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

Optimal. Leaf size=124 \[ \frac{e (g x)^{m+2} \, _2F_1\left (1,\frac{m-3}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(g x)^{m+1} \, _2F_1\left (1,\frac{m-4}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d g (m+1) \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.237815, antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{e \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^5 g (m+1) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.5333, size = 134, normalized size = 1.08 \[ \frac{\left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{7} g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} + \frac{e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{8} g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} \]

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)**(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**7*g*sqrt(1 - e**2*x**2/d**2)*(m + 1)) + 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**8*g**2*sqr
t(1 - e**2*x**2/d**2)*(m + 2))

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Mathematica [A]  time = 0.145272, size = 121, normalized size = 0.98 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (e (m+1) x \, _2F_1\left (\frac{7}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )+d (m+2) \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )\right )}{d^6 (m+1) (m+2) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{ \left ( gx \right ) ^{m} \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)*(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. \[{\rm integral}\left (-\frac{\left (g x\right )^{m}}{{\left (e^{5} x^{5} - d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} + 2 \, d^{3} e^{2} x^{2} + d^{4} e x - d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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