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

Optimal. Leaf size=213 \[ \frac{4 (d+e x) (g x)^{m+1}}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e (7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(1-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3 g (m+1) \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.430359, antiderivative size = 213, 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 \[ \frac{4 (d+e x) (g x)^{m+1}}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e (7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(1-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3 g (m+1) \sqrt{d^2-e^2 x^2}} \]

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

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Rubi in Sympy [A]  time = 71.1337, size = 280, normalized size = 1.31 \[ \frac{\left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{5} g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} + \frac{3 e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{6} g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} + \frac{3 e^{2} \left (g x\right )^{m + 3} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{7} g^{3} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 3\right )} + \frac{e^{3} \left (g x\right )^{m + 4} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{8} g^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 4\right )} \]

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((7/2, m/2 + 1/2), (m/2 + 3/2,), e**2
*x**2/d**2)/(d**5*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((7/2, m/2 + 1), (m/2 + 2,), e**2*x**2/d**2)/(d**6*g**2*s
qrt(1 - e**2*x**2/d**2)*(m + 2)) + 3*e**2*(g*x)**(m + 3)*sqrt(d**2 - e**2*x**2)*
hyper((7/2, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2/d**2)/(d**7*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((7/2, m/2
+ 2), (m/2 + 3,), e**2*x**2/d**2)/(d**8*g**4*sqrt(1 - e**2*x**2/d**2)*(m + 4))

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Mathematica [A]  time = 0.35739, size = 240, normalized size = 1.13 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (3 d^2 e \left (m^3+8 m^2+19 m+12\right ) x \, _2F_1\left (\frac{7}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )+(m+2) \left (d (m+4) \left (d^2 (m+3) \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )+3 e^2 (m+1) x^2 \, _2F_1\left (\frac{7}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )\right )+e^3 \left (m^2+4 m+3\right ) x^3 \, _2F_1\left (\frac{7}{2},\frac{m}{2}+2;\frac{m}{2}+3;\frac{e^2 x^2}{d^2}\right )\right )\right )}{d^6 (m+1) (m+2) (m+3) (m+4) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(3*d^2*e*(12 + 19*m + 8*m^2 + m^3)*x*Hypergeo
metric2F1[7/2, 1 + m/2, 2 + m/2, (e^2*x^2)/d^2] + (2 + m)*(e^3*(3 + 4*m + m^2)*x
^3*Hypergeometric2F1[7/2, 2 + m/2, 3 + m/2, (e^2*x^2)/d^2] + d*(4 + m)*(d^2*(3 +
m)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + 3*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)*(4 + m)*Sqrt[d^2 - e^2*x^2])

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{ \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \]

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. \[ \int \frac{{\left (e x + d\right )}^{3} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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