3.333 \(\int \frac{x^m (c+d x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{x^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0311237, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {459, 364} \[ \frac{x^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(x^m*(c + d*x^2))/(a + b*x^2),x]

[Out]

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 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 \frac{x^m \left (c+d x^2\right )}{a+b x^2} \, dx &=\frac{d x^{1+m}}{b (1+m)}-\frac{(-b c (1+m)+a d (1+m)) \int \frac{x^m}{a+b x^2} \, dx}{b (1+m)}\\ &=\frac{d x^{1+m}}{b (1+m)}+\frac{(b c-a d) x^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{a b (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.054659, size = 55, normalized size = 0.83 \[ \frac{x^{m+1} \left ((b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )+a d\right )}{a b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{2}+c \right ){x}^{m}}{b{x}^{2}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.81527, size = 190, normalized size = 2.88 \begin{align*} \frac{c m x x^{m} \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 x x^{m} \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{d m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 d x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*m*x*x**m*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*x*x**m
*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)) + d*m*x**3*x**m*lerc
hphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a*gamma(m/2 + 5/2)) + 3*d*x**3*x**m*lerchphi(
b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a*gamma(m/2 + 5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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