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

Optimal. Leaf size=45 \[ \frac{A (c x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0552703, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{A (c x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a c (m+1)} \]

Antiderivative was successfully verified.

[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))

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Rubi in Sympy [A]  time = 7.12858, size = 34, normalized size = 0.76 \[ \frac{A \left (c x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0259577, size = 43, normalized size = 0.96 \[ \frac{A x (c x)^m \, _2F_1\left (1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{A \left ( cx \right ) ^{m}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(A*(c*x)^m/(b*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*integrate((c*x)^m/(b*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x)^m*A/(b*x^2 + a), x)

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Sympy [A]  time = 2.26141, size = 97, normalized size = 2.16 \[ A \left (\frac{c^{m} m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{c^{m} x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(A*(c*x)**m/(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)) + 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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*A/(b*x^2 + a), x)