3.40 \(\int \frac{(e x)^m \left (A+B x^2\right )}{\left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=103 \[ \frac{(e x)^{m+1} (A d (3-m)+B c (m+1)) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{4 c^3 d e (m+1)}-\frac{(e x)^{m+1} (B c-A d)}{4 c d e \left (c+d x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.152584, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(e x)^{m+1} (A d (3-m)+B c (m+1)) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{4 c^3 d e (m+1)}-\frac{(e x)^{m+1} (B c-A d)}{4 c d e \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(A + B*x^2))/(c + d*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.6257, size = 80, normalized size = 0.78 \[ \frac{\left (e x\right )^{m + 1} \left (A d - B c\right )}{4 c d e \left (c + d x^{2}\right )^{2}} + \frac{\left (e x\right )^{m + 1} \left (A d \left (- m + 3\right ) + B c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{4 c^{3} d e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x**2+A)/(d*x**2+c)**3,x)

[Out]

(e*x)**(m + 1)*(A*d - B*c)/(4*c*d*e*(c + d*x**2)**2) + (e*x)**(m + 1)*(A*d*(-m +
3) + B*c*(m + 1))*hyper((2, m/2 + 1/2), (m/2 + 3/2,), -d*x**2/c)/(4*c**3*d*e*(m
+ 1))

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Mathematica [A]  time = 0.0907193, size = 81, normalized size = 0.79 \[ \frac{x (e x)^m \left ((A d-B c) \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )+B c \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )\right )}{c^3 d (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2)^3,x]

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) }{ \left ( d{x}^{2}+c \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x^2+A)/(d*x^2+c)^3,x)

[Out]

int((e*x)^m*(B*x^2+A)/(d*x^2+c)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)*(e*x)^m/(d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3), 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)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c)^3, x)