3.248 \(\int x^m (c+a^2 c x^2)^3 \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=270 \[ -\frac{a c^3 x^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}-\frac{3 a^3 c^3 x^{m+4} \text{Hypergeometric2F1}\left (1,\frac{m+4}{2},\frac{m+6}{2},-a^2 x^2\right )}{m^2+7 m+12}-\frac{3 a^5 c^3 x^{m+6} \text{Hypergeometric2F1}\left (1,\frac{m+6}{2},\frac{m+8}{2},-a^2 x^2\right )}{(m+5) (m+6)}-\frac{a^7 c^3 x^{m+8} \text{Hypergeometric2F1}\left (1,\frac{m+8}{2},\frac{m+10}{2},-a^2 x^2\right )}{(m+7) (m+8)}+\frac{3 a^2 c^3 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{3 a^4 c^3 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{a^6 c^3 x^{m+7} \tan ^{-1}(a x)}{m+7}+\frac{c^3 x^{m+1} \tan ^{-1}(a x)}{m+1} \]

[Out]

(c^3*x^(1 + m)*ArcTan[a*x])/(1 + m) + (3*a^2*c^3*x^(3 + m)*ArcTan[a*x])/(3 + m) + (3*a^4*c^3*x^(5 + m)*ArcTan[
a*x])/(5 + m) + (a^6*c^3*x^(7 + m)*ArcTan[a*x])/(7 + m) - (a*c^3*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4
+ m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*c^3*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x
^2)])/(12 + 7*m + m^2) - (3*a^5*c^3*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)
*(6 + m)) - (a^7*c^3*x^(8 + m)*Hypergeometric2F1[1, (8 + m)/2, (10 + m)/2, -(a^2*x^2)])/((7 + m)*(8 + m))

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Rubi [A]  time = 0.225201, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4948, 4852, 364} \[ -\frac{a c^3 x^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m^2+3 m+2}-\frac{3 a^3 c^3 x^{m+4} \, _2F_1\left (1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right )}{m^2+7 m+12}-\frac{3 a^5 c^3 x^{m+6} \, _2F_1\left (1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right )}{(m+5) (m+6)}-\frac{a^7 c^3 x^{m+8} \, _2F_1\left (1,\frac{m+8}{2};\frac{m+10}{2};-a^2 x^2\right )}{(m+7) (m+8)}+\frac{3 a^2 c^3 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{3 a^4 c^3 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{a^6 c^3 x^{m+7} \tan ^{-1}(a x)}{m+7}+\frac{c^3 x^{m+1} \tan ^{-1}(a x)}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x],x]

[Out]

(c^3*x^(1 + m)*ArcTan[a*x])/(1 + m) + (3*a^2*c^3*x^(3 + m)*ArcTan[a*x])/(3 + m) + (3*a^4*c^3*x^(5 + m)*ArcTan[
a*x])/(5 + m) + (a^6*c^3*x^(7 + m)*ArcTan[a*x])/(7 + m) - (a*c^3*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4
+ m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*c^3*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x
^2)])/(12 + 7*m + m^2) - (3*a^5*c^3*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)
*(6 + m)) - (a^7*c^3*x^(8 + m)*Hypergeometric2F1[1, (8 + m)/2, (10 + m)/2, -(a^2*x^2)])/((7 + m)*(8 + m))

Rule 4948

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

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 x^m \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\int \left (c^3 x^m \tan ^{-1}(a x)+3 a^2 c^3 x^{2+m} \tan ^{-1}(a x)+3 a^4 c^3 x^{4+m} \tan ^{-1}(a x)+a^6 c^3 x^{6+m} \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int x^m \tan ^{-1}(a x) \, dx+\left (3 a^2 c^3\right ) \int x^{2+m} \tan ^{-1}(a x) \, dx+\left (3 a^4 c^3\right ) \int x^{4+m} \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^{6+m} \tan ^{-1}(a x) \, dx\\ &=\frac{c^3 x^{1+m} \tan ^{-1}(a x)}{1+m}+\frac{3 a^2 c^3 x^{3+m} \tan ^{-1}(a x)}{3+m}+\frac{3 a^4 c^3 x^{5+m} \tan ^{-1}(a x)}{5+m}+\frac{a^6 c^3 x^{7+m} \tan ^{-1}(a x)}{7+m}-\frac{\left (a c^3\right ) \int \frac{x^{1+m}}{1+a^2 x^2} \, dx}{1+m}-\frac{\left (3 a^3 c^3\right ) \int \frac{x^{3+m}}{1+a^2 x^2} \, dx}{3+m}-\frac{\left (3 a^5 c^3\right ) \int \frac{x^{5+m}}{1+a^2 x^2} \, dx}{5+m}-\frac{\left (a^7 c^3\right ) \int \frac{x^{7+m}}{1+a^2 x^2} \, dx}{7+m}\\ &=\frac{c^3 x^{1+m} \tan ^{-1}(a x)}{1+m}+\frac{3 a^2 c^3 x^{3+m} \tan ^{-1}(a x)}{3+m}+\frac{3 a^4 c^3 x^{5+m} \tan ^{-1}(a x)}{5+m}+\frac{a^6 c^3 x^{7+m} \tan ^{-1}(a x)}{7+m}-\frac{a c^3 x^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}-\frac{3 a^3 c^3 x^{4+m} \, _2F_1\left (1,\frac{4+m}{2};\frac{6+m}{2};-a^2 x^2\right )}{12+7 m+m^2}-\frac{3 a^5 c^3 x^{6+m} \, _2F_1\left (1,\frac{6+m}{2};\frac{8+m}{2};-a^2 x^2\right )}{(5+m) (6+m)}-\frac{a^7 c^3 x^{8+m} \, _2F_1\left (1,\frac{8+m}{2};\frac{10+m}{2};-a^2 x^2\right )}{(7+m) (8+m)}\\ \end{align*}

Mathematica [A]  time = 0.338003, size = 234, normalized size = 0.87 \[ c^3 x^{m+1} \left (-\frac{3 a^3 x^3 \text{Hypergeometric2F1}\left (1,\frac{m+4}{2},\frac{m+6}{2},-a^2 x^2\right )}{m^2+7 m+12}-\frac{a x \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}-\frac{a^7 x^7 \text{Hypergeometric2F1}\left (1,\frac{m}{2}+4,\frac{m}{2}+5,-a^2 x^2\right )}{(m+7) (m+8)}-\frac{3 a^5 x^5 \text{Hypergeometric2F1}\left (1,\frac{m+6}{2},\frac{m+8}{2},-a^2 x^2\right )}{(m+5) (m+6)}+\frac{a^6 x^6 \tan ^{-1}(a x)}{m+7}+\frac{3 a^4 x^4 \tan ^{-1}(a x)}{m+5}+\frac{3 a^2 x^2 \tan ^{-1}(a x)}{m+3}+\frac{\tan ^{-1}(a x)}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x],x]

[Out]

c^3*x^(1 + m)*(ArcTan[a*x]/(1 + m) + (3*a^2*x^2*ArcTan[a*x])/(3 + m) + (3*a^4*x^4*ArcTan[a*x])/(5 + m) + (a^6*
x^6*ArcTan[a*x])/(7 + m) - (a^7*x^7*Hypergeometric2F1[1, 4 + m/2, 5 + m/2, -(a^2*x^2)])/((7 + m)*(8 + m)) - (a
*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*x^3*Hypergeometric2F1[1, (
4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (3*a^5*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a
^2*x^2)])/((5 + m)*(6 + m)))

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Maple [C]  time = 0.747, size = 600, normalized size = 2.2 \begin{align*}{\frac{{a}^{-1-m}{c}^{3}}{4} \left ( -4\,{\frac{{x}^{m}{a}^{m} \left ({a}^{6}{m}^{3}{x}^{6}+6\,{a}^{6}{m}^{2}{x}^{6}+8\,m{x}^{6}{a}^{6}-{a}^{4}{m}^{3}{x}^{4}-8\,{a}^{4}{m}^{2}{x}^{4}-12\,m{x}^{4}{a}^{4}+{a}^{2}{m}^{3}{x}^{2}+10\,{a}^{2}{m}^{2}{x}^{2}+24\,m{x}^{2}{a}^{2}-{m}^{3}-12\,{m}^{2}-44\,m-48 \right ) }{ \left ( 7+m \right ) m \left ( 2+m \right ) \left ( 4+m \right ) \left ( 6+m \right ) }}+8\,{\frac{{x}^{8+m}{a}^{8+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 14+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m} \left ( -8-m \right ){\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{ \left ( 8+m \right ) \left ( 7+m \right ) }} \right ) }+{\frac{3\,{a}^{-1-m}{c}^{3}}{4} \left ( -4\,{\frac{{x}^{m}{a}^{m} \left ({a}^{4}{m}^{2}{x}^{4}+2\,m{x}^{4}{a}^{4}-{a}^{2}{m}^{2}{x}^{2}-4\,m{x}^{2}{a}^{2}+{m}^{2}+6\,m+8 \right ) }{ \left ( 5+m \right ) m \left ( 2+m \right ) \left ( 4+m \right ) }}+8\,{\frac{{x}^{6+m}{a}^{6+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 10+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m}{\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{5+m}} \right ) }+{\frac{3\,{a}^{-1-m}{c}^{3}}{4} \left ( -4\,{\frac{{x}^{m}{a}^{m} \left ( m{x}^{2}{a}^{2}-m-2 \right ) }{ \left ( 3+m \right ) m \left ( 2+m \right ) }}+8\,{\frac{{x}^{4+m}{a}^{4+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 6+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m} \left ( -4-m \right ){\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) }} \right ) }+{\frac{{a}^{-1-m}{c}^{3}}{4} \left ( 4\,{\frac{{x}^{m}{a}^{m} \left ( -m-2 \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) m}}+8\,{\frac{{x}^{2+m}{a}^{2+m}\arctan \left ( \sqrt{{a}^{2}{x}^{2}} \right ) }{ \left ( 2+2\,m \right ) \sqrt{{a}^{2}{x}^{2}}}}+2\,{\frac{{x}^{m}{a}^{m}{\it LerchPhi} \left ( -{a}^{2}{x}^{2},1,m/2 \right ) }{1+m}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x)

[Out]

1/4*a^(-1-m)*c^3*(-4*x^m*a^m*(a^6*m^3*x^6+6*a^6*m^2*x^6+8*a^6*m*x^6-a^4*m^3*x^4-8*a^4*m^2*x^4-12*a^4*m*x^4+a^2
*m^3*x^2+10*a^2*m^2*x^2+24*a^2*m*x^2-m^3-12*m^2-44*m-48)/(7+m)/m/(2+m)/(4+m)/(6+m)+8*x^(8+m)*a^(8+m)/(14+2*m)/
(a^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2))+2/(8+m)*x^m*a^m*(-8-m)/(7+m)*LerchPhi(-a^2*x^2,1,1/2*m))+3/4*a^(-1-m)*
c^3*(-4*x^m*a^m*(a^4*m^2*x^4+2*a^4*m*x^4-a^2*m^2*x^2-4*a^2*m*x^2+m^2+6*m+8)/(5+m)/m/(2+m)/(4+m)+8*x^(6+m)*a^(6
+m)/(10+2*m)/(a^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2))+2*x^m*a^m/(5+m)*LerchPhi(-a^2*x^2,1,1/2*m))+3/4*a^(-1-m)*
c^3*(-4*x^m*a^m*(a^2*m*x^2-m-2)/(3+m)/m/(2+m)+8*x^(4+m)*a^(4+m)/(6+2*m)/(a^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2)
)+2/(4+m)*x^m*a^m*(-4-m)/(3+m)*LerchPhi(-a^2*x^2,1,1/2*m))+1/4*a^(-1-m)*c^3*(4/(2+m)*x^m*a^m*(-m-2)/(1+m)/m+8*
x^(2+m)*a^(2+m)/(2+2*m)/(a^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2))+2*x^m*a^m/(1+m)*LerchPhi(-a^2*x^2,1,1/2*m))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} x^{m} \arctan \left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="fricas")

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*x^m*arctan(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int x^{m} \operatorname{atan}{\left (a x \right )}\, dx + \int 3 a^{2} x^{2} x^{m} \operatorname{atan}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} x^{m} \operatorname{atan}{\left (a x \right )}\, dx + \int a^{6} x^{6} x^{m} \operatorname{atan}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a**2*c*x**2+c)**3*atan(a*x),x)

[Out]

c**3*(Integral(x**m*atan(a*x), x) + Integral(3*a**2*x**2*x**m*atan(a*x), x) + Integral(3*a**4*x**4*x**m*atan(a
*x), x) + Integral(a**6*x**6*x**m*atan(a*x), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^3*x^m*arctan(a*x), x)