\(\int \frac {A (c x)^m}{a+b x^2} \, dx\) [58]

Optimal result

Integrand size = 16, antiderivative size = 45 \[ \int \frac {A (c x)^m}{a+b x^2} \, dx=\frac {A (c x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a c (1+m)} \]

[Out]

A*(c*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a/c/(1+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, 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 = {12, 371} \[ \int \frac {A (c x)^m}{a+b x^2} \, dx=\frac {A (c x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a c (m+1)} \]

[In]

Int[(A*(c*x)^m)/(a + b*x^2),x]

[Out]

(A*(c*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*c*(1 + m))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

Rubi steps \begin{align*} \text {integral}& = A \int \frac {(c x)^m}{a+b x^2} \, dx \\ & = \frac {A (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a c (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {A (c x)^m}{a+b x^2} \, dx=\frac {A x (c x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {b x^2}{a}\right )}{a (1+m)} \]

[In]

Integrate[(A*(c*x)^m)/(a + b*x^2),x]

[Out]

(A*x*(c*x)^m*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a*(1 + m))

Maple [F]

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

[In]

int(A*(c*x)^m/(b*x^2+a),x)

[Out]

int(A*(c*x)^m/(b*x^2+a),x)

Fricas [F]

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

[In]

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

[Out]

integral((c*x)^m*A/(b*x^2 + a), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.16 \[ \int \frac {A (c x)^m}{a+b x^2} \, dx=A \left (\frac {c^{m} m x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c^{m} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}\right ) \]

[In]

integrate(A*(c*x)**m/(b*x**2+a),x)

[Out]

A*(c**m*m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2))
+ c**m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)))

Maxima [F]

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

[In]

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

[Out]

A*integrate((c*x)^m/(b*x^2 + a), x)

Giac [F]

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

[In]

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

[Out]

integrate((c*x)^m*A/(b*x^2 + a), x)

Mupad [F(-1)]

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

[In]

int((A*(c*x)^m)/(a + b*x^2),x)

[Out]

int((A*(c*x)^m)/(a + b*x^2), x)