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

Optimal. Leaf size=230 \[ -\frac{b^2 x^{m+1} (a d (5-m)-b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 (m+1) (b c-a d)^3}-\frac{d^2 x^{m+1} (a d (1-m)-b c (5-m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{2 c^2 (m+1) (b c-a d)^3}+\frac{d x^{m+1} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

[Out]

(d*(b*c + a*d)*x^(1 + m))/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x^(1 + m))/(2*a
*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^2*(a*d*(5 - m) - b*(c - c*m))*x^(1 +
m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^
3*(1 + m)) - (d^2*(a*d*(1 - m) - b*c*(5 - m))*x^(1 + m)*Hypergeometric2F1[1, (1
+ m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*(b*c - a*d)^3*(1 + m))

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Rubi [A]  time = 0.963905, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^2 x^{m+1} (a d (5-m)-b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 (m+1) (b c-a d)^3}-\frac{d^2 x^{m+1} (a d (1-m)-b c (5-m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{2 c^2 (m+1) (b c-a d)^3}+\frac{d x^{m+1} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*c + a*d)*x^(1 + m))/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x^(1 + m))/(2*a
*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^2*(a*d*(5 - m) - b*(c - c*m))*x^(1 +
m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^
3*(1 + m)) - (d^2*(a*d*(1 - m) - b*c*(5 - m))*x^(1 + m)*Hypergeometric2F1[1, (1
+ m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*(b*c - a*d)^3*(1 + m))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.480614, size = 195, normalized size = 0.85 \[ \frac{a c (m+3) x^{m+1} F_1\left (\frac{m+1}{2};2,2;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{(m+1) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (a c (m+3) F_1\left (\frac{m+1}{2};2,2;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-4 x^2 \left (a d F_1\left (\frac{m+3}{2};2,3;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{m+3}{2};3,2;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*c*(3 + m)*x^(1 + m)*AppellF1[(1 + m)/2, 2, 2, (3 + m)/2, -((b*x^2)/a), -((d*x
^2)/c)])/((1 + m)*(a + b*x^2)^2*(c + d*x^2)^2*(a*c*(3 + m)*AppellF1[(1 + m)/2, 2
, 2, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] - 4*x^2*(a*d*AppellF1[(3 + m)/2, 2,
3, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[(3 + m)/2, 3, 2, (5 + m
)/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) ^{2} \left ( d{x}^{2}+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b^{2} d^{2} x^{8} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b^2*d^2*x^8 + 2*(b^2*c*d + a*b*d^2)*x^6 + (b^2*c^2 + 4*a*b*c*d + a
^2*d^2)*x^4 + a^2*c^2 + 2*(a*b*c^2 + a^2*c*d)*x^2), 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(x**m/(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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