Integrand size = 24, antiderivative size = 614 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\frac {\sqrt {e x} (a d+b c x)}{4 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}+\frac {\sqrt {e x} \left (7 a^2 d^3+5 b^2 c^3 x-a b c d (c-13 d x)\right )}{16 a^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {2 \sqrt {c} d^{9/2} \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\left (b c^2+a d^2\right )^3}-\frac {\left (5 b^{5/2} c^5-3 \sqrt {a} b^2 c^4 d+18 a b^{3/2} c^3 d^2-14 a^{3/2} b c^2 d^3+45 a^2 \sqrt {b} c d^4+21 a^{5/2} d^5\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (5 b^{5/2} c^5-3 \sqrt {a} b^2 c^4 d+18 a b^{3/2} c^3 d^2-14 a^{3/2} b c^2 d^3+45 a^2 \sqrt {b} c d^4+21 a^{5/2} d^5\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\left (5 b^{5/2} c^5+3 \sqrt {a} b^2 c^4 d+18 a b^{3/2} c^3 d^2+14 a^{3/2} b c^2 d^3+45 a^2 \sqrt {b} c d^4-21 a^{5/2} d^5\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3} \] Output:
1/4*(e*x)^(1/2)*(b*c*x+a*d)/a/(a*d^2+b*c^2)/(b*x^2+a)^2+1/16*(e*x)^(1/2)*( 7*a^2*d^3+5*b^2*c^3*x-a*b*c*d*(-13*d*x+c))/a^2/(a*d^2+b*c^2)^2/(b*x^2+a)-2 *c^(1/2)*d^(9/2)*e^(1/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/(a*d^ 2+b*c^2)^3-1/64*(5*b^(5/2)*c^5-3*a^(1/2)*b^2*c^4*d+18*a*b^(3/2)*c^3*d^2-14 *a^(3/2)*b*c^2*d^3+45*a^2*b^(1/2)*c*d^4+21*a^(5/2)*d^5)*e^(1/2)*arctan(1-2 ^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(9/4)/b^(1/4)/(a*d^2 +b*c^2)^3+1/64*(5*b^(5/2)*c^5-3*a^(1/2)*b^2*c^4*d+18*a*b^(3/2)*c^3*d^2-14* a^(3/2)*b*c^2*d^3+45*a^2*b^(1/2)*c*d^4+21*a^(5/2)*d^5)*e^(1/2)*arctan(1+2^ (1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(9/4)/b^(1/4)/(a*d^2+ b*c^2)^3-1/64*(5*b^(5/2)*c^5+3*a^(1/2)*b^2*c^4*d+18*a*b^(3/2)*c^3*d^2+14*a ^(3/2)*b*c^2*d^3+45*a^2*b^(1/2)*c*d^4-21*a^(5/2)*d^5)*e^(1/2)*arctanh(2^(1 /2)*a^(1/4)*b^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(9/ 4)/b^(1/4)/(a*d^2+b*c^2)^3
Time = 2.13 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\frac {\sqrt {e x} \left (\frac {4 \left (b c^2+a d^2\right ) \left (11 a^3 d^3+5 b^3 c^3 x^3+a^2 b d \left (3 c^2+17 c d x+7 d^2 x^2\right )+a b^2 c x \left (9 c^2-c d x+13 d^2 x^2\right )\right )}{a^2 \left (a+b x^2\right )^2}-\frac {128 \sqrt {c} d^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{\sqrt {x}}-\frac {\sqrt {2} \left (5 b^{5/2} c^5-3 \sqrt {a} b^2 c^4 d+18 a b^{3/2} c^3 d^2-14 a^{3/2} b c^2 d^3+45 a^2 \sqrt {b} c d^4+21 a^{5/2} d^5\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} \sqrt [4]{b} \sqrt {x}}-\frac {\sqrt {2} \left (5 b^{5/2} c^5+3 \sqrt {a} b^2 c^4 d+18 a b^{3/2} c^3 d^2+14 a^{3/2} b c^2 d^3+45 a^2 \sqrt {b} c d^4-21 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} \sqrt [4]{b} \sqrt {x}}\right )}{64 \left (b c^2+a d^2\right )^3} \] Input:
Integrate[Sqrt[e*x]/((c + d*x)*(a + b*x^2)^3),x]
Output:
(Sqrt[e*x]*((4*(b*c^2 + a*d^2)*(11*a^3*d^3 + 5*b^3*c^3*x^3 + a^2*b*d*(3*c^ 2 + 17*c*d*x + 7*d^2*x^2) + a*b^2*c*x*(9*c^2 - c*d*x + 13*d^2*x^2)))/(a^2* (a + b*x^2)^2) - (128*Sqrt[c]*d^(9/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/S qrt[x] - (Sqrt[2]*(5*b^(5/2)*c^5 - 3*Sqrt[a]*b^2*c^4*d + 18*a*b^(3/2)*c^3* d^2 - 14*a^(3/2)*b*c^2*d^3 + 45*a^2*Sqrt[b]*c*d^4 + 21*a^(5/2)*d^5)*ArcTan [(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(9/4)*b^(1/4 )*Sqrt[x]) - (Sqrt[2]*(5*b^(5/2)*c^5 + 3*Sqrt[a]*b^2*c^4*d + 18*a*b^(3/2)* c^3*d^2 + 14*a^(3/2)*b*c^2*d^3 + 45*a^2*Sqrt[b]*c*d^4 - 21*a^(5/2)*d^5)*Ar cTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(9/4)*b ^(1/4)*Sqrt[x])))/(64*(b*c^2 + a*d^2)^3)
Leaf count is larger than twice the leaf count of optimal. \(1326\) vs. \(2(614)=1228\).
Time = 3.15 (sec) , antiderivative size = 1326, normalized size of antiderivative = 2.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {615, 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.
| \(\displaystyle \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^3 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (-\frac {b d^2 \sqrt {e x} (d x-c)}{\left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}+\frac {b \sqrt {e x} (c-d x)}{\left (a+b x^2\right )^3 \left (a d^2+b c^2\right )}+\frac {d^6 \sqrt {e x}}{(c+d x) \left (a d^2+b c^2\right )^3}-\frac {b d^4 \sqrt {e x} (d x-c)}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {c} \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{9/2}}{\left (b c^2+a d^2\right )^3}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^4}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^4}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{4 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{4 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{8 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{8 \sqrt {2} a^{5/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}+\frac {\sqrt {e x} (a d+b c x) d^2}{2 a \left (b c^2+a d^2\right )^2 \left (b x^2+a\right )}-\frac {\left (5 \sqrt {b} c-3 \sqrt {a} d\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )}+\frac {\left (5 \sqrt {b} c-3 \sqrt {a} d\right ) \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )}+\frac {\left (5 \sqrt {b} c+3 \sqrt {a} d\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{64 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )}-\frac {\left (5 \sqrt {b} c+3 \sqrt {a} d\right ) \sqrt {e} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{64 \sqrt {2} a^{9/4} \sqrt [4]{b} \left (b c^2+a d^2\right )}-\frac {\sqrt {e x} (a d-5 b c x)}{16 a^2 \left (b c^2+a d^2\right ) \left (b x^2+a\right )}+\frac {\sqrt {e x} (a d+b c x)}{4 a \left (b c^2+a d^2\right ) \left (b x^2+a\right )^2}\) |
Input:
Int[Sqrt[e*x]/((c + d*x)*(a + b*x^2)^3),x]
Output:
(Sqrt[e*x]*(a*d + b*c*x))/(4*a*(b*c^2 + a*d^2)*(a + b*x^2)^2) - (Sqrt[e*x] *(a*d - 5*b*c*x))/(16*a^2*(b*c^2 + a*d^2)*(a + b*x^2)) + (d^2*Sqrt[e*x]*(a *d + b*c*x))/(2*a*(b*c^2 + a*d^2)^2*(a + b*x^2)) - (2*Sqrt[c]*d^(9/2)*Sqrt [e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^2)^3 - (d^ 4*(Sqrt[b]*c + Sqrt[a]*d)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/( a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2)^3) - (d^2*(Sqr t[b]*c - Sqrt[a]*d)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4 )*Sqrt[e])])/(4*Sqrt[2]*a^(5/4)*b^(1/4)*(b*c^2 + a*d^2)^2) - ((5*Sqrt[b]*c - 3*Sqrt[a]*d)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sq rt[e])])/(32*Sqrt[2]*a^(9/4)*b^(1/4)*(b*c^2 + a*d^2)) + (d^4*(Sqrt[b]*c + Sqrt[a]*d)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e] )])/(Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2)^3) + (d^2*(Sqrt[b]*c - Sqrt[a ]*d)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(4 *Sqrt[2]*a^(5/4)*b^(1/4)*(b*c^2 + a*d^2)^2) + ((5*Sqrt[b]*c - 3*Sqrt[a]*d) *Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sq rt[2]*a^(9/4)*b^(1/4)*(b*c^2 + a*d^2)) + (d^4*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt [e]*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt [e*x]])/(2*Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2)^3) + (d^2*(Sqrt[b]*c + Sqrt[a]*d)*Sqrt[e]*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e]*x - Sqrt[2]*a^(1/ 4)*b^(1/4)*Sqrt[e*x]])/(8*Sqrt[2]*a^(5/4)*b^(1/4)*(b*c^2 + a*d^2)^2) + ...
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Time = 0.76 (sec) , antiderivative size = 553, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\left (11 \sqrt {e x}\, \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, \left (a^{3} d^{3}+\frac {3 d \left (\frac {7}{3} d^{2} x^{2}+\frac {17}{3} c d x +c^{2}\right ) b \,a^{2}}{11}+\frac {9 x \,b^{2} \left (\frac {13}{9} d^{2} x^{2}-\frac {1}{9} c d x +c^{2}\right ) c a}{11}+\frac {5 b^{3} c^{3} x^{3}}{11}\right )-32 a^{2} c \,d^{5} e \left (b \,x^{2}+a \right )^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{16}+\frac {45 \left (\frac {7 d \left (a^{2} d^{4}-\frac {2}{3} b \,c^{2} d^{2} a -\frac {1}{7} b^{2} c^{4}\right ) \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {a \,e^{2}}{b}}}{15}+\left (a^{2} d^{4}+\frac {2}{5} b \,c^{2} d^{2} a +\frac {1}{9} b^{2} c^{4}\right ) e c \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right ) \sqrt {d e c}\, \sqrt {2}\, \left (b \,x^{2}+a \right )^{2}}{128}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, \left (b \,x^{2}+a \right )^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3} a^{2}}\) | \(553\) |
| derivativedivides | \(2 e^{6} \left (\frac {\frac {\frac {b^{2} c \left (13 a^{2} d^{4}+18 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a^{2}}+\frac {b d e \left (7 a^{2} d^{4}+6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a}+\frac {b c \,e^{2} \left (17 a^{2} d^{4}+26 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{32 a}+\frac {d \,e^{3} \left (11 a^{2} d^{4}+14 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (21 d^{5} e \,a^{3}-14 a^{2} c^{2} e \,d^{3} b -3 a \,c^{4} e d \,b^{2}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (45 a^{2} c \,d^{4} b +18 a \,b^{2} c^{3} d^{2}+5 b^{3} c^{5}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{32 a^{2}}}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{3}}-\frac {c \,d^{5} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{3} \sqrt {d e c}}\right )\) | \(596\) |
| default | \(2 e^{6} \left (\frac {\frac {\frac {b^{2} c \left (13 a^{2} d^{4}+18 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a^{2}}+\frac {b d e \left (7 a^{2} d^{4}+6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a}+\frac {b c \,e^{2} \left (17 a^{2} d^{4}+26 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{32 a}+\frac {d \,e^{3} \left (11 a^{2} d^{4}+14 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (21 d^{5} e \,a^{3}-14 a^{2} c^{2} e \,d^{3} b -3 a \,c^{4} e d \,b^{2}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (45 a^{2} c \,d^{4} b +18 a \,b^{2} c^{3} d^{2}+5 b^{3} c^{5}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{32 a^{2}}}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{3}}-\frac {c \,d^{5} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{3} \sqrt {d e c}}\right )\) | \(596\) |
Input:
int((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
45/128/(d*e*c)^(1/2)*(8/45*(11*(e*x)^(1/2)*(a*d^2+b*c^2)*(d*e*c)^(1/2)*(a^ 3*d^3+3/11*d*(7/3*d^2*x^2+17/3*c*d*x+c^2)*b*a^2+9/11*x*b^2*(13/9*d^2*x^2-1 /9*c*d*x+c^2)*c*a+5/11*b^3*c^3*x^3)-32*a^2*c*d^5*e*(b*x^2+a)^2*arctan(d*(e *x)^(1/2)/(d*e*c)^(1/2)))*(a*e^2/b)^(1/4)+(7/15*d*(a^2*d^4-2/3*b*c^2*d^2*a -1/7*b^2*c^4)*(ln((e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2) )/(e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))-2*arctan((-2^ (1/2)*(e*x)^(1/2)+(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4))+2*arctan((2^(1/2)*(e*x )^(1/2)+(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4)))*(a*e^2/b)^(1/2)+(a^2*d^4+2/5*b* c^2*d^2*a+1/9*b^2*c^4)*e*c*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a *e^2/b)^(1/2))/(e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))- 2*arctan((-2^(1/2)*(e*x)^(1/2)+(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4))+2*arctan( (2^(1/2)*(e*x)^(1/2)+(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4))))*(d*e*c)^(1/2)*2^( 1/2)*(b*x^2+a)^2)/(a*e^2/b)^(1/4)/(b*x^2+a)^2/(a*d^2+b*c^2)^3/a^2
Leaf count of result is larger than twice the leaf count of optimal. 8082 vs. \(2 (493) = 986\).
Time = 107.10 (sec) , antiderivative size = 16176, normalized size of antiderivative = 26.35 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(1/2)/(d*x+c)/(b*x**2+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 1179 vs. \(2 (493) = 986\).
Time = 0.23 (sec) , antiderivative size = 1179, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")
Output:
-1/64*(128*c*d^5*e^2*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b^3*c^6 + 3*a*b^2*c ^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(c*d*e)) + 2*(3*(a*b^3*e^2)^(1/4)* a*b^3*c^4*d*e + 14*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e - 21*(a*b^3*e^2)^(1 /4)*a^3*b*d^5*e - 5*(a*b^3*e^2)^(3/4)*b^2*c^5 - 18*(a*b^3*e^2)^(3/4)*a*b*c ^3*d^2 - 45*(a*b^3*e^2)^(3/4)*a^2*c*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*e^ 2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a^3*b^5*c^6 + 3*sqrt(2 )*a^4*b^4*c^4*d^2 + 3*sqrt(2)*a^5*b^3*c^2*d^4 + sqrt(2)*a^6*b^2*d^6) + 2*( 3*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e + 14*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e - 21*(a*b^3*e^2)^(1/4)*a^3*b*d^5*e - 5*(a*b^3*e^2)^(3/4)*b^2*c^5 - 18*(a* b^3*e^2)^(3/4)*a*b*c^3*d^2 - 45*(a*b^3*e^2)^(3/4)*a^2*c*d^4)*arctan(-1/2*s qrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a ^3*b^5*c^6 + 3*sqrt(2)*a^4*b^4*c^4*d^2 + 3*sqrt(2)*a^5*b^3*c^2*d^4 + sqrt( 2)*a^6*b^2*d^6) + (3*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e + 14*(a*b^3*e^2)^(1/4 )*a^2*b^2*c^2*d^3*e - 21*(a*b^3*e^2)^(1/4)*a^3*b*d^5*e + 5*(a*b^3*e^2)^(3/ 4)*b^2*c^5 + 18*(a*b^3*e^2)^(3/4)*a*b*c^3*d^2 + 45*(a*b^3*e^2)^(3/4)*a^2*c *d^4)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2 )*a^3*b^5*c^6 + 3*sqrt(2)*a^4*b^4*c^4*d^2 + 3*sqrt(2)*a^5*b^3*c^2*d^4 + sq rt(2)*a^6*b^2*d^6) - (3*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e + 14*(a*b^3*e^2)^( 1/4)*a^2*b^2*c^2*d^3*e - 21*(a*b^3*e^2)^(1/4)*a^3*b*d^5*e + 5*(a*b^3*e^2)^ (3/4)*b^2*c^5 + 18*(a*b^3*e^2)^(3/4)*a*b*c^3*d^2 + 45*(a*b^3*e^2)^(3/4)...
Time = 11.58 (sec) , antiderivative size = 6111, normalized size of antiderivative = 9.95 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int((e*x)^(1/2)/((a + b*x^2)^3*(c + d*x)),x)
Output:
symsum(log(- root(15502147584*a^15*b^7*c^12*d^12*g^6 + 1107296256*a^19*b^3 *c^4*d^20*g^6 + 1107296256*a^11*b^11*c^20*d^4*g^6 + 13287555072*a^16*b^6*c ^10*d^14*g^6 + 13287555072*a^14*b^8*c^14*d^10*g^6 + 201326592*a^20*b^2*c^2 *d^22*g^6 + 201326592*a^10*b^12*c^22*d^2*g^6 + 8304721920*a^17*b^5*c^8*d^1 6*g^6 + 8304721920*a^13*b^9*c^16*d^8*g^6 + 3690987520*a^18*b^4*c^6*d^18*g^ 6 + 3690987520*a^12*b^10*c^18*d^6*g^6 + 16777216*a^9*b^13*c^24*g^6 + 16777 216*a^21*b*d^24*g^6 + 32260096*a^15*b*c*d^21*e*g^4 + 553517056*a^12*b^4*c^ 7*d^15*e*g^4 + 454508544*a^13*b^3*c^5*d^17*e*g^4 + 319586304*a^11*b^5*c^9* d^13*e*g^4 + 189431808*a^14*b^2*c^3*d^19*e*g^4 - 106135552*a^9*b^7*c^13*d^ 9*e*g^4 - 67239936*a^8*b^8*c^15*d^7*e*g^4 - 20496384*a^7*b^9*c^17*d^5*e*g^ 4 + 7274496*a^10*b^6*c^11*d^11*e*g^4 - 3506176*a^6*b^10*c^19*d^3*e*g^4 - 2 45760*a^5*b^11*c^21*d*e*g^4 + 3457074*a^6*b^4*c^8*d^12*e^2*g^2 + 3219516*a ^5*b^5*c^10*d^10*e^2*g^2 + 1446834*a^4*b^6*c^12*d^8*e^2*g^2 + 993528*a^7*b ^3*c^6*d^14*e^2*g^2 - 573219*a^8*b^2*c^4*d^16*e^2*g^2 + 408824*a^3*b^7*c^1 4*d^6*e^2*g^2 + 78621*a^2*b^8*c^16*d^4*e^2*g^2 + 16750314*a^9*b*c^2*d^18*e ^2*g^2 + 9450*a*b^9*c^18*d^2*e^2*g^2 + 194481*a^10*d^20*e^2*g^2 + 625*b^10 *c^20*e^2*g^2 + 100548*a^3*b*c^3*d^15*e^3 + 5700*a*b^3*c^7*d^11*e^3 + 3504 6*a^2*b^2*c^5*d^13*e^3 + 625*b^4*c^9*d^9*e^3 + 194481*a^4*c*d^17*e^3, g, k )*(root(15502147584*a^15*b^7*c^12*d^12*g^6 + 1107296256*a^19*b^3*c^4*d^20* g^6 + 1107296256*a^11*b^11*c^20*d^4*g^6 + 13287555072*a^16*b^6*c^10*d^1...
Time = 0.25 (sec) , antiderivative size = 3333, normalized size of antiderivative = 5.43 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x)
Output:
(sqrt(e)*( - 90*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b*c*d**4 - 36*b**(1 /4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/ (b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b**2*c**3*d**2 - 180*b**(1/4)*a**(3/4)*s qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 1/4)*sqrt(2)))*a**3*b**2*c*d**4*x**2 - 10*b**(1/4)*a**(3/4)*sqrt(2)*atan(( b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)) )*a**2*b**3*c**5 - 72*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**3*c**3*d** 2*x**2 - 90*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2* sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**3*c*d**4*x**4 - 20*b **(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt( b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**4*c**5*x**2 - 36*b**(1/4)*a**(3/4)*s qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 1/4)*sqrt(2)))*a*b**4*c**3*d**2*x**4 - 10*b**(1/4)*a**(3/4)*sqrt(2)*atan(( b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)) )*b**5*c**5*x**4 - 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5*d**5 + 28*b** (3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b) )/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4*b*c**2*d**3 - 84*b**(3/4)*a**(1/4)*...