3.1174 \(\int e^{3 \tanh ^{-1}(a x)} x^m (c-a^2 c x^2)^p \, dx\)

Optimal. Leaf size=251 \[ \frac{(4 m+2 p+3) x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{m+1}{2},\frac{3}{2}-p,\frac{m+3}{2},a^2 x^2\right )}{(m+1) (m+2 p)}+\frac{a (4 m+6 p+5) x^{m+2} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{m+2}{2},\frac{3}{2}-p,\frac{m+4}{2},a^2 x^2\right )}{(m+2) (m+2 p+1)}-\frac{3 x^{m+1} \left (c-a^2 c x^2\right )^p}{\sqrt{1-a^2 x^2} (m+2 p)}-\frac{a x^{m+2} \left (c-a^2 c x^2\right )^p}{\sqrt{1-a^2 x^2} (m+2 p+1)} \]

[Out]

(-3*x^(1 + m)*(c - a^2*c*x^2)^p)/((m + 2*p)*Sqrt[1 - a^2*x^2]) - (a*x^(2 + m)*(c - a^2*c*x^2)^p)/((1 + m + 2*p
)*Sqrt[1 - a^2*x^2]) + ((3 + 4*m + 2*p)*x^(1 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(1 + m)/2, 3/2 - p, (3 +
m)/2, a^2*x^2])/((1 + m)*(m + 2*p)*(1 - a^2*x^2)^p) + (a*(5 + 4*m + 6*p)*x^(2 + m)*(c - a^2*c*x^2)^p*Hypergeo
metric2F1[(2 + m)/2, 3/2 - p, (4 + m)/2, a^2*x^2])/((2 + m)*(1 + m + 2*p)*(1 - a^2*x^2)^p)

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Rubi [A]  time = 0.416018, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6153, 6148, 1809, 808, 364} \[ \frac{(4 m+2 p+3) x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{m+1}{2},\frac{3}{2}-p;\frac{m+3}{2};a^2 x^2\right )}{(m+1) (m+2 p)}+\frac{a (4 m+6 p+5) x^{m+2} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{m+2}{2},\frac{3}{2}-p;\frac{m+4}{2};a^2 x^2\right )}{(m+2) (m+2 p+1)}-\frac{3 x^{m+1} \left (c-a^2 c x^2\right )^p}{\sqrt{1-a^2 x^2} (m+2 p)}-\frac{a x^{m+2} \left (c-a^2 c x^2\right )^p}{\sqrt{1-a^2 x^2} (m+2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^p,x]

[Out]

(-3*x^(1 + m)*(c - a^2*c*x^2)^p)/((m + 2*p)*Sqrt[1 - a^2*x^2]) - (a*x^(2 + m)*(c - a^2*c*x^2)^p)/((1 + m + 2*p
)*Sqrt[1 - a^2*x^2]) + ((3 + 4*m + 2*p)*x^(1 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(1 + m)/2, 3/2 - p, (3 +
m)/2, a^2*x^2])/((1 + m)*(m + 2*p)*(1 - a^2*x^2)^p) + (a*(5 + 4*m + 6*p)*x^(2 + m)*(c - a^2*c*x^2)^p*Hypergeo
metric2F1[(2 + m)/2, 3/2 - p, (4 + m)/2, a^2*x^2])/((2 + m)*(1 + m + 2*p)*(1 - a^2*x^2)^p)

Rule 6153

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

Rule 6148

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 808

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 364

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

Rubi steps

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

Mathematica [A]  time = 0.166521, size = 186, normalized size = 0.74 \[ x^{m+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac{\text{Hypergeometric2F1}\left (\frac{m+1}{2},\frac{3}{2}-p,\frac{m+3}{2},a^2 x^2\right )}{m+1}+a x \left (\frac{3 \text{Hypergeometric2F1}\left (\frac{m+2}{2},\frac{3}{2}-p,\frac{m+4}{2},a^2 x^2\right )}{m+2}+a x \left (\frac{3 \text{Hypergeometric2F1}\left (\frac{m+3}{2},\frac{3}{2}-p,\frac{m+5}{2},a^2 x^2\right )}{m+3}+\frac{a x \text{Hypergeometric2F1}\left (\frac{m+4}{2},\frac{3}{2}-p,\frac{m+6}{2},a^2 x^2\right )}{m+4}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^p,x]

[Out]

(x^(1 + m)*(c - a^2*c*x^2)^p*(Hypergeometric2F1[(1 + m)/2, 3/2 - p, (3 + m)/2, a^2*x^2]/(1 + m) + a*x*((3*Hype
rgeometric2F1[(2 + m)/2, 3/2 - p, (4 + m)/2, a^2*x^2])/(2 + m) + a*x*((3*Hypergeometric2F1[(3 + m)/2, 3/2 - p,
(5 + m)/2, a^2*x^2])/(3 + m) + (a*x*Hypergeometric2F1[(4 + m)/2, 3/2 - p, (6 + m)/2, a^2*x^2])/(4 + m)))))/(1
- a^2*x^2)^p

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Maple [F]  time = 0.406, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) ^{3}{x}^{m} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^p,x)

[Out]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (-a^{2} c x^{2} + c\right )}^{p} x^{m}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^p,x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3*(-a^2*c*x^2 + c)^p*x^m/(-a^2*x^2 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^p,x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p*x^m/(a^2*x^2 - 2*a*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**m*(-a**2*c*x**2+c)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (-a^{2} c x^{2} + c\right )}^{p} x^{m}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^p,x, algorithm="giac")

[Out]

integrate((a*x + 1)^3*(-a^2*c*x^2 + c)^p*x^m/(-a^2*x^2 + 1)^(3/2), x)