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

Optimal. Leaf size=162 \[ \frac{B (e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{d e (m+1)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} (B c-A d) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c d e (m+1)} \]

[Out]

-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2,
-((b*x^2)/a), -((d*x^2)/c)])/(c*d*e*(1 + m)*(1 + (b*x^2)/a)^p)) + (B*(e*x)^(1 +
m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(d*
e*(1 + m)*(1 + (b*x^2)/a)^p)

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Rubi [A]  time = 0.445689, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{B (e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{d e (m+1)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} (B c-A d) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c d e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2,
-((b*x^2)/a), -((d*x^2)/c)])/(c*d*e*(1 + m)*(1 + (b*x^2)/a)^p)) + (B*(e*x)^(1 +
m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(d*
e*(1 + m)*(1 + (b*x^2)/a)^p)

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Rubi in Sympy [A]  time = 66.4841, size = 124, normalized size = 0.77 \[ \frac{B \left (e x\right )^{m + 1} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{d e \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (A d - B c\right ) \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},1,- p,\frac{m}{2} + \frac{3}{2},- \frac{d x^{2}}{c},- \frac{b x^{2}}{a} \right )}}{c d e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(e*x)**(m + 1)*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, m/2 + 1/2), (m/
2 + 3/2,), -b*x**2/a)/(d*e*(m + 1)) + (e*x)**(m + 1)*(1 + b*x**2/a)**(-p)*(a + b
*x**2)**p*(A*d - B*c)*appellf1(m/2 + 1/2, 1, -p, m/2 + 3/2, -d*x**2/c, -b*x**2/a
)/(c*d*e*(m + 1))

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Mathematica [B]  time = 0.756713, size = 446, normalized size = 2.75 \[ \frac{x (e x)^m \left (a+b x^2\right )^p \left (a A c d (m+3) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-a B c^2 (m+3) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+B \left (c+d x^2\right ) \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (2 x^2 \left (b c p F_1\left (\frac{m+3}{2};1-p,1;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-a d F_1\left (\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+a c (m+3) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}{d (m+1) \left (c+d x^2\right ) \left (2 x^2 \left (b c p F_1\left (\frac{m+3}{2};1-p,1;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-a d F_1\left (\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+a c (m+3) F_1\left (\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(e*x)^m*(a + b*x^2)^p*(-(a*B*c^2*(3 + m)*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2
, -((b*x^2)/a), -((d*x^2)/c)]) + a*A*c*d*(3 + m)*AppellF1[(1 + m)/2, -p, 1, (3 +
m)/2, -((b*x^2)/a), -((d*x^2)/c)] + (B*(c + d*x^2)*(a*c*(3 + m)*AppellF1[(1 + m
)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^2*(b*c*p*AppellF1[(3 +
m)/2, 1 - p, 1, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] - a*d*AppellF1[(3 + m)/2,
-p, 2, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)]))*Hypergeometric2F1[(1 + m)/2, -p
, (3 + m)/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/(d*(1 + m)*(c + d*x^2)*(a*c*(3 +
m)*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^2*(b
*c*p*AppellF1[(3 + m)/2, 1 - p, 1, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] - a*d*
AppellF1[(3 + m)/2, -p, 2, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( B{x}^{2}+A \right ) }{d{x}^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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