\(\int (d+e x)^m \sqrt {d^2-e^2 x^2} \, dx\) [954]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int (d+e x)^m \sqrt {d^2-e^2 x^2} \, dx=\frac {(d-e x) (d+e x)^{1+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (1,3+m,\frac {5}{2}+m,\frac {d+e x}{2 d}\right )}{d e (3+2 m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24, 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} \[ \int (d+e x)^m \sqrt {d^2-e^2 x^2} \, dx=-\frac {2^{m+\frac {3}{2}} \left (d^2-e^2 x^2\right )^{3/2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {3}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m-\frac {1}{2},\frac {5}{2},\frac {d-e x}{2 d}\right )}{3 d e} \]

[In]

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

[Out]

-1/3*(2^(3/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-3/2 - m)*(d^2 - e^2*x^2)^(3/2)*Hypergeometric2F1[3/2, -1/2 - m,
5/2, (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 \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {3}{2}-m} \left (d^2-e^2 x^2\right )^{3/2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {1}{2}+m} \sqrt {d^2-d e x} \, dx}{\left (d^2-d e x\right )^{3/2}} \\ & = -\frac {2^{\frac {3}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {3}{2}-m} \left (d^2-e^2 x^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};\frac {d-e x}{2 d}\right )}{3 d e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int (d+e x)^m \sqrt {d^2-e^2 x^2} \, dx=-\frac {2^{\frac {3}{2}+m} (d-e x) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {1}{2}-m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-m,\frac {5}{2},\frac {d-e x}{2 d}\right )}{3 e} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

int((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x)

[Out]

int((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x)

Fricas [F]

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

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((e*x+d)**m*(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**m, x)

Maxima [F]

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

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((d^2 - e^2*x^2)^(1/2)*(d + e*x)^m,x)

[Out]

int((d^2 - e^2*x^2)^(1/2)*(d + e*x)^m, x)