Integrand size = 24, antiderivative size = 617 \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {b \sqrt {e x} (c-d x)}{4 a \left (b c^2+a d^2\right ) e \left (a+b x^2\right )^2}+\frac {b \sqrt {e x} \left (a d^2 (15 c-13 d x)+b c^2 (7 c-5 d x)\right )}{16 a^2 \left (b c^2+a d^2\right )^2 e \left (a+b x^2\right )}+\frac {2 d^{11/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c} \left (b c^2+a d^2\right )^3 \sqrt {e}}-\frac {\sqrt [4]{b} \left (21 b^{5/2} c^5-5 \sqrt {a} b^2 c^4 d+66 a b^{3/2} c^3 d^2-18 a^{3/2} b c^2 d^3+77 a^2 \sqrt {b} c d^4-45 a^{5/2} d^5\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {\sqrt [4]{b} \left (21 b^{5/2} c^5-5 \sqrt {a} b^2 c^4 d+66 a b^{3/2} c^3 d^2-18 a^{3/2} b c^2 d^3+77 a^2 \sqrt {b} c d^4-45 a^{5/2} d^5\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {\sqrt [4]{b} \left (21 b^{5/2} c^5+5 \sqrt {a} b^2 c^4 d+66 a b^{3/2} c^3 d^2+18 a^{3/2} b c^2 d^3+77 a^2 \sqrt {b} c d^4+45 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^3 \sqrt {e}} \] Output:
1/4*b*(e*x)^(1/2)*(-d*x+c)/a/(a*d^2+b*c^2)/e/(b*x^2+a)^2+1/16*b*(e*x)^(1/2 )*(a*d^2*(-13*d*x+15*c)+b*c^2*(-5*d*x+7*c))/a^2/(a*d^2+b*c^2)^2/e/(b*x^2+a )+2*d^(11/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(1/2)/(a*d^2+b* c^2)^3/e^(1/2)-1/64*b^(1/4)*(21*b^(5/2)*c^5-5*a^(1/2)*b^2*c^4*d+66*a*b^(3/ 2)*c^3*d^2-18*a^(3/2)*b*c^2*d^3+77*a^2*b^(1/2)*c*d^4-45*a^(5/2)*d^5)*arcta n(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(11/4)/(a*d^2+b *c^2)^3/e^(1/2)+1/64*b^(1/4)*(21*b^(5/2)*c^5-5*a^(1/2)*b^2*c^4*d+66*a*b^(3 /2)*c^3*d^2-18*a^(3/2)*b*c^2*d^3+77*a^2*b^(1/2)*c*d^4-45*a^(5/2)*d^5)*arct an(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(11/4)/(a*d^2+ b*c^2)^3/e^(1/2)+1/64*b^(1/4)*(21*b^(5/2)*c^5+5*a^(1/2)*b^2*c^4*d+66*a*b^( 3/2)*c^3*d^2+18*a^(3/2)*b*c^2*d^3+77*a^2*b^(1/2)*c*d^4+45*a^(5/2)*d^5)*arc tanh(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^(11/4)/(a*d^2+b*c^2)^3/e^(1/2)
Time = 2.16 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-\frac {4 b \left (b c^2+a d^2\right ) \sqrt {x} \left (b^2 c^2 x^2 (-7 c+5 d x)+a^2 d^2 (-19 c+17 d x)+a b \left (-11 c^3+9 c^2 d x-15 c d^2 x^2+13 d^3 x^3\right )\right )}{a^2 \left (a+b x^2\right )^2}+\frac {128 d^{11/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} \left (-21 b^{5/2} c^5+5 \sqrt {a} b^2 c^4 d-66 a b^{3/2} c^3 d^2+18 a^{3/2} b c^2 d^3-77 a^2 \sqrt {b} c d^4+45 a^{5/2} d^5\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{11/4}}+\frac {\sqrt {2} \sqrt [4]{b} \left (21 b^{5/2} c^5+5 \sqrt {a} b^2 c^4 d+66 a b^{3/2} c^3 d^2+18 a^{3/2} b c^2 d^3+77 a^2 \sqrt {b} c d^4+45 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{11/4}}\right )}{64 \left (b c^2+a d^2\right )^3 \sqrt {e x}} \] Input:
Integrate[1/(Sqrt[e*x]*(c + d*x)*(a + b*x^2)^3),x]
Output:
(Sqrt[x]*((-4*b*(b*c^2 + a*d^2)*Sqrt[x]*(b^2*c^2*x^2*(-7*c + 5*d*x) + a^2* d^2*(-19*c + 17*d*x) + a*b*(-11*c^3 + 9*c^2*d*x - 15*c*d^2*x^2 + 13*d^3*x^ 3)))/(a^2*(a + b*x^2)^2) + (128*d^(11/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] )/Sqrt[c] + (Sqrt[2]*b^(1/4)*(-21*b^(5/2)*c^5 + 5*Sqrt[a]*b^2*c^4*d - 66*a *b^(3/2)*c^3*d^2 + 18*a^(3/2)*b*c^2*d^3 - 77*a^2*Sqrt[b]*c*d^4 + 45*a^(5/2 )*d^5)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^ (11/4) + (Sqrt[2]*b^(1/4)*(21*b^(5/2)*c^5 + 5*Sqrt[a]*b^2*c^4*d + 66*a*b^( 3/2)*c^3*d^2 + 18*a^(3/2)*b*c^2*d^3 + 77*a^2*Sqrt[b]*c*d^4 + 45*a^(5/2)*d^ 5)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(11 /4)))/(64*(b*c^2 + a*d^2)^3*Sqrt[e*x])
Leaf count is larger than twice the leaf count of optimal. \(1337\) vs. \(2(617)=1234\).
Time = 3.11 (sec) , antiderivative size = 1337, normalized size of antiderivative = 2.17, 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.
| \(\displaystyle \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^3 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (-\frac {b d^2 (d x-c)}{\sqrt {e x} \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}+\frac {b (c-d x)}{\sqrt {e x} \left (a+b x^2\right )^3 \left (a d^2+b c^2\right )}+\frac {d^6}{\sqrt {e x} (c+d x) \left (a d^2+b c^2\right )^3}-\frac {b d^4 (d x-c)}{\sqrt {e x} \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{11/2}}{\sqrt {c} \left (b c^2+a d^2\right )^3 \sqrt {e}}-\frac {\sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}-\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right ) \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} a^{3/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right ) \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} a^{3/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}-\frac {\sqrt [4]{b} \left (3 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{4 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {\sqrt [4]{b} \left (3 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{4 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}-\frac {\sqrt [4]{b} \left (3 \sqrt {b} c+\sqrt {a} d\right ) \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^{7/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {\sqrt [4]{b} \left (3 \sqrt {b} c+\sqrt {a} d\right ) \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^{7/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {b \sqrt {e x} (c-d x) d^2}{2 a \left (b c^2+a d^2\right )^2 e \left (b x^2+a\right )}-\frac {\sqrt [4]{b} \left (21 \sqrt {b} c-5 \sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right ) \sqrt {e}}+\frac {\sqrt [4]{b} \left (21 \sqrt {b} c-5 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right ) \sqrt {e}}-\frac {\sqrt [4]{b} \left (21 \sqrt {b} c+5 \sqrt {a} d\right ) \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^{11/4} \left (b c^2+a d^2\right ) \sqrt {e}}+\frac {\sqrt [4]{b} \left (21 \sqrt {b} c+5 \sqrt {a} d\right ) \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^{11/4} \left (b c^2+a d^2\right ) \sqrt {e}}+\frac {b \sqrt {e x} (7 c-5 d x)}{16 a^2 \left (b c^2+a d^2\right ) e \left (b x^2+a\right )}+\frac {b \sqrt {e x} (c-d x)}{4 a \left (b c^2+a d^2\right ) e \left (b x^2+a\right )^2}\) |
Input:
Int[1/(Sqrt[e*x]*(c + d*x)*(a + b*x^2)^3),x]
Output:
(b*Sqrt[e*x]*(c - d*x))/(4*a*(b*c^2 + a*d^2)*e*(a + b*x^2)^2) + (b*Sqrt[e* x]*(7*c - 5*d*x))/(16*a^2*(b*c^2 + a*d^2)*e*(a + b*x^2)) + (b*d^2*Sqrt[e*x ]*(c - d*x))/(2*a*(b*c^2 + a*d^2)^2*e*(a + b*x^2)) + (2*d^(11/2)*ArcTan[(S qrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(Sqrt[c]*(b*c^2 + a*d^2)^3*Sqrt[e]) - (b^(1/4)*d^4*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e* x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(3/4)*(b*c^2 + a*d^2)^3*Sqrt[e]) - (b^( 1/4)*d^2*(3*Sqrt[b]*c - Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/ (a^(1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(7/4)*(b*c^2 + a*d^2)^2*Sqrt[e]) - (b^(1/ 4)*(21*Sqrt[b]*c - 5*Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^ (1/4)*Sqrt[e])])/(32*Sqrt[2]*a^(11/4)*(b*c^2 + a*d^2)*Sqrt[e]) + (b^(1/4)* d^4*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4 )*Sqrt[e])])/(Sqrt[2]*a^(3/4)*(b*c^2 + a*d^2)^3*Sqrt[e]) + (b^(1/4)*d^2*(3 *Sqrt[b]*c - Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sq rt[e])])/(4*Sqrt[2]*a^(7/4)*(b*c^2 + a*d^2)^2*Sqrt[e]) + (b^(1/4)*(21*Sqrt [b]*c - 5*Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[ e])])/(32*Sqrt[2]*a^(11/4)*(b*c^2 + a*d^2)*Sqrt[e]) - (b^(1/4)*d^4*(Sqrt[b ]*c + Sqrt[a]*d)*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^(3/4)*(b*c^2 + a*d^2)^3*Sqrt[e]) - (b^(1 /4)*d^2*(3*Sqrt[b]*c + Sqrt[a]*d)*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^(7/4)*(b*c^2 + a*d^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 = 0.78 (sec) , antiderivative size = 553, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\frac {19 \left (\sqrt {e x}\, b \left (a \,d^{2}+b \,c^{2}\right ) \left (d^{2} \left (-\frac {17 d x}{19}+c \right ) a^{2}+\frac {11 b \left (-\frac {13}{11} d^{3} x^{3}+\frac {15}{11} c \,d^{2} x^{2}-\frac {9}{11} c^{2} d x +c^{3}\right ) a}{19}+\frac {7 x^{2} \left (-\frac {5 d x}{7}+c \right ) b^{2} c^{2}}{19}\right ) \sqrt {d e c}+\frac {32 a^{2} d^{6} e \left (b \,x^{2}+a \right )^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{19}\right ) a \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{16}+\frac {77 \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \sqrt {d e c}\, \left (\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 ) b \left (a^{2} d^{4}+\frac {6}{7} b \,c^{2} d^{2} a +\frac {3}{11} b^{2} c^{4}\right ) c \sqrt {\frac {a \,e^{2}}{b}}-\frac {90 d \left (a^{2} d^{4}+\frac {2}{5} b \,c^{2} d^{2} a +\frac {1}{9} b^{2} c^{4}\right ) e \left (\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 )+\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 )+\frac {\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}\right ) a}{77}\right )}{128}}{e \left (b \,x^{2}+a \right )^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3} a^{3} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}}\) | \(553\) |
| derivativedivides | \(2 e^{6} \left (\frac {d^{6} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{3} \sqrt {d e c}}+\frac {b \left (\frac {-\frac {b d \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 c e \left (15 a^{2} d^{4}+22 b \,c^{2} d^{2} a +7 b^{2} c^{4}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a^{2}}-\frac {d \,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 {c \,e^{3} \left (19 a^{2} d^{4}+30 b \,c^{2} d^{2} a +11 b^{2} c^{4}\right ) \sqrt {e x}}{32 a}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (77 c e \,a^{2} d^{4}+66 a b \,d^{2} e \,c^{3}+21 b^{2} e \,c^{5}\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} d^{5}-18 d^{3} a \,c^{2} b -5 b^{2} c^{4} d \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}}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\right )\) | \(589\) |
| default | \(2 e^{6} \left (\frac {d^{6} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{3} \sqrt {d e c}}+\frac {b \left (\frac {-\frac {b d \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 c e \left (15 a^{2} d^{4}+22 b \,c^{2} d^{2} a +7 b^{2} c^{4}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a^{2}}-\frac {d \,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 {c \,e^{3} \left (19 a^{2} d^{4}+30 b \,c^{2} d^{2} a +11 b^{2} c^{4}\right ) \sqrt {e x}}{32 a}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (77 c e \,a^{2} d^{4}+66 a b \,d^{2} e \,c^{3}+21 b^{2} e \,c^{5}\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} d^{5}-18 d^{3} a \,c^{2} b -5 b^{2} c^{4} d \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}}\right )}{e^{6} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\right )\) | \(589\) |
Input:
int(1/(e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
77/128/(d*e*c)^(1/2)*(152/77*((e*x)^(1/2)*b*(a*d^2+b*c^2)*(d^2*(-17/19*d*x +c)*a^2+11/19*b*(-13/11*d^3*x^3+15/11*c*d^2*x^2-9/11*c^2*d*x+c^3)*a+7/19*x ^2*(-5/7*d*x+c)*b^2*c^2)*(d*e*c)^(1/2)+32/19*a^2*d^6*e*(b*x^2+a)^2*arctan( d*(e*x)^(1/2)/(d*e*c)^(1/2)))*a*(a*e^2/b)^(1/4)+2^(1/2)*(b*x^2+a)^2*(d*e*c )^(1/2)*((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)))*b*(a^2*d^4+6/7*b*c^2*d^2*a+3/11*b^2*c ^4)*c*(a*e^2/b)^(1/2)-90/77*d*(a^2*d^4+2/5*b*c^2*d^2*a+1/9*b^2*c^4)*e*(arc tan((2^(1/2)*(e*x)^(1/2)-(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4))+arctan((2^(1/2) *(e*x)^(1/2)+(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4))+1/2*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))))*a))/(a*e^2/b)^(1/4)/e/(b*x^2+a)^2/(a*d^2+b*c^2)^3/ a^3
Timed out. \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)**(1/2)/(d*x+c)/(b*x**2+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(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. 1215 vs. \(2 (496) = 992\).
Time = 0.21 (sec) , antiderivative size = 1215, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x)^(1/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")
Output:
2*d^6*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)) + 1/32*(21*(a*b^3*e^2)^(1/4)*b^4*c^5*e + 66*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2*e + 77*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^4 *e - 5*(a*b^3*e^2)^(3/4)*b^2*c^4*d - 18*(a*b^3*e^2)^(3/4)*a*b*c^2*d^3 - 45 *(a*b^3*e^2)^(3/4)*a^2*d^5)*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*e^2 + 3*sqrt(2)*a^4*b^4 *c^4*d^2*e^2 + 3*sqrt(2)*a^5*b^3*c^2*d^4*e^2 + sqrt(2)*a^6*b^2*d^6*e^2) + 1/32*(21*(a*b^3*e^2)^(1/4)*b^4*c^5*e + 66*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2* e + 77*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^4*e - 5*(a*b^3*e^2)^(3/4)*b^2*c^4*d - 18*(a*b^3*e^2)^(3/4)*a*b*c^2*d^3 - 45*(a*b^3*e^2)^(3/4)*a^2*d^5)*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*e^2 + 3*sqrt(2)*a^4*b^4*c^4*d^2*e^2 + 3*sqrt(2)*a^5*b^3*c^ 2*d^4*e^2 + sqrt(2)*a^6*b^2*d^6*e^2) + 1/64*(21*(a*b^3*e^2)^(1/4)*b^4*c^5* e + 66*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2*e + 77*(a*b^3*e^2)^(1/4)*a^2*b^2*c* d^4*e + 5*(a*b^3*e^2)^(3/4)*b^2*c^4*d + 18*(a*b^3*e^2)^(3/4)*a*b*c^2*d^3 + 45*(a*b^3*e^2)^(3/4)*a^2*d^5)*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*e^2 + 3*sqrt(2)*a^4*b^4*c^4*d^2*e^2 + 3*sqrt(2)*a^5*b^3*c^2*d^4*e^2 + sqrt(2)*a^6*b^2*d^6*e^2) - 1/64*(21*(a* b^3*e^2)^(1/4)*b^4*c^5*e + 66*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2*e + 77*(a*b^ 3*e^2)^(1/4)*a^2*b^2*c*d^4*e + 5*(a*b^3*e^2)^(3/4)*b^2*c^4*d + 18*(a*b^...
Time = 12.87 (sec) , antiderivative size = 32185, normalized size of antiderivative = 52.16 \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/((e*x)^(1/2)*(a + b*x^2)^3*(c + d*x)),x)
Output:
(((11*b^2*c^3*e^3 + 19*a*b*c*d^2*e^3)*(e*x)^(1/2))/(16*a*(a^2*d^4 + b^2*c^ 4 + 2*a*b*c^2*d^2)) - ((9*b^2*c^2*d*e^2 + 17*a*b*d^3*e^2)*(e*x)^(3/2))/(16 *a*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - (b*(e*x)^(7/2)*(5*b^2*c^2*d + 13 *a*b*d^3))/(16*a^2*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (b*e*(e*x)^(5/2) *(7*b^2*c^3 + 15*a*b*c*d^2))/(16*a^2*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2))) /(a^2*e^4 + b^2*e^4*x^4 + 2*a*b*e^4*x^2) - atan(((((522144*a^2*b^14*c^19*d ^3*e^9 + 5513120*a^3*b^13*c^17*d^5*e^9 + 26633856*a^4*b^12*c^15*d^7*e^9 + 76183680*a^5*b^11*c^13*d^9*e^9 + 142456000*a^6*b^10*c^11*d^11*e^9 + 189052 608*a^7*b^9*c^9*d^13*e^9 + 201120896*a^8*b^8*c^7*d^15*e^9 + 187175552*a^9* b^7*c^5*d^17*e^9 + 124989600*a^10*b^6*c^3*d^19*e^9 + 37797536*a^11*b^5*c*d ^21*e^9)/(32768*(a^16*d^16 + a^8*b^8*c^16 + 8*a^15*b*c^2*d^14 + 8*a^9*b^7* c^14*d^2 + 28*a^10*b^6*c^12*d^4 + 56*a^11*b^5*c^10*d^6 + 70*a^12*b^4*c^8*d ^8 + 56*a^13*b^3*c^6*d^10 + 28*a^14*b^2*c^4*d^12)) - (((16777216*a^18*b^4* d^24*e^10 - 2752512*a^7*b^15*c^22*d^2*e^10 - 23855104*a^8*b^14*c^20*d^4*e^ 10 - 86638592*a^9*b^13*c^18*d^6*e^10 - 147849216*a^10*b^12*c^16*d^8*e^10 - 18087936*a^11*b^11*c^14*d^10*e^10 + 473432064*a^12*b^10*c^12*d^12*e^10 + 1117519872*a^13*b^9*c^10*d^14*e^10 + 1387266048*a^14*b^8*c^8*d^16*e^10 + 1 065222144*a^15*b^7*c^6*d^18*e^10 + 508821504*a^16*b^6*c^4*d^20*e^10 + 1393 29536*a^17*b^5*c^2*d^22*e^10)/(32768*(a^16*d^16 + a^8*b^8*c^16 + 8*a^15*b* c^2*d^14 + 8*a^9*b^7*c^14*d^2 + 28*a^10*b^6*c^12*d^4 + 56*a^11*b^5*c^10...
Time = 0.31 (sec) , antiderivative size = 3371, normalized size of antiderivative = 5.46 \[ \int \frac {1}{\sqrt {e x} (c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(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*c*d**5 + 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*c**3*d**3 + 180*b**(1/4)*a**(3/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr t(2)))*a**3*b*c*d**5*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**2 *c**5*d + 72*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**2*c**3*d**3*x**2 + 90*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*s qrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*c*d**5*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**3*c**5*d*x**2 + 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)*sq rt(2)))*a*b**3*c**3*d**3*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**4* c**5*d*x**4 - 154*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*c**2*d**4 - 132*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**3*b*c**4*d**2 - 308*b**(3/4)*a**(1/...