Integrand size = 24, antiderivative size = 703 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{5/4} (b c+3 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {3 b^{5/4} (b c+3 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 b^{5/4} (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4} \]
-3/4*d*x^(3/2)/(-a*d+b*c)^2/(d*x^2+c)^2-1/2*x^(3/2)/(-a*d+b*c)/(b*x^2+a)/( d*x^2+c)^2-3/16*d*(a*d+7*b*c)*x^(3/2)/c/(-a*d+b*c)^3/(d*x^2+c)-3/8*b^(5/4) *(3*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(1/4)/(-a*d+b*c)^ 4*2^(1/2)+3/8*b^(5/4)*(3*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4) )/a^(1/4)/(-a*d+b*c)^4*2^(1/2)+3/64*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^ 2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(5/4)/(-a*d+b*c)^4*2^(1/2)- 3/64*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^( 1/2)/c^(1/4))/c^(5/4)/(-a*d+b*c)^4*2^(1/2)+3/16*b^(5/4)*(3*a*d+b*c)*ln(a^( 1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/(-a*d+b*c)^4*2^(1/ 2)-3/16*b^(5/4)*(3*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x ^(1/2))/a^(1/4)/(-a*d+b*c)^4*2^(1/2)-3/128*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15 *b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(5/4)/(- a*d+b*c)^4*2^(1/2)+3/128*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^2)*ln(c^(1/ 2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(5/4)/(-a*d+b*c)^4*2^(1/2)
Time = 2.28 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.56 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {-\frac {4 (b c-a d) x^{3/2} \left (a^2 d^2 \left (-c+3 d x^2\right )+a b d \left (17 c^2+12 c d x^2+3 d^2 x^4\right )+b^2 c \left (8 c^2+33 c d x^2+21 d^2 x^4\right )\right )}{c \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {24 \sqrt {2} b^{5/4} (b c+3 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}+\frac {3 \sqrt {2} \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{5/4}}-\frac {24 \sqrt {2} b^{5/4} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}+\frac {3 \sqrt {2} \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{5/4}}}{64 (b c-a d)^4} \]
((-4*(b*c - a*d)*x^(3/2)*(a^2*d^2*(-c + 3*d*x^2) + a*b*d*(17*c^2 + 12*c*d* x^2 + 3*d^2*x^4) + b^2*c*(8*c^2 + 33*c*d*x^2 + 21*d^2*x^4)))/(c*(a + b*x^2 )*(c + d*x^2)^2) - (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[(Sqrt[a] - Sqr t[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(1/4) + (3*Sqrt[2]*d^(1/4)*( 15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c ^(1/4)*d^(1/4)*Sqrt[x])])/c^(5/4) - (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcT anh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(1/4) + (3 *Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1 /4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/c^(5/4))/(64*(b*c - a*d)^4)
Time = 0.96 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 971, 27, 1049, 27, 1049, 1054, 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.
| \(\displaystyle \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^3}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle 2 \left (\frac {\int \frac {3 x \left (c-3 d x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {3 \int \frac {x \left (c-3 d x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\int \frac {4 c x \left (-5 b d x^2+2 b c+a d\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}-\frac {d x^{3/2}}{2 \left (c+d x^2\right )^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\int \frac {x \left (-5 b d x^2+2 b c+a d\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{2 (b c-a d)}-\frac {d x^{3/2}}{2 \left (c+d x^2\right )^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\frac {\int \frac {x \left (8 b^2 c^2+17 a b d c-a^2 d^2-b d (7 b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {d x^{3/2} (a d+7 b c)}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 (b c-a d)}-\frac {d x^{3/2}}{2 \left (c+d x^2\right )^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\frac {\int \left (\frac {8 c (b c+3 a d) x b^2}{(b c-a d) \left (b x^2+a\right )}+\frac {d \left (-15 b^2 c^2-18 a b d c+a^2 d^2\right ) x}{(b c-a d) \left (d x^2+c\right )}\right )d\sqrt {x}}{4 c (b c-a d)}-\frac {d x^{3/2} (a d+7 b c)}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 (b c-a d)}-\frac {d x^{3/2}}{2 \left (c+d x^2\right )^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\frac {\frac {\sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {2 \sqrt {2} b^{5/4} c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (3 a d+b c)}{\sqrt [4]{a} (b c-a d)}+\frac {2 \sqrt {2} b^{5/4} c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (3 a d+b c)}{\sqrt [4]{a} (b c-a d)}+\frac {\sqrt {2} b^{5/4} c (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a} (b c-a d)}-\frac {\sqrt {2} b^{5/4} c (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a} (b c-a d)}}{4 c (b c-a d)}-\frac {d x^{3/2} (a d+7 b c)}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 (b c-a d)}-\frac {d x^{3/2}}{2 \left (c+d x^2\right )^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {x^{3/2}}{4 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
2*(-1/4*x^(3/2)/((b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (3*(-1/2*(d*x^(3 /2))/((b*c - a*d)*(c + d*x^2)^2) + (-1/4*(d*(7*b*c + a*d)*x^(3/2))/(c*(b*c - a*d)*(c + d*x^2)) + ((-2*Sqrt[2]*b^(5/4)*c*(b*c + 3*a*d)*ArcTan[1 - (Sq rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(b*c - a*d)) + (2*Sqrt[2]*b^(5/4 )*c*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)* (b*c - a*d)) + (d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sq rt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) - (d^(1/4 )*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x]) /c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) + (Sqrt[2]*b^(5/4)*c*(b*c + 3*a *d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*( b*c - a*d)) - (Sqrt[2]*b^(5/4)*c*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/ 4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d)) - (d^(1/4)*(15*b^2* c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)) + (d^(1/4)*(15*b^2*c^2 + 18* a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d] *x])/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)))/(4*c*(b*c - a*d)))/(2*(b*c - a*d)))) /(4*(b*c - a*d)))
3.5.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[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_)), 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] + Simp[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]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> 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]
Time = 7.00 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.52
| method | result | size |
| derivativedivides | \(\frac {2 d \left (\frac {\frac {d \left (3 a^{2} d^{2}+10 a b c d -13 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c}+\left (-\frac {1}{32} a^{2} d^{2}+\frac {9}{16} a b c d -\frac {17}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (a^{2} d^{2}-18 a b c d -15 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 b^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 a d}{4}+\frac {3 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\) | \(368\) |
| default | \(\frac {2 d \left (\frac {\frac {d \left (3 a^{2} d^{2}+10 a b c d -13 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c}+\left (-\frac {1}{32} a^{2} d^{2}+\frac {9}{16} a b c d -\frac {17}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (a^{2} d^{2}-18 a b c d -15 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 b^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 a d}{4}+\frac {3 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\) | \(368\) |
2*d/(a*d-b*c)^4*((1/32*d*(3*a^2*d^2+10*a*b*c*d-13*b^2*c^2)/c*x^(7/2)+(-1/3 2*a^2*d^2+9/16*a*b*c*d-17/32*b^2*c^2)*x^(3/2))/(d*x^2+c)^2+3/256*(a^2*d^2- 18*a*b*c*d-15*b^2*c^2)/c/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)*x^(1/2)* 2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan (2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))+ 2*b^2/(a*d-b*c)^4*((1/4*a*d-1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(9/4*a*d+3/4*b* c)/b/(a/b)^(1/4)*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*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 122.75 (sec) , antiderivative size = 8884, normalized size of antiderivative = 12.64 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 791, normalized size of antiderivative = 1.13 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {3 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {3 \, {\left (15 \, b^{2} c^{2} d + 18 \, a b c d^{2} - a^{2} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )}} - \frac {3 \, {\left (7 \, b^{2} c d^{2} + a b d^{3}\right )} x^{\frac {11}{2}} + 3 \, {\left (11 \, b^{2} c^{2} d + 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {7}{2}} + {\left (8 \, b^{2} c^{3} + 17 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (a b^{3} c^{6} - 3 \, a^{2} b^{2} c^{5} d + 3 \, a^{3} b c^{4} d^{2} - a^{4} c^{3} d^{3} + {\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x^{4} + {\left (b^{4} c^{6} - a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 5 \, a^{3} b c^{3} d^{3} - 2 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )}} \]
3/16*(b^3*c + 3*a*b^2*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^ (1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*s qrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt( b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt( 2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^( 3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a) )/(a^(1/4)*b^(3/4)))/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3* b*c*d^3 + a^4*d^4) - 3/128*(15*b^2*c^2*d + 18*a*b*c*d^2 - a^2*d^3)*(2*sqrt (2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt( sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2* sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d) ))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*s qrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^( 1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4) - 1/16*(3 *(7*b^2*c*d^2 + a*b*d^3)*x^(11/2) + 3*(11*b^2*c^2*d + 4*a*b*c*d^2 + a^2*d^ 3)*x^(7/2) + (8*b^2*c^3 + 17*a*b*c^2*d - a^2*c*d^2)*x^(3/2))/(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4*c^3*d^3 + (b^4*c^4*d^2 - 3*a*b^3*c ^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^6 + (2*b^4*c^5*d - 5*a*b^3*c^4 *d^2 + 3*a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4 - a^4*c*d^5)*x^4 + (b^4*c^6 - ...
Leaf count of result is larger than twice the leaf count of optimal. 1238 vs. \(2 (547) = 1094\).
Time = 0.56 (sec) , antiderivative size = 1238, normalized size of antiderivative = 1.76 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/2*b^2*x^(3/2)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x ^2 + a)) + 3/4*((a*b^3)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2 )*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^5*c^4 - 4*sq rt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) + 3/4*((a*b^3)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*arcta n(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b ^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a ^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) - 3/32*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c *d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*( c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^6*d^2 - 4*sqrt(2)*a*b^ 3*c^5*d^3 + 6*sqrt(2)*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)* a^4*c^2*d^6) - 3/32*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt( x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^6*d^2 - 4*sqrt(2)*a*b^3*c^5*d^3 + 6*sqrt(2 )*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 3/8*( (a*b^3)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a ^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) + 3/8*((a*b^ 3)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^...
Time = 12.80 (sec) , antiderivative size = 44169, normalized size of antiderivative = 62.83 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
2*atan((((((864*a*b^27*c^23*d^4 - (27*a^24*b^4*d^27)/16 + (1863*a^23*b^5*c *d^26)/16 - 5184*a^2*b^26*c^22*d^5 - (132597*a^3*b^25*c^21*d^6)/16 + (2587 113*a^4*b^24*c^20*d^7)/16 - (4585005*a^5*b^23*c^19*d^8)/8 + (5105997*a^6*b ^22*c^18*d^9)/8 + (22410891*a^7*b^21*c^17*d^10)/16 - (93270447*a^8*b^20*c^ 16*d^11)/16 + (13320261*a^9*b^19*c^15*d^12)/2 + (12854835*a^10*b^18*c^14*d ^13)/2 - (279642213*a^11*b^17*c^13*d^14)/8 + (501573033*a^12*b^16*c^12*d^1 5)/8 - (274240863*a^13*b^15*c^11*d^16)/4 + (196146927*a^14*b^14*c^10*d^17) /4 - (166924665*a^15*b^13*c^9*d^18)/8 + (14462037*a^16*b^12*c^8*d^19)/8 + (8300637*a^17*b^11*c^7*d^20)/2 - (6325749*a^18*b^10*c^6*d^21)/2 + (1972374 3*a^19*b^9*c^5*d^22)/16 - (4658715*a^20*b^8*c^4*d^23)/16 + (327267*a^21*b^ 7*c^3*d^24)/8 - (24867*a^22*b^6*c^2*d^25)/8)*1i)/(b^21*c^23 - a^21*c^2*d^2 1 + 21*a^20*b*c^3*d^20 + 210*a^2*b^19*c^21*d^2 - 1330*a^3*b^18*c^20*d^3 + 5985*a^4*b^17*c^19*d^4 - 20349*a^5*b^16*c^18*d^5 + 54264*a^6*b^15*c^17*d^6 - 116280*a^7*b^14*c^16*d^7 + 203490*a^8*b^13*c^15*d^8 - 293930*a^9*b^12*c ^14*d^9 + 352716*a^10*b^11*c^13*d^10 - 352716*a^11*b^10*c^12*d^11 + 293930 *a^12*b^9*c^11*d^12 - 203490*a^13*b^8*c^10*d^13 + 116280*a^14*b^7*c^9*d^14 - 54264*a^15*b^6*c^8*d^15 + 20349*a^16*b^5*c^7*d^16 - 5985*a^17*b^4*c^6*d ^17 + 1330*a^18*b^3*c^5*d^18 - 210*a^19*b^2*c^4*d^19 - 21*a*b^20*c^22*d) - (9*x^(1/2)*(-(81*a^8*d^9 + 4100625*b^8*c^8*d + 19683000*a*b^7*c^7*d^2 + 3 4335900*a^2*b^6*c^6*d^3 + 24406920*a^3*b^5*c^5*d^4 + 3888486*a^4*b^4*c^...