3.334 \(\int \frac{x^m}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=102 \[ \frac{b x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.130967, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.2751, size = 76, normalized size = 0.75 \[ \frac{d x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{c \left (m + 1\right ) \left (a d - b c\right )} - \frac{b x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a \left (m + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0761294, size = 85, normalized size = 0.83 \[ \frac{x^{m+1} \left (a d \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )-b c \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b*d*x^4 + (b*c + a*d)*x^2 + a*c), x)

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Sympy [A]  time = 93.4697, size = 354, normalized size = 3.47 \[ \frac{a m x^{m} \Phi \left (\frac{a e^{i \pi }}{b x^{2}}, 1, - \frac{m}{2} + \frac{3}{2}\right ) \Gamma ^{2}\left (- \frac{m}{2} + \frac{3}{2}\right )}{x^{3} \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} - \frac{3 a x^{m} \Phi \left (\frac{a e^{i \pi }}{b x^{2}}, 1, - \frac{m}{2} + \frac{3}{2}\right ) \Gamma ^{2}\left (- \frac{m}{2} + \frac{3}{2}\right )}{x^{3} \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} + \frac{b m x^{m} \Phi \left (\frac{c e^{i \pi }}{d x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )}{x \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} - \frac{b x^{m} \Phi \left (\frac{c e^{i \pi }}{d x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )}{x \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*m*x**m*lerchphi(a*exp_polar(I*pi)/(b*x**2), 1, -m/2 + 3/2)*gamma(-m/2 + 3/2)**
2/(x**3*(4*a*b*d*gamma(-m/2 + 3/2)*gamma(-m/2 + 5/2) - 4*b**2*c*gamma(-m/2 + 3/2
)*gamma(-m/2 + 5/2))) - 3*a*x**m*lerchphi(a*exp_polar(I*pi)/(b*x**2), 1, -m/2 +
3/2)*gamma(-m/2 + 3/2)**2/(x**3*(4*a*b*d*gamma(-m/2 + 3/2)*gamma(-m/2 + 5/2) - 4
*b**2*c*gamma(-m/2 + 3/2)*gamma(-m/2 + 5/2))) + b*m*x**m*lerchphi(c*exp_polar(I*
pi)/(d*x**2), 1, -m/2 + 1/2)*gamma(-m/2 + 1/2)*gamma(-m/2 + 5/2)/(x*(4*a*b*d*gam
ma(-m/2 + 3/2)*gamma(-m/2 + 5/2) - 4*b**2*c*gamma(-m/2 + 3/2)*gamma(-m/2 + 5/2))
) - b*x**m*lerchphi(c*exp_polar(I*pi)/(d*x**2), 1, -m/2 + 1/2)*gamma(-m/2 + 1/2)
*gamma(-m/2 + 5/2)/(x*(4*a*b*d*gamma(-m/2 + 3/2)*gamma(-m/2 + 5/2) - 4*b**2*c*ga
mma(-m/2 + 3/2)*gamma(-m/2 + 5/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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