3.310 \(\int \frac{(g x)^m (d^2-e^2 x^2)^p}{d+e x} \, dx\)

Optimal. Leaf size=163 \[ \frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},1-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.135751, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {892, 82, 126, 365, 364} \[ \frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},1-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 892

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + c*x^
2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (
c*x)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !Int
egerQ[p] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rule 82

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[a, Int[(a + b*
x)^n*(c + d*x)^n*(f*x)^p, x], x] + Dist[b/f, Int[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b,
c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] &&  !RationalQ[p] &&  !IGtQ[m, 0] && NeQ[m +
n + p + 2, 0]

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[((a + b*x)^Fra
cPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a
, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(g x)^m \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\left ((d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d-e x)^p (d+e x)^{-1+p} \, dx\\ &=\left (d (d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d-e x)^{-1+p} (d+e x)^{-1+p} \, dx-\frac{\left (e (d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^{1+m} (d-e x)^{-1+p} (d+e x)^{-1+p} \, dx}{g}\\ &=d \int (g x)^m \left (d^2-e^2 x^2\right )^{-1+p} \, dx-\frac{e \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p} \, dx}{g}\\ &=\frac{\left (\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d}-\frac{\left (e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac{e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d^2 g}\\ &=\frac{(g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1+m}{2},1-p;\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{d g (1+m)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{2+m}{2},1-p;\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (2+m)}\\ \end{align*}

Mathematica [A]  time = 0.0536167, size = 124, normalized size = 0.76 \[ \frac{x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d (m+2) \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )-e (m+1) x \, _2F_1\left (\frac{m}{2}+1,1-p;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )\right )}{d^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.674, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx \right ) ^{m} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 11.8517, size = 337, normalized size = 2.07 \begin{align*} - \frac{0^{p} d d^{2 p} g^{m} m x^{m} \Phi \left (\frac{d^{2}}{e^{2} x^{2}}, 1, \frac{1}{2} - \frac{m}{2}\right ) \Gamma \left (\frac{1}{2} - \frac{m}{2}\right )}{4 e^{2} x \Gamma \left (\frac{3}{2} - \frac{m}{2}\right )} + \frac{0^{p} d d^{2 p} g^{m} x^{m} \Phi \left (\frac{d^{2}}{e^{2} x^{2}}, 1, \frac{1}{2} - \frac{m}{2}\right ) \Gamma \left (\frac{1}{2} - \frac{m}{2}\right )}{4 e^{2} x \Gamma \left (\frac{3}{2} - \frac{m}{2}\right )} + \frac{0^{p} d^{2 p} g^{m} m x^{m} \Phi \left (\frac{d^{2}}{e^{2} x^{2}}, 1, \frac{m e^{i \pi }}{2}\right ) \Gamma \left (- \frac{m}{2}\right )}{4 e \Gamma \left (1 - \frac{m}{2}\right )} + \frac{d e^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac{m}{2} - p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, - \frac{m}{2} - p + \frac{1}{2} \\ - \frac{m}{2} - p + \frac{3}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x \Gamma \left (p + 1\right ) \Gamma \left (- \frac{m}{2} - p + \frac{3}{2}\right )} - \frac{e^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac{m}{2} - p\right ){{}_{2}F_{1}\left (\begin{matrix} 1 - p, - \frac{m}{2} - p \\ - \frac{m}{2} - p + 1 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (p + 1\right ) \Gamma \left (- \frac{m}{2} - p + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-0**p*d*d**(2*p)*g**m*m*x**m*lerchphi(d**2/(e**2*x**2), 1, 1/2 - m/2)*gamma(1/2 - m/2)/(4*e**2*x*gamma(3/2 - m
/2)) + 0**p*d*d**(2*p)*g**m*x**m*lerchphi(d**2/(e**2*x**2), 1, 1/2 - m/2)*gamma(1/2 - m/2)/(4*e**2*x*gamma(3/2
- m/2)) + 0**p*d**(2*p)*g**m*m*x**m*lerchphi(d**2/(e**2*x**2), 1, m*exp_polar(I*pi)/2)*gamma(-m/2)/(4*e*gamma
(1 - m/2)) + d*e**(2*p)*g**m*p*x**m*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p + 1/2)*hyper((1 - p, -m/2 - p
+ 1/2), (-m/2 - p + 3/2,), d**2/(e**2*x**2))/(2*e**2*x*gamma(p + 1)*gamma(-m/2 - p + 3/2)) - e**(2*p)*g**m*p*
x**m*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p)*hyper((1 - p, -m/2 - p), (-m/2 - p + 1,), d**2/(e**2*x**2))
/(2*e*gamma(p + 1)*gamma(-m/2 - p + 1))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^p/(e*x+d),x, algorithm="giac")

[Out]

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