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

Optimal. Leaf size=234 \[ \frac{b x^{m+1} (b c (3-m)-a d (7-m))}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{b x^{m+1} \left (a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (m^2-6 m+5\right )+b^2 c^2 \left (m^2-4 m+3\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{8 a^3 (m+1) (b c-a d)^3}-\frac{d^3 x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)^3}+\frac{b x^{m+1}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]

[Out]

(b*x^(1 + m))/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(b*c*(3 - m) - a*d*(7 - m))*x
^(1 + m))/(8*a^2*(b*c - a*d)^2*(a + b*x^2)) + (b*(a^2*d^2*(15 - 8*m + m^2) - 2*a
*b*c*d*(5 - 6*m + m^2) + b^2*c^2*(3 - 4*m + m^2))*x^(1 + m)*Hypergeometric2F1[1,
(1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^3*(1 + m)) - (d^3*x^(1
+ m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(b*c - a*d)^3*
(1 + m))

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Rubi [A]  time = 1.00289, antiderivative size = 234, 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 \[ \frac{b x^{m+1} (b c (3-m)-a d (7-m))}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{b x^{m+1} \left (a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (m^2-6 m+5\right )+b^2 c^2 \left (m^2-4 m+3\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{8 a^3 (m+1) (b c-a d)^3}-\frac{d^3 x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)^3}+\frac{b x^{m+1}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^(1 + m))/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(b*c*(3 - m) - a*d*(7 - m))*x
^(1 + m))/(8*a^2*(b*c - a*d)^2*(a + b*x^2)) + (b*(a^2*d^2*(15 - 8*m + m^2) - 2*a
*b*c*d*(5 - 6*m + m^2) + b^2*c^2*(3 - 4*m + m^2))*x^(1 + m)*Hypergeometric2F1[1,
(1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^3*(1 + m)) - (d^3*x^(1
+ m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(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)**3/(d*x**2+c),x)

[Out]

Timed out

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Mathematica [C]  time = 0.41484, size = 196, normalized size = 0.84 \[ \frac{a c (m+3) x^{m+1} F_1\left (\frac{m+1}{2};3,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{(m+1) \left (a+b x^2\right )^3 \left (c+d x^2\right ) \left (a c (m+3) F_1\left (\frac{m+1}{2};3,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-2 x^2 \left (a d F_1\left (\frac{m+3}{2};3,2;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{m+3}{2};4,1;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) ^{3} \left ( d{x}^{2}+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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