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

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 = 0.107302, 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 \[ \frac{x^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)} \]

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))

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Rubi in Sympy [A]  time = 12.9328, size = 49, normalized size = 0.74 \[ \frac{d x^{m + 1}}{b \left (m + 1\right )} - \frac{x^{m + 1} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(d*x**2+c)/(b*x**2+a),x)

[Out]

d*x**(m + 1)/(b*(m + 1)) - x**(m + 1)*(a*d - b*c)*hyper((1, m/2 + 1/2), (m/2 + 3
/2,), -b*x**2/a)/(a*b*(m + 1))

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Mathematica [A]  time = 0.06636, 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 = 0.046, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( d{x}^{2}+c \right ) }{b{x}^{2}+a}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*x^m/(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. \[{\rm integral}\left (\frac{{\left (d x^{2} + c\right )} x^{m}}{b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*x^m/(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 [A]  time = 15.4561, size = 190, normalized size = 2.88 \[ \frac{c m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{c x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{d m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 d x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \]

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*lerchphi(b*x**2*exp_pola
r(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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} x^{m}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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