3.61 \(\int \frac{(c x)^m \left (A+B x+C x^2\right )}{a+b x^2} \, dx\)

Optimal. Leaf size=121 \[ \frac{(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b c (m+1)}+\frac{B (c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c^2 (m+2)}+\frac{C (c x)^{m+1}}{b c (m+1)} \]

[Out]

(C*(c*x)^(1 + m))/(b*c*(1 + m)) + ((A*b - a*C)*(c*x)^(1 + m)*Hypergeometric2F1[1
, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*b*c*(1 + m)) + (B*(c*x)^(2 + m)*Hyperg
eometric2F1[1, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)])/(a*c^2*(2 + m))

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Rubi [A]  time = 0.279132, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b c (m+1)}+\frac{B (c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c^2 (m+2)}+\frac{C (c x)^{m+1}}{b c (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[((c*x)^m*(A + B*x + C*x^2))/(a + b*x^2),x]

[Out]

(C*(c*x)^(1 + m))/(b*c*(1 + m)) + ((A*b - a*C)*(c*x)^(1 + m)*Hypergeometric2F1[1
, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*b*c*(1 + m)) + (B*(c*x)^(2 + m)*Hyperg
eometric2F1[1, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)])/(a*c^2*(2 + m))

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Rubi in Sympy [A]  time = 39.393, size = 90, normalized size = 0.74 \[ \frac{B \left (c x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c^{2} \left (m + 2\right )} + \frac{C \left (c x\right )^{m + 1}}{b c \left (m + 1\right )} + \frac{\left (c x\right )^{m + 1} \left (A b - C a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a b c \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.103921, size = 100, normalized size = 0.83 \[ \frac{x (c x)^m \left (b B (m+1) x \, _2F_1\left (1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right )-(m+2) \left ((a C-A b) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )-a C\right )\right )}{a b (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x)^m*(A + B*x + C*x^2))/(a + b*x^2),x]

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{m} \left ( C{x}^{2}+Bx+A \right ) }{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(C*x^2+B*x+A)/(b*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*(c*x)^m/(b*x^2 + a), x)

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Sympy [A]  time = 9.67938, size = 298, normalized size = 2.46 \[ \frac{A c^{m} m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A c^{m} x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{B c^{m} m x^{2} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{4 a \Gamma \left (\frac{m}{2} + 2\right )} + \frac{B c^{m} x^{2} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{2 a \Gamma \left (\frac{m}{2} + 2\right )} + \frac{C c^{m} m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 C c^{m} x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**m*(C*x**2+B*x+A)/(b*x**2+a),x)

[Out]

A*c**m*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)) + A*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)) + B*c**m*m*x**2*x**m*lerchphi(
b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(4*a*gamma(m/2 + 2)) + B*c*
*m*x**2*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a*
gamma(m/2 + 2)) + C*c**m*m*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)) + 3*C*c**m*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. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \left (c x\right )^{m}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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