3.239 \(\int \frac{(g x)^m}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac{4 (d-e x) (g x)^{m+1}}{11 g \left (d^2-e^2 x^2\right )^{11/2}}-\frac{e (25-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^{10} g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^9 g (m+1) \sqrt{d^2-e^2 x^2}} \]

[Out]

(4*(g*x)^(1 + m)*(d - e*x))/(11*g*(d^2 - e^2*x^2)^(11/2)) + ((7 - 4*m)*(g*x)^(1
+ m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[11/2, (1 + m)/2, (3 + m)/2, (e^2*
x^2)/d^2])/(11*d^9*g*(1 + m)*Sqrt[d^2 - e^2*x^2]) - (e*(25 - 4*m)*(g*x)^(2 + m)*
Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[11/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/
d^2])/(11*d^10*g^2*(2 + m)*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.506485, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{4 (d-e x) (g x)^{m+1}}{11 g \left (d^2-e^2 x^2\right )^{11/2}}-\frac{e (25-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^{10} g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^9 g (m+1) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(g*x)^m/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(4*(g*x)^(1 + m)*(d - e*x))/(11*g*(d^2 - e^2*x^2)^(11/2)) + ((7 - 4*m)*(g*x)^(1
+ m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[11/2, (1 + m)/2, (3 + m)/2, (e^2*
x^2)/d^2])/(11*d^9*g*(1 + m)*Sqrt[d^2 - e^2*x^2]) - (e*(25 - 4*m)*(g*x)^(2 + m)*
Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[11/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/
d^2])/(11*d^10*g^2*(2 + m)*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 115.693, size = 280, normalized size = 1.31 \[ \frac{\left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{13}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{11} g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} - \frac{3 e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{13}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{12} g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} + \frac{3 e^{2} \left (g x\right )^{m + 3} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{13}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{13} g^{3} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 3\right )} - \frac{e^{3} \left (g x\right )^{m + 4} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{13}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{14} g^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m/(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

(g*x)**(m + 1)*sqrt(d**2 - e**2*x**2)*hyper((13/2, m/2 + 1/2), (m/2 + 3/2,), e**
2*x**2/d**2)/(d**11*g*sqrt(1 - e**2*x**2/d**2)*(m + 1)) - 3*e*(g*x)**(m + 2)*sqr
t(d**2 - e**2*x**2)*hyper((13/2, m/2 + 1), (m/2 + 2,), e**2*x**2/d**2)/(d**12*g*
*2*sqrt(1 - e**2*x**2/d**2)*(m + 2)) + 3*e**2*(g*x)**(m + 3)*sqrt(d**2 - e**2*x*
*2)*hyper((13/2, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2/d**2)/(d**13*g**3*sqrt(1 -
e**2*x**2/d**2)*(m + 3)) - e**3*(g*x)**(m + 4)*sqrt(d**2 - e**2*x**2)*hyper((13/
2, m/2 + 2), (m/2 + 3,), e**2*x**2/d**2)/(d**14*g**4*sqrt(1 - e**2*x**2/d**2)*(m
+ 4))

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Mathematica [C]  time = 1.04234, size = 157, normalized size = 0.73 \[ \frac{2 d (m+2) x (g x)^m F_1\left (m+1;\frac{7}{2},\frac{13}{2};m+2;\frac{e x}{d},-\frac{e x}{d}\right )}{(m+1) (d-e x)^{7/2} (d+e x)^{13/2} \left (2 d (m+2) F_1\left (m+1;\frac{7}{2},\frac{13}{2};m+2;\frac{e x}{d},-\frac{e x}{d}\right )+e x \left (7 F_1\left (m+2;\frac{9}{2},\frac{13}{2};m+3;\frac{e x}{d},-\frac{e x}{d}\right )-13 F_1\left (m+2;\frac{7}{2},\frac{15}{2};m+3;\frac{e x}{d},-\frac{e x}{d}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(g*x)^m/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(2*d*(2 + m)*x*(g*x)^m*AppellF1[1 + m, 7/2, 13/2, 2 + m, (e*x)/d, -((e*x)/d)])/(
(1 + m)*(d - e*x)^(7/2)*(d + e*x)^(13/2)*(2*d*(2 + m)*AppellF1[1 + m, 7/2, 13/2,
2 + m, (e*x)/d, -((e*x)/d)] + e*x*(-13*AppellF1[2 + m, 7/2, 15/2, 3 + m, (e*x)/
d, -((e*x)/d)] + 7*AppellF1[2 + m, 9/2, 13/2, 3 + m, (e*x)/d, -((e*x)/d)])))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m}}{ \left ( ex+d \right ) ^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m/(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

int((g*x)^m/(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)^m/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

integrate((g*x)^m/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\left (g x\right )^{m}}{{\left (e^{9} x^{9} + 3 \, d e^{8} x^{8} - 8 \, d^{3} e^{6} x^{6} - 6 \, d^{4} e^{5} x^{5} + 6 \, d^{5} e^{4} x^{4} + 8 \, d^{6} e^{3} x^{3} - 3 \, d^{8} e x - d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)^m/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

integral(-(g*x)^m/((e^9*x^9 + 3*d*e^8*x^8 - 8*d^3*e^6*x^6 - 6*d^4*e^5*x^5 + 6*d^
5*e^4*x^4 + 8*d^6*e^3*x^3 - 3*d^8*e*x - d^9)*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)**3/(-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. \[ \int \frac{\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)^m/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

integrate((g*x)^m/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^3), x)