\(\int (d+e x)^m (d^2-e^2 x^2)^{5/2} \, dx\) [952]

Optimal result

Integrand size = 24, antiderivative size = 83 \[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {2^{\frac {7}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {7}{2}-m} \left (d^2-e^2 x^2\right )^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {d-e x}{2 d}\right )}{7 d e} \]

[Out]

-1/7*2^(7/2+m)*(e*x+d)^m*(1+e*x/d)^(-7/2-m)*(-e^2*x^2+d^2)^(7/2)*hypergeom([7/2, -5/2-m],[9/2],1/2*(-e*x+d)/d)
/d/e

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {694, 692, 71} \[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {2^{m+\frac {7}{2}} \left (d^2-e^2 x^2\right )^{7/2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {7}{2}} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m-\frac {5}{2},\frac {9}{2},\frac {d-e x}{2 d}\right )}{7 d e} \]

[In]

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

[Out]

-1/7*(2^(7/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-7/2 - m)*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F1[7/2, -5/2 - m,
9/2, (d - e*x)/(2*d)])/(d*e)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(m - 1)*((a + c*x^2)^(p + 1)/((1
+ e*(x/d))^(p + 1)*(a/d + (c*x)/e)^(p + 1))), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a,
c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (
IntegerQ[3*p] || IntegerQ[4*p]))

Rule 694

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^IntPart[m]*((d + e*x)^FracPart[m]
/(1 + e*(x/d))^FracPart[m]), Int[(1 + e*(x/d))^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && EqQ[c*d
^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] || GtQ[d, 0])

Rubi steps \begin{align*} \text {integral}& = \left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \left (1+\frac {e x}{d}\right )^m \left (d^2-e^2 x^2\right )^{5/2} \, dx \\ & = \frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {7}{2}-m} \left (d^2-e^2 x^2\right )^{7/2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {5}{2}+m} \left (d^2-d e x\right )^{5/2} \, dx}{\left (d^2-d e x\right )^{7/2}} \\ & = -\frac {2^{\frac {7}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {7}{2}-m} \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac {7}{2},-\frac {5}{2}-m;\frac {9}{2};\frac {d-e x}{2 d}\right )}{7 d e} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 1.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.73 \[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {1}{2}-m} \left (-10 d^2 e^3 x^3 \sqrt {d-e x} \sqrt {d+e x} \operatorname {AppellF1}\left (3,-\frac {1}{2},-\frac {1}{2}-m,4,\frac {e x}{d},-\frac {e x}{d}\right )+3 e^5 x^5 \sqrt {d-e x} \sqrt {d+e x} \operatorname {AppellF1}\left (5,-\frac {1}{2},-\frac {1}{2}-m,6,\frac {e x}{d},-\frac {e x}{d}\right )-5\ 2^{\frac {3}{2}+m} d^4 (d-e x) \sqrt {1-\frac {e x}{d}} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-m,\frac {5}{2},\frac {d-e x}{2 d}\right )\right )}{15 e \sqrt {1-\frac {e x}{d}}} \]

[In]

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

[Out]

((d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*(-10*d^2*e^3*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[3, -1/2, -1/2 - m,
4, (e*x)/d, -((e*x)/d)] + 3*e^5*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[5, -1/2, -1/2 - m, 6, (e*x)/d, -((e*
x)/d)] - 5*2^(3/2 + m)*d^4*(d - e*x)*Sqrt[1 - (e*x)/d]*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/
2, (d - e*x)/(2*d)]))/(15*e*Sqrt[1 - (e*x)/d])

Maple [F]

\[\int \left (e x +d \right )^{m} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{m} \,d x } \]

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

integral((e^4*x^4 - 2*d^2*e^2*x^2 + d^4)*sqrt(-e^2*x^2 + d^2)*(e*x + d)^m, x)

Sympy [F]

\[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{m}\, dx \]

[In]

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

[Out]

Integral((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**m, x)

Maxima [F]

\[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{m} \,d x } \]

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{m} \,d x } \]

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^m \,d x \]

[In]

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

[Out]

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