\(\int \frac {x^m (c+d x^2)^2}{(a+b x^2)^2} \, dx\) [338]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int \frac {x^m \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 x^{1+m}}{b^2 (1+m)}+\frac {(b c-a d)^2 x^{1+m}}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-a d) (a d (3+m)+b (c-c m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{2 a^2 b^2 (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {474, 470, 371} \[ \int \frac {x^m \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {x^{m+1} (b c-a d) (a d (m+3)+b (c-c m)) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{2 a^2 b^2 (m+1)}+\frac {x^{m+1} (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x^{m+1}}{b^2 (m+1)} \]

[In]

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

[Out]

(d^2*x^(1 + m))/(b^2*(1 + m)) + ((b*c - a*d)^2*x^(1 + m))/(2*a*b^2*(a + b*x^2)) + ((b*c - a*d)*(a*d*(3 + m) +
b*(c - c*m))*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b^2*(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 470

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

Rule 474

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 x^{1+m}}{2 a b^2 \left (a+b x^2\right )}-\frac {\int \frac {x^m \left (-2 b^2 c^2+(b c-a d)^2 (1+m)-2 a b d^2 x^2\right )}{a+b x^2} \, dx}{2 a b^2} \\ & = \frac {d^2 x^{1+m}}{b^2 (1+m)}+\frac {(b c-a d)^2 x^{1+m}}{2 a b^2 \left (a+b x^2\right )}--\frac {\left (-2 a^2 b d^2 (1+m)-b (1+m) \left (-2 b^2 c^2+(b c-a d)^2 (1+m)\right )\right ) \int \frac {x^m}{a+b x^2} \, dx}{2 a b^3 (1+m)} \\ & = \frac {d^2 x^{1+m}}{b^2 (1+m)}+\frac {(b c-a d)^2 x^{1+m}}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-a d) (b c (1-m)+a d (3+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^2 (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 = 1.62 (sec) , antiderivative size = 895, normalized size of antiderivative = 7.46 \[ \int \frac {x^m \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {x^{1+m} \left (a \left (105+71 m+15 m^2+m^3\right ) \left (c^2 \left (9-5 m+3 m^2+m^3\right )+2 c d (1+m)^3 x^2+d^2 (1+m)^3 x^4\right ) \Phi \left (-\frac {b x^2}{a},1,\frac {1+m}{2}\right )-2 a \left (105+71 m+15 m^2+m^3\right ) \left (c^2 (3+m)^3+2 c d \left (31+31 m+9 m^2+m^3\right ) x^2+d^2 (3+m)^3 x^4\right ) \Phi \left (-\frac {b x^2}{a},1,\frac {3+m}{2}\right )+13125 a c^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+16750 a c^2 m \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+8775 a c^2 m^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+2420 a c^2 m^3 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+371 a c^2 m^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+30 a c^2 m^5 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+a c^2 m^6 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+26250 a c d x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+33500 a c d m x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+17550 a c d m^2 x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+4840 a c d m^3 x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+742 a c d m^4 x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+60 a c d m^5 x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+2 a c d m^6 x^2 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+10605 a d^2 x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+14206 a d^2 m x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+7847 a d^2 m^2 x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+2276 a d^2 m^3 x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+363 a d^2 m^4 x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+30 a d^2 m^5 x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )+a d^2 m^6 x^4 \Phi \left (-\frac {b x^2}{a},1,\frac {5+m}{2}\right )-128 b c^2 x^2 \, _4F_3\left (2,2,2,\frac {3}{2}+\frac {m}{2};1,1,\frac {9}{2}+\frac {m}{2};-\frac {b x^2}{a}\right )-256 b c d x^4 \, _4F_3\left (2,2,2,\frac {3}{2}+\frac {m}{2};1,1,\frac {9}{2}+\frac {m}{2};-\frac {b x^2}{a}\right )-128 b d^2 x^6 \, _4F_3\left (2,2,2,\frac {3}{2}+\frac {m}{2};1,1,\frac {9}{2}+\frac {m}{2};-\frac {b x^2}{a}\right )\right )}{32 a^3 (3+m) (5+m) (7+m)} \]

[In]

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

[Out]

(x^(1 + m)*(a*(105 + 71*m + 15*m^2 + m^3)*(c^2*(9 - 5*m + 3*m^2 + m^3) + 2*c*d*(1 + m)^3*x^2 + d^2*(1 + m)^3*x
^4)*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] - 2*a*(105 + 71*m + 15*m^2 + m^3)*(c^2*(3 + m)^3 + 2*c*d*(31 +
31*m + 9*m^2 + m^3)*x^2 + d^2*(3 + m)^3*x^4)*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + 13125*a*c^2*Hurwit
zLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 16750*a*c^2*m*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8775*a*c^2
*m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2420*a*c^2*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2]
+ 371*a*c^2*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 30*a*c^2*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (5
+ m)/2] + a*c^2*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26250*a*c*d*x^2*HurwitzLerchPhi[-((b*x^2)/a
), 1, (5 + m)/2] + 33500*a*c*d*m*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 17550*a*c*d*m^2*x^2*Hurwitz
LerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 4840*a*c*d*m^3*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 742*a*
c*d*m^4*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 60*a*c*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5
+ m)/2] + 2*a*c*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 10605*a*d^2*x^4*HurwitzLerchPhi[-((b*
x^2)/a), 1, (5 + m)/2] + 14206*a*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 7847*a*d^2*m^2*x^4*Hu
rwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2276*a*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 3
63*a*d^2*m^4*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 30*a*d^2*m^5*x^4*HurwitzLerchPhi[-((b*x^2)/a),
1, (5 + m)/2] + a*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] - 128*b*c^2*x^2*HypergeometricPFQ[{2
, 2, 2, 3/2 + m/2}, {1, 1, 9/2 + m/2}, -((b*x^2)/a)] - 256*b*c*d*x^4*HypergeometricPFQ[{2, 2, 2, 3/2 + m/2}, {
1, 1, 9/2 + m/2}, -((b*x^2)/a)] - 128*b*d^2*x^6*HypergeometricPFQ[{2, 2, 2, 3/2 + m/2}, {1, 1, 9/2 + m/2}, -((
b*x^2)/a)]))/(32*a^3*(3 + m)*(5 + m)*(7 + m))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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