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

Optimal. Leaf size=120 \[ -\frac{x^{m+1} (b c-a d) (a d (1-m)+b c (m+3)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{2 c^2 d^2 (m+1)}+\frac{x^{m+1} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^{m+1}}{d^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.279261, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{x^{m+1} (b c-a d) (a d (1-m)+b c (m+3)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{2 c^2 d^2 (m+1)}+\frac{x^{m+1} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x^{m+1}}{d^2 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.9547, size = 104, normalized size = 0.87 \[ \frac{b^{2} x^{m + 1}}{d^{2} \left (m + 1\right )} + \frac{x^{m + 1} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} + \frac{x^{m + 1} \left (a d - b c\right ) \left (- a d m + a d + b c m + 3 b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{2 c^{2} d^{2} \left (m + 1\right )} \]

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]

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

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Mathematica [A]  time = 0.162157, size = 118, normalized size = 0.98 \[ \frac{x^{m+1} \left (\frac{a^2 \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{m+1}+b x^2 \left (\frac{2 a \, _2F_1\left (2,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right )}{m+3}+\frac{b x^2 \, _2F_1\left (2,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right )}{m+5}\right )\right )}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, 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. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} x^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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