3.342 \(\int \frac{x^m}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=325 \[ -\frac{b^3 x^{m+1} (a d (7-m)-b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 (m+1) (b c-a d)^4}+\frac{d x^{m+1} \left (-a^2 d^2 (3-m)+a b c d (11-m)+4 b^2 c^2\right )}{8 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^2 x^{m+1} \left (a^2 d^2 \left (m^2-4 m+3\right )-2 a b c d \left (m^2-8 m+7\right )+b^2 c^2 \left (m^2-12 m+35\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{8 c^3 (m+1) (b c-a d)^4}+\frac{d x^{m+1} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(d*(2*b*c + a*d)*x^(1 + m))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(1 + m))/
(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2 - a^2*d^2*(3 - m) +
a*b*c*d*(11 - m))*x^(1 + m))/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^3*(a*d*(7
- 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)^4*(1 + m)) + (d^2*(b^2*c^2*(35 - 12*m + m^2) - 2*a*b*
c*d*(7 - 8*m + m^2) + a^2*d^2*(3 - 4*m + m^2))*x^(1 + m)*Hypergeometric2F1[1, (1
+ m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^4*(1 + m))

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Rubi [A]  time = 1.60165, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b^3 x^{m+1} (b c (1-m)-a d (7-m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 (m+1) (b c-a d)^4}+\frac{d x^{m+1} \left (-a^2 d^2 (3-m)+a b c d (11-m)+4 b^2 c^2\right )}{8 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^2 x^{m+1} \left (a^2 d^2 \left (m^2-4 m+3\right )-2 a b c d \left (m^2-8 m+7\right )+b^2 c^2 \left (m^2-12 m+35\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{8 c^3 (m+1) (b c-a d)^4}+\frac{d x^{m+1} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{b x^{m+1}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(d*(2*b*c + a*d)*x^(1 + m))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(1 + m))/
(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2 - a^2*d^2*(3 - m) +
a*b*c*d*(11 - m))*x^(1 + m))/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) + (b^3*(b*c*(1
- m) - a*d*(7 - m))*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^
2)/a)])/(2*a^2*(b*c - a*d)^4*(1 + m)) + (d^2*(b^2*c^2*(35 - 12*m + m^2) - 2*a*b*
c*d*(7 - 8*m + m^2) + a^2*d^2*(3 - 4*m + m^2))*x^(1 + m)*Hypergeometric2F1[1, (1
+ m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^4*(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)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.663452, size = 197, normalized size = 0.61 \[ \frac{a c (m+3) x^{m+1} F_1\left (\frac{m+1}{2};2,3;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{(m+1) \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (a c (m+3) F_1\left (\frac{m+1}{2};2,3;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-2 x^2 \left (3 a d F_1\left (\frac{m+3}{2};2,4;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+2 b c F_1\left (\frac{m+3}{2};3,3;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^m/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

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

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) ^{2} \left ( d{x}^{2}+c \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

int(x^m/(b*x^2+a)^2/(d*x^2+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

integrate(x^m/((b*x^2 + a)^2*(d*x^2 + c)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b^{2} d^{3} x^{10} +{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} +{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + a^{2} c^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} +{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

integral(x^m/(b^2*d^3*x^10 + (3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + (3*b^2*c^2*d + 6*a*
b*c*d^2 + a^2*d^3)*x^6 + a^2*c^3 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + (
2*a*b*c^3 + 3*a^2*c^2*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

integrate(x^m/((b*x^2 + a)^2*(d*x^2 + c)^3), x)