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

Optimal. Leaf size=216 \[ \frac{2 (d+e x) (g x)^{m+1}}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 e (3-m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^5 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(3-2 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 g (m+1) \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.426606, antiderivative size = 216, 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 \[ \frac{2 (d+e x) (g x)^{m+1}}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 e (3-m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^5 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(3-2 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 g (m+1) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.2686, size = 209, normalized size = 0.97 \[ \frac{\left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{6} g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} + \frac{2 e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{7} g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} + \frac{e^{2} \left (g x\right )^{m + 3} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{8} g^{3} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 3\right )} \]

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)**(7/2),x)

[Out]

(g*x)**(m + 1)*sqrt(d**2 - e**2*x**2)*hyper((7/2, m/2 + 1/2), (m/2 + 3/2,), e**2
*x**2/d**2)/(d**6*g*sqrt(1 - e**2*x**2/d**2)*(m + 1)) + 2*e*(g*x)**(m + 2)*sqrt(
d**2 - e**2*x**2)*hyper((7/2, m/2 + 1), (m/2 + 2,), e**2*x**2/d**2)/(d**7*g**2*s
qrt(1 - e**2*x**2/d**2)*(m + 2)) + e**2*(g*x)**(m + 3)*sqrt(d**2 - e**2*x**2)*hy
per((7/2, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2/d**2)/(d**8*g**3*sqrt(1 - e**2*x**
2/d**2)*(m + 3))

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Mathematica [A]  time = 0.216172, size = 178, normalized size = 0.82 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (2 d e \left (m^2+4 m+3\right ) x \, _2F_1\left (\frac{7}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )+(m+2) \left (d^2 (m+3) \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )+e^2 (m+1) x^2 \, _2F_1\left (\frac{7}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )\right )\right )}{d^6 (m+1) (m+2) (m+3) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{ \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \]

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)^(7/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(7/2),x)

[Out]

Integral((g*x)**m*(d + e*x)**2/(-(-d + e*x)*(d + e*x))**(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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