3.1.39 \(\int \frac {(e x)^m (a+b x^2) (A+B x^2)}{(c+d x^2)^3} \, dx\) [39]
Optimal. Leaf size=208 \[ -\frac {(b c-a d) (e x)^{1+m} \left (A+B x^2\right )}{4 c d e \left (c+d x^2\right )^2}+\frac {(b c (A d (1+m)-B c (3+m))+a d (A d (3-m)-B (c-c m))) (e x)^{1+m}}{8 c^2 d^2 e \left (c+d x^2\right )}+\frac {(a d (1-m) (A d (3-m)+B c (1+m))+b c (1+m) (A d (1-m)+B c (3+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{8 c^3 d^2 e (1+m)} \]
[Out]
-1/4*(-a*d+b*c)*(e*x)^(1+m)*(B*x^2+A)/c/d/e/(d*x^2+c)^2+1/8*(b*c*(A*d*(1+m)-B*c*(3+m))+a*d*(A*d*(3-m)-B*(-c*m+
c)))*(e*x)^(1+m)/c^2/d^2/e/(d*x^2+c)+1/8*(a*d*(1-m)*(A*d*(3-m)+B*c*(1+m))+b*c*(1+m)*(A*d*(1-m)+B*c*(3+m)))*(e*
x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-d*x^2/c)/c^3/d^2/e/(1+m)
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Rubi [A]
time = 0.21, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {591, 468, 371}
\begin {gather*} \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 (1-m) (A d (3-m)+B c (m+1))+b c (m+1) (A d (1-m)+B c (m+3)))}{8 c^3 d^2 e (m+1)}+\frac {(e x)^{m+1} (a d (A d (3-m)-B (c-c m))+b c (A d (m+1)-B c (m+3)))}{8 c^2 d^2 e \left (c+d x^2\right )}-\frac {\left (A+B x^2\right ) (e x)^{m+1} (b c-a d)}{4 c d e \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^3,x]
[Out]
-1/4*((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^2))/(c*d*e*(c + d*x^2)^2) + ((b*c*(A*d*(1 + m) - B*c*(3 + m)) + a*d*(
A*d*(3 - m) - B*(c - c*m)))*(e*x)^(1 + m))/(8*c^2*d^2*e*(c + d*x^2)) + ((a*d*(1 - m)*(A*d*(3 - m) + B*c*(1 + m
)) + b*c*(1 + m)*(A*d*(1 - m) + B*c*(3 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^
2)/c)])/(8*c^3*d^2*e*(1 + m))
Rule 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 468
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))
Rule 591
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{\left (c+d x^2\right )^3} \, dx &=-\frac {(b c-a d) (e x)^{1+m} \left (A+B x^2\right )}{4 c d e \left (c+d x^2\right )^2}-\frac {\int \frac {(e x)^m \left (-A (a d (3-m)+b c (1+m))-B (a d (1-m)+b c (3+m)) x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d}\\ &=-\frac {(b c-a d) (e x)^{1+m} \left (A+B x^2\right )}{4 c d e \left (c+d x^2\right )^2}+\frac {(b c (A d (1+m)-B c (3+m))+a d (A d (3-m)-B (c-c m))) (e x)^{1+m}}{8 c^2 d^2 e \left (c+d x^2\right )}+\frac {(a d (1-m) (A d (3-m)+B c (1+m))+b c (1+m) (A d (1-m)+B c (3+m))) \int \frac {(e x)^m}{c+d x^2} \, dx}{8 c^2 d^2}\\ &=-\frac {(b c-a d) (e x)^{1+m} \left (A+B x^2\right )}{4 c d e \left (c+d x^2\right )^2}+\frac {(b c (A d (1+m)-B c (3+m))+a d (A d (3-m)-B (c-c m))) (e x)^{1+m}}{8 c^2 d^2 e \left (c+d x^2\right )}+\frac {(a d (1-m) (A d (3-m)+B c (1+m))+b c (1+m) (A d (1-m)+B c (3+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{8 c^3 d^2 e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 133, normalized size = 0.64 \begin {gather*} \frac {x (e x)^m \left (b B c^2 \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )+c (-2 b B c+A b d+a B d) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )+(b c-a d) (B c-A d) \, _2F_1\left (3,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )\right )}{c^3 d^2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^3,x]
[Out]
(x*(e*x)^m*(b*B*c^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + c*(-2*b*B*c + A*b*d + a*B*d)*Hy
pergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/
2, (3 + m)/2, -((d*x^2)/c)]))/(c^3*d^2*(1 + m))
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (b \,x^{2}+a \right ) \left (B \,x^{2}+A \right )}{\left (d \,x^{2}+c \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c)^3,x)
[Out]
int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c)^3,x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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)^3,x, algorithm="maxima")
[Out]
integrate((B*x^2 + A)*(b*x^2 + a)*(x*e)^m/(d*x^2 + c)^3, x)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)^3,x, algorithm="fricas")
[Out]
integral((B*b*x^4 + (B*a + A*b)*x^2 + A*a)*(x*e)^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 [C] Result contains complex when optimal does not.
time = 96.35, size = 6368, normalized size = 30.62 \begin {gather*} \text {Too large to display} \end {gather*}
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)**3,x)
[Out]
A*a*(c**2*e**m*m**3*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/
2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 3*c**2*e**m*m**2*x*x**m*ler
chphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamm
a(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 2*c**2*e**m*m**2*x*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m
/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - c**2*e**m*m*x*x**m*lerchph
i(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/
2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 8*c**2*e**m*m*x*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/
2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 3*c**2*e**m*x*x**m*lerchphi(d*x**
2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2
) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 10*c**2*e**m*x*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*
c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 2*c*d*e**m*m**3*x**3*x**m*lerchphi(d*x**2
*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2)
+ 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 6*c*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)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m
/2 + 3/2)) - 2*c*d*e**m*m**2*x**3*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 +
3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - 2*c*d*e**m*m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2
+ 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma
(m/2 + 3/2)) + 4*c*d*e**m*m*x**3*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 +
3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 6*c*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)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/
2 + 3/2)) + 6*c*d*e**m*x**3*x**m*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2)
+ 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + d**2*e**m*m**3*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1
/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2
+ 3/2)) - 3*d**2*e**m*m**2*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c*
*5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) - d**2*e**m*m*x**5
*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*
x**2*gamma(m/2 + 3/2) + 32*c**3*d**2*x**4*gamma(m/2 + 3/2)) + 3*d**2*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(
I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*c**5*gamma(m/2 + 3/2) + 64*c**4*d*x**2*gamma(m/2 + 3/2) + 32*c**3*
d**2*x**4*gamma(m/2 + 3/2))) + A*b*(c**2*e**m*m**3*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*
gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5
/2)) + 3*c**2*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)/(32*c**5*g
amma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 2*c**2*e**m*m**2*x**
3*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(
m/2 + 5/2)) - c**2*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)/(32*c**5
*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 3*c**2*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)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x*
*2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + 18*c**2*e**m*x**3*x**m*gamma(m/2 + 3/2)/(32*c**5*g
amma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + 2*c*d*e**m*m**3*x**5
*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*
x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) + 6*c*d*e**m*m**2*x**5*x**m*lerchphi(d*x**2*exp_po
lar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32*c
**3*d**2*x**4*gamma(m/2 + 5/2)) - 2*c*d*e**m*m**2*x**5*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c*
*4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 2*c*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)/(32*c**5*gamma(m/2 + 5/2) + 64*c**4*d*x**2*gamma(m/2 + 5/2) + 32
*c**3*d**2*x**4*gamma(m/2 + 5/2)) - 4*c*d*e**m*m*x**5*x**m*gamma(m/2 + 3/2)/(32*c**5*gamma(m/2 + 5/2) + 64*c**
4*d*x**2*gamma(m/2 + 5/2) + 32*c**3*d**2*x**4*g...
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)^3,x, algorithm="giac")
[Out]
integrate((B*x^2 + A)*(b*x^2 + a)*(x*e)^m/(d*x^2 + c)^3, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,\left (b\,x^2+a\right )}{{\left (d\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x^2)*(e*x)^m*(a + b*x^2))/(c + d*x^2)^3,x)
[Out]
int(((A + B*x^2)*(e*x)^m*(a + b*x^2))/(c + d*x^2)^3, x)
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