3.347 \(\int \frac{(c x)^{1+m}}{a+b x^2} \, dx\)

Optimal. Leaf size=44 \[ \frac{(c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.0423558, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{(c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c (m+2)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(1 + m)/(a + b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.18764, size = 29, normalized size = 0.66 \[ \frac{\left (c x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(1+m)/(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.0383368, size = 45, normalized size = 1.02 \[ \frac{c x^2 (c x)^m \, _2F_1\left (1,\frac{m+2}{2};\frac{m+2}{2}+1;-\frac{b x^2}{a}\right )}{a (m+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(1 + m)/(a + b*x^2),x]

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{1+m}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(1+m)/(b*x^2+a),x)

[Out]

int((c*x)^(1+m)/(b*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m + 1}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m + 1)/(b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((c*x)^(m + 1)/(b*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (c x\right )^{m + 1}}{b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m + 1)/(b*x^2 + a),x, algorithm="fricas")

[Out]

integral((c*x)^(m + 1)/(b*x^2 + a), x)

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Sympy [A]  time = 19.4873, size = 92, normalized size = 2.09 \[ \frac{c c^{m} m x^{2} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{4 a \Gamma \left (\frac{m}{2} + 2\right )} + \frac{c c^{m} x^{2} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{2 a \Gamma \left (\frac{m}{2} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(1+m)/(b*x**2+a),x)

[Out]

c*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)) + c*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))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m + 1}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m + 1)/(b*x^2 + a),x, algorithm="giac")

[Out]

integrate((c*x)^(m + 1)/(b*x^2 + a), x)