3.232 \(\int \frac{(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=250 \[ -\frac{d^2 e (4 m+11) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+3) \sqrt{d^2-e^2 x^2}}+\frac{e \sqrt{d^2-e^2 x^2} (g x)^{m+2}}{g^2 (m+3)}-\frac{3 d \sqrt{d^2-e^2 x^2} (g x)^{m+1}}{g (m+2)}+\frac{d^3 (4 m+5) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+2) \sqrt{d^2-e^2 x^2}} \]

[Out]

(-3*d*(g*x)^(1 + m)*Sqrt[d^2 - e^2*x^2])/(g*(2 + m)) + (e*(g*x)^(2 + m)*Sqrt[d^2
- e^2*x^2])/(g^2*(3 + m)) + (d^3*(5 + 4*m)*(g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2
]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(g*(1 + m)*(2 + m
)*Sqrt[d^2 - e^2*x^2]) - (d^2*e*(11 + 4*m)*(g*x)^(2 + m)*Sqrt[1 - (e^2*x^2)/d^2]
*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(g^2*(2 + m)*(3 +
m)*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.736592, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{d^2 e (4 m+11) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+3) \sqrt{d^2-e^2 x^2}}+\frac{e \sqrt{d^2-e^2 x^2} (g x)^{m+2}}{g^2 (m+3)}-\frac{3 d \sqrt{d^2-e^2 x^2} (g x)^{m+1}}{g (m+2)}+\frac{d^3 (4 m+5) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+2) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((g*x)^m*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(-3*d*(g*x)^(1 + m)*Sqrt[d^2 - e^2*x^2])/(g*(2 + m)) + (e*(g*x)^(2 + m)*Sqrt[d^2
- e^2*x^2])/(g^2*(3 + m)) + (d^3*(5 + 4*m)*(g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2
]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(g*(1 + m)*(2 + m
)*Sqrt[d^2 - e^2*x^2]) - (d^2*e*(11 + 4*m)*(g*x)^(2 + m)*Sqrt[1 - (e^2*x^2)/d^2]
*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(g^2*(2 + m)*(3 +
m)*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 113.819, size = 274, normalized size = 1.1 \[ \frac{d \left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} - \frac{3 e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} + \frac{3 e^{2} \left (g x\right )^{m + 3} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d g^{3} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 3\right )} - \frac{e^{3} \left (g x\right )^{m + 4} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{2} g^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**3,x)

[Out]

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

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Mathematica [C]  time = 1.27142, size = 272, normalized size = 1.09 \[ \frac{x (g x)^m \left (\frac{8 d^3 (m+2)^2 \sqrt{d-e x} F_1\left (m+1;-\frac{1}{2},\frac{1}{2};m+2;\frac{e x}{d},-\frac{e x}{d}\right )}{\sqrt{d+e x} \left (2 d (m+2) F_1\left (m+1;-\frac{1}{2},\frac{1}{2};m+2;\frac{e x}{d},-\frac{e x}{d}\right )-e x \left (\, _2F_1\left (\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )+F_1\left (m+2;-\frac{1}{2},\frac{3}{2};m+3;\frac{e x}{d},-\frac{e x}{d}\right )\right )\right )}+\frac{\sqrt{d^2-e^2 x^2} \left (e (m+1) x \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )-3 d (m+2) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )\right )}{\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{(m+1) (m+2)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((g*x)^m*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(x*(g*x)^m*((Sqrt[d^2 - e^2*x^2]*(e*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2
+ m/2, (e^2*x^2)/d^2] - 3*d*(2 + m)*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/
2, (e^2*x^2)/d^2]))/Sqrt[1 - (e^2*x^2)/d^2] + (8*d^3*(2 + m)^2*Sqrt[d - e*x]*App
ellF1[1 + m, -1/2, 1/2, 2 + m, (e*x)/d, -((e*x)/d)])/(Sqrt[d + e*x]*(2*d*(2 + m)
*AppellF1[1 + m, -1/2, 1/2, 2 + m, (e*x)/d, -((e*x)/d)] - e*x*(AppellF1[2 + m, -
1/2, 3/2, 3 + m, (e*x)/d, -((e*x)/d)] + HypergeometricPFQ[{1/2, 1 + m/2}, {2 + m
/2}, (e^2*x^2)/d^2])))))/((1 + m)*(2 + m))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m}}{ \left ( ex+d \right ) ^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^3,x)

[Out]

int((g*x)^m*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m/(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m/(e*x + d)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m/(e*x + d)^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m/(e*x + d)^3,x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m/(e*x + d)^3, x)