((-1/2, m/2 + 3/
2), (m/2 + 5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/gamma(m/2 + 5/2) - d**3*e**3
*g**m*x**4*x**m*gamma(m/2 + 2)*hyper((-1/2, m/2 + 2), (m/2 + 3,), e**2*x**2*exp_
polar(2*I*pi)/d**2)/gamma(m/2 + 3) + d**2*e**4*g**m*x**5*x**m*gamma(m/2 + 5/2)*h
yper((-1/2, m/2 + 5/2), (m/2 + 7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma
(m/2 + 7/2)) + d*e**5*g**m*x**6*x**m*gamma(m/2 + 3)*hyper((-1/2, m/2 + 3), (m/2
+ 4,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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