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

Optimal. Leaf size=125 \[ \frac{(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c e (m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.339741, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c e (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.5012, size = 94, normalized size = 0.75 \[ \frac{\left (e x\right )^{m + 1} \left (A d - B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{c e \left (m + 1\right ) \left (a d - b c\right )} - \frac{\left (e x\right )^{m + 1} \left (A b - B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a e \left (m + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.162532, size = 100, normalized size = 0.8 \[ \frac{x (e x)^m \left ((a B c-A b c) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )+a (A d-B c) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*((-(A*b*c) + a*B*c)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x
^2)/a)] + a*(-(B*c) + 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 = 0.074, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) }{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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