\(\int \frac {x^m (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [337]

Optimal result

Integrand size = 22, antiderivative size = 201 \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^{1+m}}{2 a b^3 (1+m)}-\frac {d^2 (b c (3+m)-a d (5+m)) x^{3+m}}{2 a b^2 (3+m)}+\frac {(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^2 (a d (5+m)+b (c-c m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{2 a^2 b^3 (1+m)} \]

[Out]

-1/2*d*(2*b^2*c^2*(1+m)-3*a*b*c*d*(3+m)+a^2*d^2*(5+m))*x^(1+m)/a/b^3/(1+m)-1/2*d^2*(b*c*(3+m)-a*d*(5+m))*x^(3+
m)/a/b^2/(3+m)+1/2*(-a*d+b*c)*x^(1+m)*(d*x^2+c)^2/a/b/(b*x^2+a)+1/2*(-a*d+b*c)^2*(a*d*(5+m)+b*(-c*m+c))*x^(1+m
)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^2/b^3/(1+m)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {479, 584, 371} \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{2 a^2 b^3 (m+1)}-\frac {d x^{m+1} \left (a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right )}{2 a b^3 (m+1)}-\frac {d^2 x^{m+3} (b c (m+3)-a d (m+5))}{2 a b^2 (m+3)}+\frac {x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[In]

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

[Out]

-1/2*(d*(2*b^2*c^2*(1 + m) - 3*a*b*c*d*(3 + m) + a^2*d^2*(5 + m))*x^(1 + m))/(a*b^3*(1 + m)) - (d^2*(b*c*(3 +
m) - a*d*(5 + m))*x^(3 + m))/(2*a*b^2*(3 + m)) + ((b*c - a*d)*x^(1 + m)*(c + d*x^2)^2)/(2*a*b*(a + b*x^2)) + (
(b*c - a*d)^2*(a*d*(5 + m) + b*(c - c*m))*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/
(2*a^2*b^3*(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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\int \frac {x^m \left (c+d x^2\right ) \left (-c (b c (1-m)+a d (1+m))+d (b c (3+m)-a d (5+m)) x^2\right )}{a+b x^2} \, dx}{2 a b} \\ & = \frac {(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\int \left (\frac {d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^m}{b^2}+\frac {d^2 (b c (3+m)-a d (5+m)) x^{2+m}}{b}+\frac {\left (-b^3 c^3-3 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3+b^3 c^3 m-3 a b^2 c^2 d m+3 a^2 b c d^2 m-a^3 d^3 m\right ) x^m}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 a b} \\ & = -\frac {d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^{1+m}}{2 a b^3 (1+m)}-\frac {d^2 (b c (3+m)-a d (5+m)) x^{3+m}}{2 a b^2 (3+m)}+\frac {(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\left (-b^3 c^3-3 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3+b^3 c^3 m-3 a b^2 c^2 d m+3 a^2 b c d^2 m-a^3 d^3 m\right ) \int \frac {x^m}{a+b x^2} \, dx}{2 a b^3} \\ & = -\frac {d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^{1+m}}{2 a b^3 (1+m)}-\frac {d^2 (b c (3+m)-a d (5+m)) x^{3+m}}{2 a b^2 (3+m)}+\frac {(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^2 (b c (1-m)+a d (5+m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b^3 (1+m)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 3.60 (sec) , antiderivative size = 2524, normalized size of antiderivative = 12.56 \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

-1/192*(x^(1 + m)*(a*(945 + 744*m + 206*m^2 + 24*m^3 + m^4)*(c^3*(-47 + 52*m + 6*m^2 + 4*m^3 + m^4) + 3*c^2*d*
(1 + m)^4*x^2 + 3*c*d^2*(1 + m)^4*x^4 + d^3*(1 + m)^4*x^6)*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] - 3*a*(
945 + 744*m + 206*m^2 + 24*m^3 + m^4)*(c^3*(3 + m)^4 + 3*c^2*d*(65 + 92*m + 54*m^2 + 12*m^3 + m^4)*x^2 + 3*c*d
^2*(3 + m)^4*x^4 + d^3*(3 + m)^4*x^6)*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + 1771875*a*c^3*HurwitzLerch
Phi[-((b*x^2)/a), 1, (5 + m)/2] + 2812500*a*c^3*m*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1927500*a*c^3*
m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 745500*a*c^3*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2]
+ 178050*a*c^3*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26892*a*c^3*m^5*HurwitzLerchPhi[-((b*x^2)/a)
, 1, (5 + m)/2] + 2508*a*c^3*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 132*a*c^3*m^7*HurwitzLerchPhi[-
((b*x^2)/a), 1, (5 + m)/2] + 3*a*c^3*m^8*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5315625*a*c^2*d*x^2*Hur
witzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8437500*a*c^2*d*m*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] +
5782500*a*c^2*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2236500*a*c^2*d*m^3*x^2*HurwitzLerchPhi
[-((b*x^2)/a), 1, (5 + m)/2] + 534150*a*c^2*d*m^4*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 80676*a*c^
2*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 7524*a*c^2*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1
, (5 + m)/2] + 396*a*c^2*d*m^7*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 9*a*c^2*d*m^8*x^2*HurwitzLerc
hPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5723865*a*c*d^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8894988*a
*c*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5978628*a*c*d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a
), 1, (5 + m)/2] + 2276532*a*c*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 538038*a*c*d^2*m^4*x^
4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 80820*a*c*d^2*m^5*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)
/2] + 7524*a*c*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 396*a*c*d^2*m^7*x^4*HurwitzLerchPhi[-
((b*x^2)/a), 1, (5 + m)/2] + 9*a*c*d^2*m^8*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1771875*a*d^3*x^6
*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2812500*a*d^3*m*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2]
+ 1927500*a*d^3*m^2*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 745500*a*d^3*m^3*x^6*HurwitzLerchPhi[-(
(b*x^2)/a), 1, (5 + m)/2] + 178050*a*d^3*m^4*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26892*a*d^3*m^5
*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2508*a*d^3*m^6*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)
/2] + 132*a*d^3*m^7*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 3*a*d^3*m^8*x^6*HurwitzLerchPhi[-((b*x^2
)/a), 1, (5 + m)/2] - 2268945*a*c^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3082884*a*c^3*m*HurwitzLerch
Phi[-((b*x^2)/a), 1, (7 + m)/2] - 1793204*a*c^3*m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 585452*a*c^3
*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 117670*a*c^3*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2
] - 14940*a*c^3*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1172*a*c^3*m^6*HurwitzLerchPhi[-((b*x^2)/a),
1, (7 + m)/2] - 52*a*c^3*m^7*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - a*c^3*m^8*HurwitzLerchPhi[-((b*x^2
)/a), 1, (7 + m)/2] - 6806835*a*c^2*d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 9248652*a*c^2*d*m*x^2*
HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 5379612*a*c^2*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)
/2] - 1756356*a*c^2*d*m^3*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 353010*a*c^2*d*m^4*x^2*HurwitzLerc
hPhi[-((b*x^2)/a), 1, (7 + m)/2] - 44820*a*c^2*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3516*a*
c^2*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 156*a*c^2*d*m^7*x^2*HurwitzLerchPhi[-((b*x^2)/a),
1, (7 + m)/2] - 3*a*c^2*d*m^8*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 6806835*a*c*d^2*x^4*HurwitzLer
chPhi[-((b*x^2)/a), 1, (7 + m)/2] - 9248652*a*c*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 537961
2*a*c*d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1756356*a*c*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x
^2)/a), 1, (7 + m)/2] - 353010*a*c*d^2*m^4*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 44820*a*c*d^2*m^5
*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3516*a*c*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 +
m)/2] - 156*a*c*d^2*m^7*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3*a*c*d^2*m^8*x^4*HurwitzLerchPhi[-(
(b*x^2)/a), 1, (7 + m)/2] - 2042145*a*d^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 2858964*a*d^3*m*x^
6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1708052*a*d^3*m^2*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)
/2] - 569804*a*d^3*m^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 116278*a*d^3*m^4*x^6*HurwitzLerchPhi[
-((b*x^2)/a), 1, (7 + m)/2] - 14892*a*d^3*m^5*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1172*a*d^3*m^6
*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 52*a*d^3*m^7*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2
] - a*d^3*m^8*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] + 1536*b*c^3*x^2*HypergeometricPFQ[{2, 2, 2, 2,
3/2 + m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)] + 4608*b*c^2*d*x^4*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2},
{1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)] + 4608*b*c*d^2*x^6*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1,
11/2 + m/2}, -((b*x^2)/a)] + 1536*b*d^3*x^8*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 + m/2},
-((b*x^2)/a)]))/(a^3*(3 + m)*(5 + m)*(7 + m)*(9 + m))

Maple [F]

\[\int \frac {x^{m} \left (d \,x^{2}+c \right )^{3}}{\left (b \,x^{2}+a \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*x^m/(b^2*x^4 + 2*a*b*x^2 + a^2), x)

Sympy [F]

\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{m} \left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(x**m*(c + d*x**2)**3/(a + b*x**2)**2, x)

Maxima [F]

\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^m\,{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

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

[Out]

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