Integrand size = 22, antiderivative size = 201 \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^{1+m}}{2 a b^3 (1+m)}-\frac {d^2 (b c (3+m)-a d (5+m)) x^{3+m}}{2 a b^2 (3+m)}+\frac {(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^2 (a d (5+m)+b (c-c m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{2 a^2 b^3 (1+m)} \]
-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)
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 \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Result too large to show} \]
-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*(94 5 + 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)*Hu rwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + 1771875*a*c^3*HurwitzLerchPhi[ -((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] + 178 050*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*HurwitzLerc hPhi[-((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*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 843 7500*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*Hurwitz LerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 396*a*c^2*d*m^7*x^2*HurwitzLerch...
Time = 0.43 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {370, 25, 437, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 370 |
\(\displaystyle \frac {x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )}-\frac {\int -\frac {x^m \left (d x^2+c\right ) \left (c (a d (m+1)+b (c-c m))-d (b c (m+3)-a d (m+5)) x^2\right )}{b x^2+a}dx}{2 a b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {x^m \left (d x^2+c\right ) \left (c (a d (m+1)+b (c-c m))-d (b c (m+3)-a d (m+5)) x^2\right )}{b x^2+a}dx}{2 a b}+\frac {x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 437 |
\(\displaystyle \frac {\int \left (-\frac {d \left (2 b^2 (m+1) c^2-3 a b d (m+3) c+a^2 d^2 (m+5)\right ) x^m}{b^2}+\frac {\left (b^3 c^3-b^3 m c^3+3 a b^2 d c^2+3 a b^2 d m c^2-9 a^2 b d^2 c-3 a^2 b d^2 m c+5 a^3 d^3+a^3 d^3 m\right ) x^m}{b^2 \left (b x^2+a\right )}-\frac {d^2 (b c (m+3)-a d (m+5)) x^{m+2}}{b}\right )dx}{2 a b}+\frac {x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {d x^{m+1} \left (a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right )}{b^2 (m+1)}+\frac {x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a b^2 (m+1)}-\frac {d^2 x^{m+3} (b c (m+3)-a d (m+5))}{b (m+3)}}{2 a b}+\frac {x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )}\) |
((b*c - a*d)*x^(1 + m)*(c + d*x^2)^2)/(2*a*b*(a + b*x^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))/(b^2*(1 + m)) ) - (d^2*(b*c*(3 + m) - a*d*(5 + m))*x^(3 + m))/(b*(3 + m)) + ((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)])/(a*b^2*(1 + m)))/(2*a*b)
3.4.37.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f , g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
\[\int \frac {x^{m} \left (d \,x^{2}+c \right )^{3}}{\left (b \,x^{2}+a \right )^{2}}d x\]
\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{m} \left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
\[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^m\,{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^2} \,d x \]