3.32 \(\int \frac{(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{\left (c+d x^2\right )^2} \, dx\)
Optimal. Leaf size=171 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+1)-B c (m+3)))}{2 c^2 d^2 e (m+1)}-\frac{\left (A+B x^2\right ) (e x)^{m+1} (b c-a d)}{2 c d e \left (c+d x^2\right )}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+3))}{2 c d^2 e (m+1)} \]
[Out]
-(B*(a*d*(1 + m) - b*c*(3 + m))*(e*x)^(1 + m))/(2*c*d^2*e*(1 + m)) - ((b*c - a*d
)*(e*x)^(1 + m)*(A + B*x^2))/(2*c*d*e*(c + d*x^2)) + ((a*d*(A*d*(1 - m) + B*c*(1
+ m)) + b*c*(A*d*(1 + m) - B*c*(3 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1
+ m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*d^2*e*(1 + m))
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Rubi [A] time = 0.606746, antiderivative size = 171, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103
\[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+1)-B c (m+3)))}{2 c^2 d^2 e (m+1)}-\frac{\left (A+B x^2\right ) (e x)^{m+1} (b c-a d)}{2 c d e \left (c+d x^2\right )}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+3))}{2 c d^2 e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^2,x]
[Out]
-(B*(a*d*(1 + m) - b*c*(3 + m))*(e*x)^(1 + m))/(2*c*d^2*e*(1 + m)) - ((b*c - a*d
)*(e*x)^(1 + m)*(A + B*x^2))/(2*c*d*e*(c + d*x^2)) + ((a*d*(A*d*(1 - m) + B*c*(1
+ m)) + b*c*(A*d*(1 + m) - B*c*(3 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1
+ m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*d^2*e*(1 + m))
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Rubi in Sympy [A] time = 44.8882, size = 148, normalized size = 0.87 \[ \frac{b \left (e x\right )^{m + 1} \left (2 A d - \left (m + 3\right ) \left (A d - B c\right )\right )}{2 c d^{2} e \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (a + b x^{2}\right ) \left (A d - B c\right )}{2 c d e \left (c + d x^{2}\right )} + \frac{\left (e x\right )^{m + 1} \left (a d \left (- A d m + A d + B c m + B c\right ) - b c \left (2 A d - \left (m + 3\right ) \left (A d - B c\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{2 c^{2} d^{2} e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x**2+a)*(B*x**2+A)/(d*x**2+c)**2,x)
[Out]
b*(e*x)**(m + 1)*(2*A*d - (m + 3)*(A*d - B*c))/(2*c*d**2*e*(m + 1)) + (e*x)**(m
+ 1)*(a + b*x**2)*(A*d - B*c)/(2*c*d*e*(c + d*x**2)) + (e*x)**(m + 1)*(a*d*(-A*d
*m + A*d + B*c*m + B*c) - b*c*(2*A*d - (m + 3)*(A*d - B*c)))*hyper((1, m/2 + 1/2
), (m/2 + 3/2,), -d*x**2/c)/(2*c**2*d**2*e*(m + 1))
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Mathematica [A] time = 0.21294, size = 121, normalized size = 0.71 \[ \frac{x (e x)^m \left (\frac{x^2 (a B+A b) \, _2F_1\left (2,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right )}{m+3}+\frac{a A \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{m+1}+\frac{b B x^4 \, _2F_1\left (2,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right )}{m+5}\right )}{c^2} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^2,x]
[Out]
(x*(e*x)^m*((a*A*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(1 +
m) + ((A*b + a*B)*x^2*Hypergeometric2F1[2, (3 + m)/2, (5 + m)/2, -((d*x^2)/c)])/
(3 + m) + (b*B*x^4*Hypergeometric2F1[2, (5 + m)/2, (7 + m)/2, -((d*x^2)/c)])/(5
+ m)))/c^2
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Maple [F] time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) \left ( B{x}^{2}+A \right ) }{ \left ( d{x}^{2}+c \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c)^2,x)
[Out]
int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c)^2, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B b x^{4} +{\left (B a + A b\right )} x^{2} + A a\right )} \left (e x\right )^{m}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c)^2,x, algorithm="fricas")
[Out]
integral((B*b*x^4 + (B*a + A*b)*x^2 + A*a)*(e*x)^m/(d^2*x^4 + 2*c*d*x^2 + c^2),
x)
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Sympy [A] time = 152.577, size = 2076, normalized size = 12.14 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(b*x**2+a)*(B*x**2+A)/(d*x**2+c)**2,x)
[Out]
A*a*(-c*e**m*m**2*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(
m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) + 2*c*e**m
*m*x*x**m*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 +
3/2)) + c*e**m*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2
+ 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) + 2*c*e**m*x*
x**m*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2))
- d*e**m*m**2*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(
m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) + d*e**m*x
**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*c*
*3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2))) + A*b*(-c*e**m*m**2*x**3*
x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*g
amma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 4*c*e**m*m*x**3*x**m*lerchph
i(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5
/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) + 2*c*e**m*m*x**3*x**m*gamma(m/2 + 3/2)/(8
*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 3*c*e**m*x**3*x**m*le
rchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/
2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) + 6*c*e**m*x**3*x**m*gamma(m/2 + 3/2)
/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - d*e**m*m**2*x**5*x
**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*ga
mma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 4*d*e**m*m*x**5*x**m*lerchphi
(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/
2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 3*d*e**m*x**5*x**m*lerchphi(d*x**2*exp_po
lar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*
x**2*gamma(m/2 + 5/2))) + B*a*(-c*e**m*m**2*x**3*x**m*lerchphi(d*x**2*exp_polar(
I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2
*gamma(m/2 + 5/2)) - 4*c*e**m*m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1,
m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 +
5/2)) + 2*c*e**m*m*x**3*x**m*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2
*d*x**2*gamma(m/2 + 5/2)) - 3*c*e**m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c
, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(
m/2 + 5/2)) + 6*c*e**m*x**3*x**m*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c
**2*d*x**2*gamma(m/2 + 5/2)) - d*e**m*m**2*x**5*x**m*lerchphi(d*x**2*exp_polar(I
*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*
gamma(m/2 + 5/2)) - 4*d*e**m*m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m
/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 +
5/2)) - 3*d*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamm
a(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2))) + B*b*(
-c*e**m*m**2*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/
2 + 5/2)/(8*c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)) - 8*c*e**m*m
*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8*
c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)) + 2*c*e**m*m*x**5*x**m*g
amma(m/2 + 5/2)/(8*c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)) - 15*
c*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/
2)/(8*c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)) + 10*c*e**m*x**5*x
**m*gamma(m/2 + 5/2)/(8*c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2))
- d*e**m*m**2*x**7*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m
/2 + 5/2)/(8*c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)) - 8*d*e**m*
m*x**7*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8
*c**3*gamma(m/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)) - 15*d*e**m*x**7*x**m*l
erchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(8*c**3*gamma(m
/2 + 7/2) + 8*c**2*d*x**2*gamma(m/2 + 7/2)))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c)^2,x, algorithm="giac")
[Out]
integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c)^2, x)