\(\int \frac {(g x)^m (d+e x)^3}{(d^2-e^2 x^2)^{7/2}} \, dx\) [233]

Optimal result

Integrand size = 29, antiderivative size = 213 \[ \int \frac {(g x)^m (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 (g x)^{1+m} (d+e x)}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(1-4 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^3 g (1+m) \sqrt {d^2-e^2 x^2}}+\frac {e (7-4 m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^4 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1820, 822, 372, 371} \[ \int \frac {(g x)^m (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 (d+e x) (g x)^{m+1}}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (7-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^4 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {(1-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^3 g (m+1) \sqrt {d^2-e^2 x^2}} \]

[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]*Hyp
ergeometric2F1[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])

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]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[(-(c*x)^(m + 1))*(f + g*x)*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/(2*a*(p + 1)), I
nt[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(m + 2*p + 3) + g*(m + 2*p + 4)*x, x], x], x]] /;
FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && LtQ[p, -1] &&  !GtQ[m, 0]

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

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.93 \[ \int \frac {(g x)^m (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {d^3 \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{1+m}+e x \left (\frac {3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{2+m}+e x \left (\frac {3 d \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {3+m}{2},\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )}{3+m}+\frac {e x \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {4+m}{2},\frac {6+m}{2},\frac {e^2 x^2}{d^2}\right )}{4+m}\right )\right )\right )}{d^6 \sqrt {d^2-e^2 x^2}} \]

[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]*((d^3*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m)
+ e*x*((3*d^2*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2
F1[7/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) + (e*x*Hypergeometric2F1[7/2, (4 + m)/2, (6 + m)/2, (e^2
*x^2)/d^2])/(4 + m)))))/(d^6*Sqrt[d^2 - e^2*x^2])

Maple [F]

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

[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)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[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)

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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