3.26 \(\int \frac{(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d e (m+1)} \]

[Out]

(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1
, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d*e*(1 + m))

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Rubi [A]  time = 0.11575, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d e (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]

[Out]

(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1
, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d*e*(1 + m))

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Rubi in Sympy [A]  time = 14.052, size = 56, normalized size = 0.73 \[ \frac{B \left (e x\right )^{m + 1}}{d e \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (A d - B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{c d e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x**2+A)/(d*x**2+c),x)

[Out]

B*(e*x)**(m + 1)/(d*e*(m + 1)) + (e*x)**(m + 1)*(A*d - B*c)*hyper((1, m/2 + 1/2)
, (m/2 + 3/2,), -d*x**2/c)/(c*d*e*(m + 1))

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Mathematica [A]  time = 0.0690555, size = 58, normalized size = 0.75 \[ -\frac{x (e x)^m \left ((B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )-B c\right )}{c d (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]

[Out]

-((x*(e*x)^m*(-(B*c) + (B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -(
(d*x^2)/c)]))/(c*d*(1 + m)))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) }{d{x}^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x^2+A)/(d*x^2+c),x)

[Out]

int((e*x)^m*(B*x^2+A)/(d*x^2+c),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)*(e*x)^m/(d*x^2 + c), x)

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Sympy [A]  time = 7.96214, size = 204, normalized size = 2.65 \[ \frac{A e^{m} m x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A e^{m} x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{B e^{m} m x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 B e^{m} x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x**2+A)/(d*x**2+c),x)

[Out]

A*e**m*m*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2
)/(4*c*gamma(m/2 + 3/2)) + A*e**m*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m
/2 + 1/2)*gamma(m/2 + 1/2)/(4*c*gamma(m/2 + 3/2)) + B*e**m*m*x**3*x**m*lerchphi(
d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/2))
+ 3*B*e**m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2
+ 3/2)/(4*c*gamma(m/2 + 5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c), x)