\(\int (d x)^m (a+b \arctan (c x^2)) \, dx\) [93]

Optimal result

Integrand size = 16, antiderivative size = 75 \[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {(d x)^{1+m} \left (a+b \arctan \left (c x^2\right )\right )}{d (1+m)}-\frac {2 b c (d x)^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{4},\frac {7+m}{4},-c^2 x^4\right )}{d^3 (1+m) (3+m)} \]

[Out]

(d*x)^(1+m)*(a+b*arctan(c*x^2))/d/(1+m)-2*b*c*(d*x)^(3+m)*hypergeom([1, 3/4+1/4*m],[7/4+1/4*m],-c^2*x^4)/d^3/(
1+m)/(3+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4958, 371} \[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {(d x)^{m+1} \left (a+b \arctan \left (c x^2\right )\right )}{d (m+1)}-\frac {2 b c (d x)^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{4},\frac {m+7}{4},-c^2 x^4\right )}{d^3 (m+1) (m+3)} \]

[In]

Int[(d*x)^m*(a + b*ArcTan[c*x^2]),x]

[Out]

((d*x)^(1 + m)*(a + b*ArcTan[c*x^2]))/(d*(1 + m)) - (2*b*c*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/4, (7 +
m)/4, -(c^2*x^4)])/(d^3*(1 + m)*(3 + m))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4958

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

Rubi steps \begin{align*} \text {integral}& = \frac {(d x)^{1+m} \left (a+b \arctan \left (c x^2\right )\right )}{d (1+m)}-\frac {(2 b c) \int \frac {(d x)^{2+m}}{1+c^2 x^4} \, dx}{d^2 (1+m)} \\ & = \frac {(d x)^{1+m} \left (a+b \arctan \left (c x^2\right )\right )}{d (1+m)}-\frac {2 b c (d x)^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{4},\frac {7+m}{4},-c^2 x^4\right )}{d^3 (1+m) (3+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {x (d x)^m \left (-\left ((3+m) \left (a+b \arctan \left (c x^2\right )\right )\right )+2 b c x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{4},\frac {7+m}{4},-c^2 x^4\right )\right )}{(1+m) (3+m)} \]

[In]

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

[Out]

-((x*(d*x)^m*(-((3 + m)*(a + b*ArcTan[c*x^2])) + 2*b*c*x^2*Hypergeometric2F1[1, (3 + m)/4, (7 + m)/4, -(c^2*x^
4)]))/((1 + m)*(3 + m)))

Maple [F]

\[\int \left (d x \right )^{m} \left (a +b \arctan \left (c \,x^{2}\right )\right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

integral((b*arctan(c*x^2) + a)*(d*x)^m, x)

Sympy [F]

\[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )\, dx \]

[In]

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

[Out]

Integral((d*x)**m*(a + b*atan(c*x**2)), x)

Maxima [F]

\[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

(d^m*x*x^m*arctan(c*x^2) - 2*(c*d^m*m + c*d^m)*integrate(x^2*x^m/((c^2*m + c^2)*x^4 + m + 1), x))*b/(m + 1) +
(d*x)^(m + 1)*a/(d*(m + 1))

Giac [F]

\[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right ) \,d x \]

[In]

int((d*x)^m*(a + b*atan(c*x^2)),x)

[Out]

int((d*x)^m*(a + b*atan(c*x^2)), x)