Integrand size = 24, antiderivative size = 386 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {(b c-13 a d) (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-13 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}} \]
-1/8*(-13*a*d+b*c)*(-a*d+b*c)^2*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/ a^(3/4)/b^(17/4)*2^(1/2)+1/8*(-13*a*d+b*c)*(-a*d+b*c)^2*arctan(1+b^(1/4)*2 ^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(17/4)*2^(1/2)-1/16*(-13*a*d+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))/a^(3/4)/b^(17 /4)*2^(1/2)+1/16*(-13*a*d+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))/a^(3/4)/b^(17/4)*2^(1/2)+1/90*d*(585*a^2*d^2-1098* a*b*c*d+497*b^2*c^2)*x^(1/2)/b^4+1/90*d*(-117*a*d+113*b*c)*(d*x^2+c)*x^(1/ 2)/b^3+13/18*d*(d*x^2+c)^2*x^(1/2)/b^2-1/2*(d*x^2+c)^3*x^(1/2)/b/(b*x^2+a)
Time = 0.50 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.66 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (585 a^3 d^3+9 a^2 b d^2 \left (-135 c+52 d x^2\right )+a b^2 d \left (675 c^2-972 c d x^2-52 d^2 x^4\right )+b^3 \left (-45 c^3+540 c^2 d x^2+108 c d^2 x^4+20 d^3 x^6\right )\right )}{a+b x^2}+\frac {45 \sqrt {2} (b c-a d)^2 (-b c+13 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {45 \sqrt {2} (b c-13 a d) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{360 b^{17/4}} \]
((4*b^(1/4)*Sqrt[x]*(585*a^3*d^3 + 9*a^2*b*d^2*(-135*c + 52*d*x^2) + a*b^2 *d*(675*c^2 - 972*c*d*x^2 - 52*d^2*x^4) + b^3*(-45*c^3 + 540*c^2*d*x^2 + 1 08*c*d^2*x^4 + 20*d^3*x^6)))/(a + b*x^2) + (45*Sqrt[2]*(b*c - a*d)^2*(-(b* c) + 13*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] )])/a^(3/4) + (45*Sqrt[2]*(b*c - 13*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*a^ (1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4))/(360*b^(17/4))
Time = 0.70 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {368, 967, 1025, 1025, 913, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^2 \left (d x^2+c\right )^3}{\left (b x^2+a\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle 2 \left (\frac {\int \frac {\left (d x^2+c\right )^2 \left (13 d x^2+c\right )}{b x^2+a}d\sqrt {x}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\left (d x^2+c\right ) \left (d (113 b c-117 a d) x^2+c (9 b c-13 a d)\right )}{b x^2+a}d\sqrt {x}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {d \left (497 b^2 c^2-1098 a b d c+585 a^2 d^2\right ) x^2+c \left (45 b^2 c^2-178 a b d c+117 a^2 d^2\right )}{b x^2+a}d\sqrt {x}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \int \frac {1}{b x^2+a}d\sqrt {x}}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{b}+\frac {45 (b c-13 a d) (b c-a d)^2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}}{5 b}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{5 b}}{9 b}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{9 b}}{4 b}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
2*(-1/4*(Sqrt[x]*(c + d*x^2)^3)/(b*(a + b*x^2)) + ((13*d*Sqrt[x]*(c + d*x^ 2)^2)/(9*b) + ((d*(113*b*c - 117*a*d)*Sqrt[x]*(c + d*x^2))/(5*b) + ((d*(49 7*b^2*c^2 - 1098*a*b*c*d + 585*a^2*d^2)*Sqrt[x])/b + (45*(b*c - 13*a*d)*(b *c - a*d)^2*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^( 1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^ (1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)* Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^( 1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) ))/b)/(5*b))/(9*b))/(4*b))
3.5.54.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
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[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1)) Int[(a + b*x ^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 2.78 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {2 d \left (5 b^{2} d^{2} x^{4}-18 x^{2} a b \,d^{2}+27 x^{2} b^{2} c d +135 a^{2} d^{2}-270 a b c d +135 b^{2} c^{2}\right ) \sqrt {x}}{45 b^{4}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (13 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{4}}\) | \(230\) |
| derivativedivides | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {9}{2}}}{9}-\frac {2 a b \,d^{2} x^{\frac {5}{2}}}{5}+\frac {3 b^{2} c d \,x^{\frac {5}{2}}}{5}+3 a^{2} d^{2} \sqrt {x}-6 a b c d \sqrt {x}+3 b^{2} c^{2} \sqrt {x}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (13 a^{3} d^{3}-27 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{4}}\) | \(269\) |
| default | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {9}{2}}}{9}-\frac {2 a b \,d^{2} x^{\frac {5}{2}}}{5}+\frac {3 b^{2} c d \,x^{\frac {5}{2}}}{5}+3 a^{2} d^{2} \sqrt {x}-6 a b c d \sqrt {x}+3 b^{2} c^{2} \sqrt {x}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (13 a^{3} d^{3}-27 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{4}}\) | \(269\) |
2/45*d*(5*b^2*d^2*x^4-18*a*b*d^2*x^2+27*b^2*c*d*x^2+135*a^2*d^2-270*a*b*c* d+135*b^2*c^2)*x^(1/2)/b^4-1/b^4*(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*(13*a*d-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*arctan(2^(1/2)/(a/ b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1764, normalized size of antiderivative = 4.57 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
-1/360*(45*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^ 10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5* b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207* a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237 276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(a*b^4*(-(b^12*c ^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3* d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/( a^3*b^17))^(1/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3 )*sqrt(x)) + 45*(I*b^5*x^2 + I*a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 14 58*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 53 5032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d ^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(I*a*b^ 4*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9 *c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b ^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a ^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a ^12*d^12)/(a^3*b^17))^(1/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^...
Leaf count of result is larger than twice the leaf count of optimal. 1833 vs. \(2 (369) = 738\).
Time = 152.06 (sec) , antiderivative size = 1833, normalized size of antiderivative = 4.75 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2 )/5 + 2*d**3*x**(9/2)/9), Eq(a, 0) & Eq(b, 0)), ((2*c**3*x**(5/2)/5 + 2*c* *2*d*x**(9/2)/3 + 6*c*d**2*x**(13/2)/13 + 2*d**3*x**(17/2)/17)/a**2, Eq(b, 0)), ((-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2* d**3*x**(9/2)/9)/b**2, Eq(a, 0)), (2340*a**4*d**3*sqrt(x)/(360*a**2*b**4 + 360*a*b**5*x**2) + 585*a**4*d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4 ))/(360*a**2*b**4 + 360*a*b**5*x**2) - 585*a**4*d**3*(-a/b)**(1/4)*log(sqr t(x) + (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 1170*a**4*d**3*( -a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 4860*a**3*b*c*d**2*sqrt(x)/(360*a**2*b**4 + 360*a*b**5*x**2) - 1215*a** 3*b*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360 *a*b**5*x**2) + 1215*a**3*b*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/ 4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 2430*a**3*b*c*d**2*(-a/b)**(1/4)*a tan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 1872*a**3*b *d**3*x**(5/2)/(360*a**2*b**4 + 360*a*b**5*x**2) + 585*a**3*b*d**3*x**2*(- a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 585*a**3*b*d**3*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(360*a* *2*b**4 + 360*a*b**5*x**2) - 1170*a**3*b*d**3*x**2*(-a/b)**(1/4)*atan(sqrt (x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 2700*a**2*b**2*c**2 *d*sqrt(x)/(360*a**2*b**4 + 360*a*b**5*x**2) + 675*a**2*b**2*c**2*d*(-a...
Time = 0.31 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.15 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {2 \, {\left (5 \, b^{2} d^{3} x^{\frac {9}{2}} + 9 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {5}{2}} + 135 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {x}\right )}}{45 \, b^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, b^{4}} \]
-1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)/(b^5*x^2 + a*b^4) + 2/45*(5*b^2*d^3*x^(9/2) + 9*(3*b^2*c*d^2 - 2*a*b*d^3)*x^(5/2) + 135*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*sqrt(x))/b^4 + 1/16*(2*sqrt(2)*(b ^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*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))) + 2*sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2* b*c*d^2 - 13*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqr t(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqr t(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*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) *(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*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)))/b^4
Time = 0.30 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.43 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \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 \, a b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \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 \, a b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \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 \, a b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \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 \, a b^{5}} - \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {2 \, {\left (5 \, b^{16} d^{3} x^{\frac {9}{2}} + 27 \, b^{16} c d^{2} x^{\frac {5}{2}} - 18 \, a b^{15} d^{3} x^{\frac {5}{2}} + 135 \, b^{16} c^{2} d \sqrt {x} - 270 \, a b^{15} c d^{2} \sqrt {x} + 135 \, a^{2} b^{14} d^{3} \sqrt {x}\right )}}{45 \, b^{18}} \]
1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a* b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqr t(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^5) + 1/8*sqrt(2)*((a*b^3)^ (1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^ 2 - 13*(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))/(a*b^5) + 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15* (a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4 )*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^5) - 1/16 *sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3 )^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b) ^(1/4) + x + sqrt(a/b))/(a*b^5) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqr t(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*b^4) + 2/45*( 5*b^16*d^3*x^(9/2) + 27*b^16*c*d^2*x^(5/2) - 18*a*b^15*d^3*x^(5/2) + 135*b ^16*c^2*d*sqrt(x) - 270*a*b^15*c*d^2*sqrt(x) + 135*a^2*b^14*d^3*sqrt(x))/b ^18
Time = 5.43 (sec) , antiderivative size = 1691, normalized size of antiderivative = 4.38 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
x^(1/2)*((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) - x^(5/2)*((4*a*d^3)/(5*b^3) - (6*c*d^2)/(5*b^2)) + (2*d^3*x^(9/ 2))/(9*b^2) + (x^(1/2)*((a^3*d^3)/2 - (b^3*c^3)/2 + (3*a*b^2*c^2*d)/2 - (3 *a^2*b*c*d^2)/2))/(a*b^4 + b^5*x^2) + (atan(((((x^(1/2)*(169*a^6*d^6 + b^6 *c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)*(13 *a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^( 21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/(8*(-a)^(3/4)*b^(17/4)) + (((x^(1 /2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1 119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c )^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c *d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/(8*(-a)^(3/ 4)*b^(17/4)))/((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 8 36*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d ^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2* b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(17/4)) - (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279* a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^ 5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*...