3.36 \(\int \frac {(e x)^m (A+B x^2)}{(a+b x^2)^3 (c+d x^2)^2} \, dx\)

Optimal. Leaf size=491 \[ -\frac {d (e x)^{m+1} \left (A \left (4 a^2 d^2+a b c d (11-m)-b^2 c^2 (3-m)\right )-a B c (a d (11-m)+b c (m+1))\right )}{8 a^2 c e \left (c+d x^2\right ) (b c-a d)^3}+\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{8 a^2 e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)^2}+\frac {b (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2-12 m+35\right )-2 a b c d \left (m^2-8 m+7\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (-m^2+4 m+5\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 e (m+1) (b c-a d)^4}+\frac {d^2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (B c (5-m)-A d (7-m)))}{2 c^2 e (m+1) (b c-a d)^4}+\frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-1/8*d*(A*(4*a^2*d^2-b^2*c^2*(3-m)+a*b*c*d*(11-m))-a*B*c*(a*d*(11-m)+b*c*(1+m)))*(e*x)^(1+m)/a^2/c/(-a*d+b*c)^
3/e/(d*x^2+c)+1/4*(A*b-B*a)*(e*x)^(1+m)/a/(-a*d+b*c)/e/(b*x^2+a)^2/(d*x^2+c)+1/8*(A*b*(b*c*(3-m)-a*d*(9-m))+a*
B*(a*d*(5-m)+b*c*(1+m)))*(e*x)^(1+m)/a^2/(-a*d+b*c)^2/e/(b*x^2+a)/(d*x^2+c)+1/8*b*(a*B*(b^2*c^2*(-m^2+1)-2*a*b
*c*d*(-m^2+4*m+5)-a^2*d^2*(m^2-8*m+15))+A*b*(a^2*d^2*(m^2-12*m+35)-2*a*b*c*d*(m^2-8*m+7)+b^2*c^2*(m^2-4*m+3)))
*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^3/(-a*d+b*c)^4/e/(1+m)+1/2*d^2*(b*c*(B*c*(5-m)-A
*d*(7-m))+a*d*(A*d*(1-m)+B*c*(1+m)))*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-d*x^2/c)/c^2/(-a*d+b*c)
^4/e/(1+m)

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Rubi [A]  time = 1.43, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {579, 584, 364} \[ \frac {b (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2-12 m+35\right )-2 a b c d \left (m^2-8 m+7\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (-m^2+4 m+5\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 e (m+1) (b c-a d)^4}-\frac {d (e x)^{m+1} \left (A \left (4 a^2 d^2+a b c d (11-m)-b^2 c^2 (3-m)\right )-a B c (a d (11-m)+b c (m+1))\right )}{8 a^2 c e \left (c+d x^2\right ) (b c-a d)^3}+\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{8 a^2 e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)^2}+\frac {d^2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (B c (5-m)-A d (7-m)))}{2 c^2 e (m+1) (b c-a d)^4}+\frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(A*(4*a^2*d^2 - b^2*c^2*(3 - m) + a*b*c*d*(11 - m)) - a*B*c*(a*d*(11 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(
8*a^2*c*(b*c - a*d)^3*e*(c + d*x^2)) + ((A*b - a*B)*(e*x)^(1 + m))/(4*a*(b*c - a*d)*e*(a + b*x^2)^2*(c + d*x^2
)) + ((A*b*(b*c*(3 - m) - a*d*(9 - m)) + a*B*(a*d*(5 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*(b*c - a*d)^2*
e*(a + b*x^2)*(c + d*x^2)) + (b*(a*B*(b^2*c^2*(1 - m^2) - 2*a*b*c*d*(5 + 4*m - m^2) - a^2*d^2*(15 - 8*m + m^2)
) + A*b*(a^2*d^2*(35 - 12*m + m^2) - 2*a*b*c*d*(7 - 8*m + m^2) + b^2*c^2*(3 - 4*m + m^2)))*(e*x)^(1 + m)*Hyper
geometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^4*e*(1 + m)) + (d^2*(b*c*(B*c*(5 - m)
- A*d*(7 - m)) + a*d*(A*d*(1 - m) + B*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((
d*x^2)/c)])/(2*c^2*(b*c - a*d)^4*e*(1 + m))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 579

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 + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx &=\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}-\frac {\int \frac {(e x)^m \left (4 a A d-A b c (3-m)-a B c (1+m)-(A b-a B) d (5-m) x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx}{4 a (b c-a d)}\\ &=\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {(A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {(e x)^m \left (-a B c (1+m) (a d (7-m)-b (c-c m))+A \left (8 a^2 d^2-a b c d \left (5-10 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )+d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 a^2 (b c-a d)^2}\\ &=-\frac {d \left (A \left (4 a^2 d^2-b^2 c^2 (3-m)+a b c d (11-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )}+\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {(A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {(e x)^m \left (-2 \left (a B c \left (4 a^2 d^2-b^2 c^2 (1-m)+a b c d (9-m)\right ) (1+m)-A \left (24 a^2 b c d^2-4 a^3 d^3 (1-m)-a b^2 c^2 d \left (11-12 m+m^2\right )+b^3 c^3 \left (3-4 m+m^2\right )\right )\right )-2 b d (1-m) \left (A \left (4 a^2 d^2-b^2 c^2 (3-m)+a b c d (11-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 a^2 c (b c-a d)^3}\\ &=-\frac {d \left (A \left (4 a^2 d^2-b^2 c^2 (3-m)+a b c d (11-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )}+\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {(A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \left (\frac {2 b c \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (5+4 m-m^2\right )-a^2 d^2 \left (15-8 m+m^2\right )\right )+A b \left (a^2 d^2 \left (35-12 m+m^2\right )-2 a b c d \left (7-8 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right ) (e x)^m}{(b c-a d) \left (a+b x^2\right )}+\frac {8 a^2 d^2 (b c (B c (5-m)-A d (7-m))+a d (A d (1-m)+B c (1+m))) (e x)^m}{(b c-a d) \left (c+d x^2\right )}\right ) \, dx}{16 a^2 c (b c-a d)^3}\\ &=-\frac {d \left (A \left (4 a^2 d^2-b^2 c^2 (3-m)+a b c d (11-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )}+\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {(A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (d^2 (b c (B c (5-m)-A d (7-m))+a d (A d (1-m)+B c (1+m)))\right ) \int \frac {(e x)^m}{c+d x^2} \, dx}{2 c (b c-a d)^4}+\frac {\left (b \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (5+4 m-m^2\right )-a^2 d^2 \left (15-8 m+m^2\right )\right )+A b \left (a^2 d^2 \left (35-12 m+m^2\right )-2 a b c d \left (7-8 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right )\right ) \int \frac {(e x)^m}{a+b x^2} \, dx}{8 a^2 (b c-a d)^4}\\ &=-\frac {d \left (A \left (4 a^2 d^2-b^2 c^2 (3-m)+a b c d (11-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )}+\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {(A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (5+4 m-m^2\right )-a^2 d^2 \left (15-8 m+m^2\right )\right )+A b \left (a^2 d^2 \left (35-12 m+m^2\right )-2 a b c d \left (7-8 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right ) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{8 a^3 (b c-a d)^4 e (1+m)}+\frac {d^2 (b c (B c (5-m)-A d (7-m))+a d (A d (1-m)+B c (1+m))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 (b c-a d)^4 e (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 265, normalized size = 0.54 \[ \frac {x (e x)^m \left (\frac {b (A b-a B) (b c-a d)^2 \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a^3}+\frac {b (b c-a d) \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) (a B d-2 A b d+b B c)}{a^2}+\frac {d^2 (b c-a d) (B c-A d) \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c^2}+\frac {d^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right ) (a B d-3 A b d+2 b B c)}{c}-\frac {b d \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right ) (a B d-3 A b d+2 b B c)}{a}\right )}{(m+1) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(-((b*d*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a) +
(d^2*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c + (b*(b*c - a*d)*
(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^2 + (d^2*(b*c - a*d)*(B*
c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^2 + (b*(A*b - a*B)*(b*c - a*d)^2*Hypergeo
metric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^3))/((b*c - a*d)^4*(1 + m))

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fricas [F]  time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{b^{3} d^{2} x^{10} + {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{8} + {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{6} + a^{3} c^{2} + {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4} + {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x^2 + A)*(e*x)^m/(b^3*d^2*x^10 + (2*b^3*c*d + 3*a*b^2*d^2)*x^8 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*
d^2)*x^6 + a^3*c^2 + (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*x^4 + (3*a^2*b*c^2 + 2*a^3*c*d)*x^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{2}+A \right ) \left (e x \right )^{m}}{\left (b \,x^{2}+a \right )^{3} \left (d \,x^{2}+c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m}{{\left (b\,x^2+a\right )}^3\,{\left (d\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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