3.3.35 \(\int \frac {(g x)^m (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [235]

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

[Out]

(g*x)^(1+m)*hypergeom([1, -2+1/2*m],[3/2+1/2*m],e^2*x^2/d^2)/d/g/(1+m)/(-e^2*x^2+d^2)^(5/2)+e*(g*x)^(2+m)*hype
rgeom([1, -3/2+1/2*m],[2+1/2*m],e^2*x^2/d^2)/d^2/g^2/(2+m)/(-e^2*x^2+d^2)^(5/2)

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Rubi [A]
time = 0.06, 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, Rules used = {822, 372, 371} \begin {gather*} \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}} \end {gather*}

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])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 822

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=d \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx+\frac {e \int \frac {(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{g}\\ &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{7/2}} \, dx}{d^5 \sqrt {d^2-e^2 x^2}}+\frac {\left (e \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^{1+m}}{\left (1-\frac {e^2 x^2}{d^2}\right )^{7/2}} \, dx}{d^6 g \sqrt {d^2-e^2 x^2}}\\ &=\frac {(g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {7}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 g (1+m) \sqrt {d^2-e^2 x^2}}+\frac {e (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {7}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^6 g^2 (2+m) \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 121, normalized size = 0.98 \begin {gather*} \frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (d (2+m) \, _2F_1\left (\frac {7}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )+e (1+m) x \, _2F_1\left (\frac {7}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )\right )}{d^6 (1+m) (2+m) \sqrt {d^2-e^2 x^2}} \end {gather*}

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]*(d*(2 + m)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*
(1 + m)*x*Hypergeometric2F1[7/2, (2 + m)/2, (4 + 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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (g x \right )^{m} \left (e x +d \right )}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="fricas")

[Out]

integral(sqrt(-x^2*e^2 + d^2)*(x*e + d)*(g*x)^m/(x^8*e^8 - 4*d^2*x^6*e^6 + 6*d^4*x^4*e^4 - 4*d^6*x^2*e^2 + d^8
), x)

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Sympy [C] Result contains complex when optimal does not.
time = 26.87, size = 117, normalized size = 0.94 \begin {gather*} \frac {g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{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} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{6} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{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} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{7} \Gamma \left (\frac {m}{2} + 2\right )} \end {gather*}

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]

g**m*x*x**m*gamma(m/2 + 1/2)*hyper((7/2, m/2 + 1/2), (m/2 + 3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*d**6*g
amma(m/2 + 3/2)) + e*g**m*x**2*x**m*gamma(m/2 + 1)*hyper((7/2, m/2 + 1), (m/2 + 2,), e**2*x**2*exp_polar(2*I*p
i)/d**2)/(2*d**7*gamma(m/2 + 2))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (g\,x\right )}^m\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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