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

Optimal. Leaf size=91 \[ \frac{A (e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a^2 e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a^2 e^2 (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.113586, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{A (e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a^2 e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a^2 e^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2885, size = 71, normalized size = 0.78 \[ \frac{A \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{a^{2} e \left (m + 1\right )} + \frac{B \left (e x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{a^{2} e^{2} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0900858, size = 82, normalized size = 0.9 \[ \frac{x (e x)^m \left (A (m+2) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )+B (m+1) x \, _2F_1\left (2,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{a^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( Bx+A \right ) }{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \left (e x\right )^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)*(e*x)^m/(c^2*x^4 + 2*a*c*x^2 + a^2), x)

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Sympy [A]  time = 133.248, size = 770, normalized size = 8.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-a*e**m*m**2*x*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/
2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*c*x**2*gamma(m/2 + 3/2)) + 2*a*e**m*m
*x*x**m*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*c*x**2*gamma(m/2 + 3/
2)) + a*e**m*x*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 +
1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*c*x**2*gamma(m/2 + 3/2)) + 2*a*e**m*x*x*
*m*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*c*x**2*gamma(m/2 + 3/2)) -
c*e**m*m**2*x**3*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/
2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*c*x**2*gamma(m/2 + 3/2)) + c*e**m*x**
3*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3
*gamma(m/2 + 3/2) + 8*a**2*c*x**2*gamma(m/2 + 3/2))) + B*(-a*e**m*m**2*x**2*x**m
*lerchphi(c*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**3*gamma(m/2
+ 2) + 8*a**2*c*x**2*gamma(m/2 + 2)) - 2*a*e**m*m*x**2*x**m*lerchphi(c*x**2*exp
_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*c*x**
2*gamma(m/2 + 2)) + 2*a*e**m*m*x**2*x**m*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) +
8*a**2*c*x**2*gamma(m/2 + 2)) + 4*a*e**m*x**2*x**m*gamma(m/2 + 1)/(8*a**3*gamma
(m/2 + 2) + 8*a**2*c*x**2*gamma(m/2 + 2)) - c*e**m*m**2*x**4*x**m*lerchphi(c*x**
2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*
c*x**2*gamma(m/2 + 2)) - 2*c*e**m*m*x**4*x**m*lerchphi(c*x**2*exp_polar(I*pi)/a,
1, m/2 + 1)*gamma(m/2 + 1)/(8*a**3*gamma(m/2 + 2) + 8*a**2*c*x**2*gamma(m/2 + 2
)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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