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

Optimal. Leaf size=247 \[ \frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (a d (m+3)+b (c-c m))+a B (b c (m+1)-a d (m+5)))}{2 a^2 b^3 e (m+1)}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+3))-a B (2 b c (m+3)-a d (m+5)))}{2 a b^3 e (m+1)}-\frac{d^2 (e x)^{m+3} (A b (m+3)-a B (m+5))}{2 a b^2 e^3 (m+3)}+\frac{\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 1.11389, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (a d (m+3)+b (c-c m))+a B (b c (m+1)-a d (m+5)))}{2 a^2 b^3 e (m+1)}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+3))-a B (2 b c (m+3)-a d (m+5)))}{2 a b^3 e (m+1)}-\frac{d^2 (e x)^{m+3} (A b (m+3)-a B (m+5))}{2 a b^2 e^3 (m+3)}+\frac{\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{2 a b e \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.496609, size = 170, normalized size = 0.69 \[ \frac{x (e x)^m \left (\frac{c x^2 (2 A d+B c) \, _2F_1\left (2,\frac{m+3}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+d x^4 \left (\frac{(A d+2 B c) \, _2F_1\left (2,\frac{m+5}{2};\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}+\frac{B d x^2 \, _2F_1\left (2,\frac{m+7}{2};\frac{m+9}{2};-\frac{b x^2}{a}\right )}{m+7}\right )+\frac{A c^2 \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*((A*c^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(1
+ m) + (c*(B*c + 2*A*d)*x^2*Hypergeometric2F1[2, (3 + m)/2, (5 + m)/2, -((b*x^2)
/a)])/(3 + m) + d*x^4*(((2*B*c + A*d)*Hypergeometric2F1[2, (5 + m)/2, (7 + m)/2,
-((b*x^2)/a)])/(5 + m) + (B*d*x^2*Hypergeometric2F1[2, (7 + m)/2, (9 + m)/2, -(
(b*x^2)/a)])/(7 + m))))/a^2

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) ^{2}}{ \left ( b{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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