Integrand size = 24, antiderivative size = 409 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {(5 b c-17 a d) (b c-a d)^2 \sqrt {x}}{2 b^5}+\frac {d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac {d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac {17 d^3 x^{13/2}}{26 b^2}-\frac {x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{21/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} b^{21/4}} \]
1/10*d*(17*a^2*d^2-39*a*b*c*d+27*b^2*c^2)*x^(5/2)/b^4+1/18*d^2*(-17*a*d+39 *b*c)*x^(9/2)/b^3+17/26*d^3*x^(13/2)/b^2-1/2*x^(5/2)*(d*x^2+c)^3/b/(b*x^2+ a)+1/8*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*arctan(1-b^(1/4)*2^(1/2)*x^(1/ 2)/a^(1/4))/b^(21/4)*2^(1/2)-1/8*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*arct an(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/b^(21/4)*2^(1/2)+1/16*a^(1/4)*(-17*a *d+5*b*c)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2 ))/b^(21/4)*2^(1/2)-1/16*a^(1/4)*(-17*a*d+5*b*c)*(-a*d+b*c)^2*ln(a^(1/2)+x *b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(21/4)*2^(1/2)+1/2*(-17*a*d+5* b*c)*(-a*d+b*c)^2*x^(1/2)/b^5
Time = 0.55 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.74 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-9945 a^4 d^3+117 a^3 b d^2 \left (195 c-68 d x^2\right )+13 a^2 b^2 d \left (-1215 c^2+1404 c d x^2+68 d^2 x^4\right )+a b^3 \left (2925 c^3-12636 c^2 d x^2-2028 c d^2 x^4-340 d^3 x^6\right )+12 b^4 x^2 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )\right )}{a+b x^2}-585 \sqrt {2} \sqrt [4]{a} (b c-a d)^2 (-5 b c+17 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+585 \sqrt {2} \sqrt [4]{a} (b c-a d)^2 (-5 b c+17 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4680 b^{21/4}} \]
((4*b^(1/4)*Sqrt[x]*(-9945*a^4*d^3 + 117*a^3*b*d^2*(195*c - 68*d*x^2) + 13 *a^2*b^2*d*(-1215*c^2 + 1404*c*d*x^2 + 68*d^2*x^4) + a*b^3*(2925*c^3 - 126 36*c^2*d*x^2 - 2028*c*d^2*x^4 - 340*d^3*x^6) + 12*b^4*x^2*(195*c^3 + 117*c ^2*d*x^2 + 65*c*d^2*x^4 + 15*d^3*x^6)))/(a + b*x^2) - 585*Sqrt[2]*a^(1/4)* (b*c - a*d)^2*(-5*b*c + 17*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1 /4)*b^(1/4)*Sqrt[x])] + 585*Sqrt[2]*a^(1/4)*(b*c - a*d)^2*(-5*b*c + 17*a*d )*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(4680* b^(21/4))
Time = 0.58 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {368, 967, 1040, 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.
| \(\displaystyle \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^4 \left (d x^2+c\right )^3}{\left (b x^2+a\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle 2 \left (\frac {\int \frac {x^2 \left (d x^2+c\right )^2 \left (17 d x^2+5 c\right )}{b x^2+a}d\sqrt {x}}{4 b}-\frac {x^{5/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1040 |
| \(\displaystyle 2 \left (\frac {\int \left (\frac {17 d^3 x^6}{b}+\frac {d^2 (39 b c-17 a d) x^4}{b^2}+\frac {d \left (27 b^2 c^2-39 a b d c+17 a^2 d^2\right ) x^2}{b^3}+\frac {(5 b c-17 a d) (b c-a d)^2}{b^4}+\frac {17 d^3 a^4-39 b c d^2 a^3+27 b^2 c^2 d a^2-5 b^3 c^3 a}{b^4 \left (b x^2+a\right )}\right )d\sqrt {x}}{4 b}-\frac {x^{5/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {\frac {d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{5 b^3}+\frac {\sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (5 b c-17 a d) (b c-a d)^2}{2 \sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (5 b c-17 a d) (b c-a d)^2}{2 \sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} b^{17/4}}+\frac {\sqrt {x} (5 b c-17 a d) (b c-a d)^2}{b^4}+\frac {d^2 x^{9/2} (39 b c-17 a d)}{9 b^2}+\frac {17 d^3 x^{13/2}}{13 b}}{4 b}-\frac {x^{5/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
2*(-1/4*(x^(5/2)*(c + d*x^2)^3)/(b*(a + b*x^2)) + (((5*b*c - 17*a*d)*(b*c - a*d)^2*Sqrt[x])/b^4 + (d*(27*b^2*c^2 - 39*a*b*c*d + 17*a^2*d^2)*x^(5/2)) /(5*b^3) + (d^2*(39*b*c - 17*a*d)*x^(9/2))/(9*b^2) + (17*d^3*x^(13/2))/(13 *b) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)* Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*b^(17/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*b^(17/4) ) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)* b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*b^(17/4)) - (a^(1/4)*(5*b*c - 17* a*d)*(b*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b] *x])/(4*Sqrt[2]*b^(17/4)))/(4*b))
3.5.52.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*n*(p + 1))), x] - Simp[e^n/(b*n*(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*( q - 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBino mialQ[a, b, c, d, e, m, n, p, q, x]
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]
Time = 2.88 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {2 \left (-45 b^{3} d^{3} x^{6}+130 a \,b^{2} d^{3} x^{4}-195 b^{3} c \,d^{2} x^{4}-351 x^{2} a^{2} b \,d^{3}+702 x^{2} a \,b^{2} c \,d^{2}-351 x^{2} b^{3} c^{2} d +2340 a^{3} d^{3}-5265 a^{2} b c \,d^{2}+3510 a \,b^{2} c^{2} d -585 b^{3} c^{3}\right ) \sqrt {x}}{585 b^{5}}+\frac {a \left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (17 a d -5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{5}}\) | \(284\) |
| derivativedivides | \(-\frac {2 \left (-\frac {d^{3} x^{\frac {13}{2}} b^{3}}{13}+\frac {2 a \,b^{2} d^{3} x^{\frac {9}{2}}}{9}-\frac {b^{3} c \,d^{2} x^{\frac {9}{2}}}{3}-\frac {3 a^{2} b \,d^{3} x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} c \,d^{2} x^{\frac {5}{2}}}{5}-\frac {3 b^{3} c^{2} d \,x^{\frac {5}{2}}}{5}+4 a^{3} d^{3} \sqrt {x}-9 a^{2} b c \,d^{2} \sqrt {x}+6 a \,b^{2} c^{2} d \sqrt {x}-b^{3} c^{3} \sqrt {x}\right )}{b^{5}}+\frac {2 a \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (17 a^{3} d^{3}-39 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{5}}\) | \(327\) |
| default | \(-\frac {2 \left (-\frac {d^{3} x^{\frac {13}{2}} b^{3}}{13}+\frac {2 a \,b^{2} d^{3} x^{\frac {9}{2}}}{9}-\frac {b^{3} c \,d^{2} x^{\frac {9}{2}}}{3}-\frac {3 a^{2} b \,d^{3} x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} c \,d^{2} x^{\frac {5}{2}}}{5}-\frac {3 b^{3} c^{2} d \,x^{\frac {5}{2}}}{5}+4 a^{3} d^{3} \sqrt {x}-9 a^{2} b c \,d^{2} \sqrt {x}+6 a \,b^{2} c^{2} d \sqrt {x}-b^{3} c^{3} \sqrt {x}\right )}{b^{5}}+\frac {2 a \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (17 a^{3} d^{3}-39 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{5}}\) | \(327\) |
-2/585*(-45*b^3*d^3*x^6+130*a*b^2*d^3*x^4-195*b^3*c*d^2*x^4-351*a^2*b*d^3* x^2+702*a*b^2*c*d^2*x^2-351*b^3*c^2*d*x^2+2340*a^3*d^3-5265*a^2*b*c*d^2+35 10*a*b^2*c^2*d-585*b^3*c^3)*x^(1/2)/b^5+a/b^5*(2*a^2*d^2-4*a*b*c*d+2*b^2*c ^2)*((-1/4*a*d+1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(17*a*d-5*b*c)*(a/b)^(1/4)/ a*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x ^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arc tan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1822, normalized size of antiderivative = 4.45 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
1/4680*(585*(b^6*x^2 + a*b^5)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d ^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5 *c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a ^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*log (b^5*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7* d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b ^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428 *a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4) - (5*b^3*c^3 - 27*a*b^2*c^2* d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x)) - 585*(-I*b^6*x^2 - I*a*b^5)*(-( 625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 71806 0*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11 369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d ^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b* c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*log(I*b^5*(-(625*a*b^12*c^12 - 13500 *a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 260 3151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c ^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*...
Timed out. \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.34 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.22 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {x}}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 39 \, a^{2} b c d^{2} - 17 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{16 \, b^{5}} + \frac {2 \, {\left (45 \, b^{3} d^{3} x^{\frac {13}{2}} + 65 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{\frac {9}{2}} + 351 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {5}{2}} + 585 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \sqrt {x}\right )}}{585 \, b^{5}} \]
1/2*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*sqrt(x)/(b^6*x ^2 + a*b^5) - 1/16*(2*sqrt(2)*(5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqr t(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*( 5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*arctan(-1/2*sqrt (2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/( sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(5*b^3*c^3 - 27*a*b^2*c^2*d + 39* a^2*b*c*d^2 - 17*a^3*d^3)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^ 2*b*c*d^2 - 17*a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*a/b^5 + 2/585*(45*b^3*d^3*x^(13/2) + 65*(3*b^ 3*c*d^2 - 2*a*b^2*d^3)*x^(9/2) + 351*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^ 3)*x^(5/2) + 585*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*sqr t(x))/b^5
Time = 0.29 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.47 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{6}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{6}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{6}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 39 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 17 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{6}} + \frac {a b^{3} c^{3} \sqrt {x} - 3 \, a^{2} b^{2} c^{2} d \sqrt {x} + 3 \, a^{3} b c d^{2} \sqrt {x} - a^{4} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {2 \, {\left (45 \, b^{24} d^{3} x^{\frac {13}{2}} + 195 \, b^{24} c d^{2} x^{\frac {9}{2}} - 130 \, a b^{23} d^{3} x^{\frac {9}{2}} + 351 \, b^{24} c^{2} d x^{\frac {5}{2}} - 702 \, a b^{23} c d^{2} x^{\frac {5}{2}} + 351 \, a^{2} b^{22} d^{3} x^{\frac {5}{2}} + 585 \, b^{24} c^{3} \sqrt {x} - 3510 \, a b^{23} c^{2} d \sqrt {x} + 5265 \, a^{2} b^{22} c d^{2} \sqrt {x} - 2340 \, a^{3} b^{21} d^{3} \sqrt {x}\right )}}{585 \, b^{26}} \]
-1/8*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39* (a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*( sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^6 - 1/8*sqrt(2)*(5*(a*b^3) ^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d ^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^6 - 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*( a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4) *a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^6 + 1/16*sqrt (2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^( 1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1 /4) + x + sqrt(a/b))/b^6 + 1/2*(a*b^3*c^3*sqrt(x) - 3*a^2*b^2*c^2*d*sqrt(x ) + 3*a^3*b*c*d^2*sqrt(x) - a^4*d^3*sqrt(x))/((b*x^2 + a)*b^5) + 2/585*(45 *b^24*d^3*x^(13/2) + 195*b^24*c*d^2*x^(9/2) - 130*a*b^23*d^3*x^(9/2) + 351 *b^24*c^2*d*x^(5/2) - 702*a*b^23*c*d^2*x^(5/2) + 351*a^2*b^22*d^3*x^(5/2) + 585*b^24*c^3*sqrt(x) - 3510*a*b^23*c^2*d*sqrt(x) + 5265*a^2*b^22*c*d^2*s qrt(x) - 2340*a^3*b^21*d^3*sqrt(x))/b^26
Time = 0.30 (sec) , antiderivative size = 1850, normalized size of antiderivative = 4.52 \[ \int \frac {x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
x^(1/2)*((2*c^3)/b^2 - (2*a*((6*c^2*d)/b^2 + (2*a*((4*a*d^3)/b^3 - (6*c*d^ 2)/b^2))/b - (2*a^2*d^3)/b^4))/b + (a^2*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/b ^2) - x^(9/2)*((4*a*d^3)/(9*b^3) - (2*c*d^2)/(3*b^2)) + x^(5/2)*((6*c^2*d) /(5*b^2) + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/(5*b) - (2*a^2*d^3)/(5*b^ 4)) - (x^(1/2)*((a^4*d^3)/2 - (a*b^3*c^3)/2 + (3*a^2*b^2*c^2*d)/2 - (3*a^3 *b*c*d^2)/2))/(a*b^5 + b^6*x^2) + (2*d^3*x^(13/2))/(13*b^2) - ((-a)^(1/4)* atan((((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5*d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c ^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 + ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2))/b^(29/4))*1i)/(8*b^(21/4)) + ((-a)^(1/4)*(a*d - b*c)^2*( 17*a*d - 5*b*c)*((x^(1/2)*(289*a^8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5* d + 1119*a^4*b^4*c^4*d^2 - 2276*a^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1 326*a^7*b*c*d^5))/b^7 - ((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5 *d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2*c^2*d - 39*a^4*b*c*d^2))/b^(29/4))*1i)/( 8*b^(21/4)))/(((-a)^(1/4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*((x^(1/2)*(289*a^ 8*d^6 + 25*a^2*b^6*c^6 - 270*a^3*b^5*c^5*d + 1119*a^4*b^4*c^4*d^2 - 2276*a ^5*b^3*c^3*d^3 + 2439*a^6*b^2*c^2*d^4 - 1326*a^7*b*c*d^5))/b^7 + ((-a)^(1/ 4)*(a*d - b*c)^2*(17*a*d - 5*b*c)*(17*a^5*d^3 - 5*a^2*b^3*c^3 + 27*a^3*b^2 *c^2*d - 39*a^4*b*c*d^2))/b^(29/4)))/(8*b^(21/4)) - ((-a)^(1/4)*(a*d - ...