3.10.62 \(\int (d+e x) (1-\frac {e^2 x^2}{d^2})^p \, dx\) [962]

Optimal. Leaf size=57 \[ -\frac {2^{1+p} d^2 \left (\frac {d-e x}{d}\right )^{1+p} \, _2F_1\left (-1-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{e (1+p)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {655, 251} \begin {gather*} d x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-\frac {d^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{p+1}}{2 e (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(1 - (e^2*x^2)/d^2)^p,x]

[Out]

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

Rule 251

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

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 56, normalized size = 0.98 \begin {gather*} -\frac {d^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{1+p}}{2 e (1+p)}+d x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(1 - (e^2*x^2)/d^2)^p,x]

[Out]

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

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Maple [A]
time = 0.45, size = 47, normalized size = 0.82

method	result	size
meijerg	\(\frac {e \,x^{2} \hypergeom \left (\left [1, -p \right ], \left [2\right ], \frac {e^{2} x^{2}}{d^{2}}\right )}{2}+d x \hypergeom \left (\left [\frac {1}{2}, -p \right ], \left [\frac {3}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(1-e^2*x^2/d^2)^p,x,method=_RETURNVERBOSE)

[Out]

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

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

[Out]

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

[Out]

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

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Sympy [A]
time = 1.54, size = 78, normalized size = 1.37 \begin {gather*} d x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: e^{2} = 0 \\- \frac {d^{2} \left (\begin {cases} \frac {\left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )} & \text {otherwise} \end {cases}\right )}{2 e^{2}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + e*Piecewise((x**2/2, Eq(e**2, 0)), (-d**2*Pie
cewise(((1 - e**2*x**2/d**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(1 - e**2*x**2/d**2), True))/(2*e**2), True))

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

[Out]

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

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Mupad [B]
time = 1.39, size = 58, normalized size = 1.02 \begin {gather*} d\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )-\frac {\left (d^2-e^2\,x^2\right )\,{\left (1-\frac {e^2\,x^2}{d^2}\right )}^p}{2\,e\,\left (p+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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