Integrand size = 24, antiderivative size = 412 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=-\frac {2}{a c^2 e \sqrt {e x}}-\frac {d^3 \sqrt {e x}}{c^2 \left (b c^2+a d^2\right ) e^2 (c+d x)}-\frac {d^{5/2} \left (7 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{5/2} \left (b c^2+a d^2\right )^2 e^{3/2}}+\frac {b^{5/4} \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^2 e^{3/2}}-\frac {b^{5/4} \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^2 e^{3/2}}+\frac {b^{5/4} \left (b c^2-2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^2 e^{3/2}} \] Output:
-2/a/c^2/e/(e*x)^(1/2)-d^3*(e*x)^(1/2)/c^2/(a*d^2+b*c^2)/e^2/(d*x+c)-d^(5/ 2)*(3*a*d^2+7*b*c^2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(5/2)/( a*d^2+b*c^2)^2/e^(3/2)+1/2*b^(5/4)*(b*c^2+2*a^(1/2)*b^(1/2)*c*d-a*d^2)*arc tan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/(a*d^2+ b*c^2)^2/e^(3/2)-1/2*b^(5/4)*(b*c^2+2*a^(1/2)*b^(1/2)*c*d-a*d^2)*arctan(1+ 2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/(a*d^2+b*c^2) ^2/e^(3/2)+1/2*b^(5/4)*(b*c^2-2*a^(1/2)*b^(1/2)*c*d-a*d^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^(5/4)/ (a*d^2+b*c^2)^2/e^(3/2)
Time = 0.65 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=\frac {x \left (-\frac {2 \left (b c^2+a d^2\right ) \left (2 b c^2 (c+d x)+a d^2 (2 c+3 d x)\right )}{a c^2 (c+d x)}-\frac {2 d^{5/2} \left (7 b c^2+3 a d^2\right ) \sqrt {x} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{5/2}}+\frac {\sqrt {2} b^{5/4} \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4}}+\frac {\sqrt {2} b^{5/4} \left (b c^2-2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4}}\right )}{2 \left (b c^2+a d^2\right )^2 (e x)^{3/2}} \] Input:
Integrate[1/((e*x)^(3/2)*(c + d*x)^2*(a + b*x^2)),x]
Output:
(x*((-2*(b*c^2 + a*d^2)*(2*b*c^2*(c + d*x) + a*d^2*(2*c + 3*d*x)))/(a*c^2* (c + d*x)) - (2*d^(5/2)*(7*b*c^2 + 3*a*d^2)*Sqrt[x]*ArcTan[(Sqrt[d]*Sqrt[x ])/Sqrt[c]])/c^(5/2) + (Sqrt[2]*b^(5/4)*(b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - a *d^2)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x ])])/a^(5/4) + (Sqrt[2]*b^(5/4)*(b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*Sq rt[x]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^ (5/4)))/(2*(b*c^2 + a*d^2)^2*(e*x)^(3/2))
Time = 1.64 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.59, 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.
| \(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right ) (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b \left (-a d^2+b c^2-2 b c d x\right )}{(e x)^{3/2} \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}+\frac {2 b c d^2}{(e x)^{3/2} (c+d x) \left (a d^2+b c^2\right )^2}+\frac {d^2}{(e x)^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^{5/4} \left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{5/4} e^{3/2} \left (a d^2+b c^2\right )^2}-\frac {b^{5/4} \left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} a^{5/4} e^{3/2} \left (a d^2+b c^2\right )^2}-\frac {b^{5/4} \left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} a^{5/4} e^{3/2} \left (a d^2+b c^2\right )^2}+\frac {b^{5/4} \left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} a^{5/4} e^{3/2} \left (a d^2+b c^2\right )^2}-\frac {4 b d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c} e^{3/2} \left (a d^2+b c^2\right )^2}-\frac {3 d^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{5/2} e^{3/2} \left (a d^2+b c^2\right )}-\frac {3 d^2}{c^2 e \sqrt {e x} \left (a d^2+b c^2\right )}-\frac {4 b d^2}{e \sqrt {e x} \left (a d^2+b c^2\right )^2}+\frac {d^2}{c e \sqrt {e x} (c+d x) \left (a d^2+b c^2\right )}-\frac {2 b \left (b c^2-a d^2\right )}{a e \sqrt {e x} \left (a d^2+b c^2\right )^2}\) |
Input:
Int[1/((e*x)^(3/2)*(c + d*x)^2*(a + b*x^2)),x]
Output:
(-4*b*d^2)/((b*c^2 + a*d^2)^2*e*Sqrt[e*x]) - (2*b*(b*c^2 - a*d^2))/(a*(b*c ^2 + a*d^2)^2*e*Sqrt[e*x]) - (3*d^2)/(c^2*(b*c^2 + a*d^2)*e*Sqrt[e*x]) + d ^2/(c*(b*c^2 + a*d^2)*e*Sqrt[e*x]*(c + d*x)) - (4*b*d^(5/2)*ArcTan[(Sqrt[d ]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(Sqrt[c]*(b*c^2 + a*d^2)^2*e^(3/2)) - (3* d^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(c^(5/2)*(b*c^2 + a *d^2)*e^(3/2)) + (b^(5/4)*(b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^2*e^(3/2)) - (b^(5/4)*(b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*Ar cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(5/4)* (b*c^2 + a*d^2)^2*e^(3/2)) - (b^(5/4)*(b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d - a*d ^2)*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^(5/4)*(b*c^2 + a*d^2)^2*e^(3/2)) + (b^(5/4)*(b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e]*x + S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^2*e^ (3/2))
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 = 0.58 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(-\frac {2}{a \,c^{2} e \sqrt {e x}}-\frac {-\frac {2 b^{2} c^{2} \left (-\frac {c d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}+\frac {\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {2 d^{3} a \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (3 a \,d^{2}+7 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2}}}{c^{2} a e}\) | \(426\) |
| derivativedivides | \(2 e^{3} \left (-\frac {d^{3} \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (3 a \,d^{2}+7 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{c^{2} e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{2}}-\frac {1}{c^{2} e^{4} a \sqrt {e x}}-\frac {b^{2} \left (\frac {c d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}+\frac {\left (-a \,d^{2}+b \,c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} e^{4} a}\right )\) | \(427\) |
| default | \(2 e^{3} \left (-\frac {d^{3} \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (3 a \,d^{2}+7 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{c^{2} e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{2}}-\frac {1}{c^{2} e^{4} a \sqrt {e x}}-\frac {b^{2} \left (\frac {c d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}+\frac {\left (-a \,d^{2}+b \,c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} e^{4} a}\right )\) | \(427\) |
| pseudoelliptic | \(\frac {-12 e \left (\left (a \,d^{2}+\frac {7 b \,c^{2}}{3}\right ) d^{3} \left (d x +c \right ) a \sqrt {e x}\, \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )+\frac {2 \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {3}{2} a x \,d^{3}+b \,c^{2} d x +a \,d^{2} c +b \,c^{3}\right )}{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+\left (d x +c \right ) \sqrt {d e c}\, b \sqrt {2}\, c^{2} \sqrt {e x}\, \left (-2 d \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) b c \sqrt {\frac {a \,e^{2}}{b}}+\left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) e \left (a \,d^{2}-b \,c^{2}\right )\right )}{4 \sqrt {e x}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, e^{2} c^{2} a \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )}\) | \(448\) |
Input:
int(1/(e*x)^(3/2)/(d*x+c)^2/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/a/c^2/e/(e*x)^(1/2)-1/c^2/a*(-2*b^2*c^2/(a*d^2+b*c^2)^2*(-1/4*c*d/e*(a* e^2/b)^(1/4)*2^(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*arct an(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4) *(e*x)^(1/2)-1))+1/8*(a*d^2-b*c^2)/b/(a*e^2/b)^(1/4)*2^(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)/(a*e^2/b)^(1/4)*(e*x)^( 1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1)))+2*d^3/(a*d^2+b*c ^2)^2*a*((1/2*a*d^2+1/2*b*c^2)*(e*x)^(1/2)/(d*e*x+c*e)+1/2*(3*a*d^2+7*b*c^ 2)/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2))))/e
Leaf count of result is larger than twice the leaf count of optimal. 4539 vs. \(2 (326) = 652\).
Time = 33.18 (sec) , antiderivative size = 9097, normalized size of antiderivative = 22.08 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)^2/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)**(3/2)/(d*x+c)**2/(b*x**2+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)^2/(b*x^2+a),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. 716 vs. \(2 (326) = 652\).
Time = 0.16 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)^2/(b*x^2+a),x, algorithm="giac")
Output:
-1/2*(2*(2*(a*b^3*e^2)^(1/4)*a*b^2*c*d*e + (a*b^3*e^2)^(3/4)*b*c^2 - (a*b^ 3*e^2)^(3/4)*a*d^2)*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^2*b^3*c^4*e^2 + 2*sqrt(2)*a^3*b^2*c^2*d^2 *e^2 + sqrt(2)*a^4*b*d^4*e^2) + 2*(2*(a*b^3*e^2)^(1/4)*a*b^2*c*d*e + (a*b^ 3*e^2)^(3/4)*b*c^2 - (a*b^3*e^2)^(3/4)*a*d^2)*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^2*b^3*c^4*e^2 + 2*sqrt(2)*a^3*b^2*c^2*d^2*e^2 + sqrt(2)*a^4*b*d^4*e^2) + (2*(a*b^3*e^2)^ (1/4)*a*b^2*c*d*e - (a*b^3*e^2)^(3/4)*b*c^2 + (a*b^3*e^2)^(3/4)*a*d^2)*log (e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^2*b^3 *c^4*e^2 + 2*sqrt(2)*a^3*b^2*c^2*d^2*e^2 + sqrt(2)*a^4*b*d^4*e^2) - (2*(a* b^3*e^2)^(1/4)*a*b^2*c*d*e - (a*b^3*e^2)^(3/4)*b*c^2 + (a*b^3*e^2)^(3/4)*a *d^2)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2 )*a^2*b^3*c^4*e^2 + 2*sqrt(2)*a^3*b^2*c^2*d^2*e^2 + sqrt(2)*a^4*b*d^4*e^2) + 2*(7*b*c^2*d^3 + 3*a*d^5)*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b^2*c^6 + 2 *a*b*c^4*d^2 + a^2*c^2*d^4)*sqrt(c*d*e)) + 2*(2*b*c^2*d*e*x + 3*a*d^3*e*x + 2*b*c^3*e + 2*a*c*d^2*e)/((a*b*c^4 + a^2*c^2*d^2)*(sqrt(e*x)*d*e*x + sqr t(e*x)*c*e)))/e
Time = 11.86 (sec) , antiderivative size = 21231, normalized size of antiderivative = 51.53 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/((e*x)^(3/2)*(a + b*x^2)*(c + d*x)^2),x)
Output:
atan((((e*x)^(1/2)*(32*a^7*b^16*c^32*d^5*e^13 - 560*a^8*b^15*c^30*d^7*e^13 - 3136*a^9*b^14*c^28*d^9*e^13 - 5632*a^10*b^13*c^26*d^11*e^13 - 2816*a^11 *b^12*c^24*d^13*e^13 + 3872*a^12*b^11*c^22*d^15*e^13 + 6720*a^13*b^10*c^20 *d^17*e^13 + 4224*a^14*b^9*c^18*d^19*e^13 + 1248*a^15*b^8*c^16*d^21*e^13 + 144*a^16*b^7*c^14*d^23*e^13) + ((a^2*d^4*(-a^5*b^5)^(1/2) + b^2*c^4*(-a^5 *b^5)^(1/2) - 4*a^3*b^4*c^3*d + 4*a^4*b^3*c*d^3 - 6*a*b*c^2*d^2*(-a^5*b^5) ^(1/2))/(4*(a^9*d^8*e^3 + a^5*b^4*c^8*e^3 + 4*a^6*b^3*c^6*d^2*e^3 + 6*a^7* b^2*c^4*d^4*e^3 + 4*a^8*b*c^2*d^6*e^3)))^(1/2)*(32*a^7*b^17*c^37*d^2*e^15 - ((e*x)^(1/2)*(64*a^8*b^17*c^39*d^2*e^16 + 512*a^9*b^16*c^37*d^4*e^16 + 7 04*a^10*b^15*c^35*d^6*e^16 + 3200*a^11*b^14*c^33*d^8*e^16 + 37632*a^12*b^1 3*c^31*d^10*e^16 + 156160*a^13*b^12*c^29*d^12*e^16 + 337792*a^14*b^11*c^27 *d^14*e^16 + 443136*a^15*b^10*c^25*d^16*e^16 + 372800*a^16*b^9*c^23*d^18*e ^16 + 202752*a^17*b^8*c^21*d^20*e^16 + 69056*a^18*b^7*c^19*d^22*e^16 + 134 40*a^19*b^6*c^17*d^24*e^16 + 1152*a^20*b^5*c^15*d^26*e^16) + ((a^2*d^4*(-a ^5*b^5)^(1/2) + b^2*c^4*(-a^5*b^5)^(1/2) - 4*a^3*b^4*c^3*d + 4*a^4*b^3*c*d ^3 - 6*a*b*c^2*d^2*(-a^5*b^5)^(1/2))/(4*(a^9*d^8*e^3 + a^5*b^4*c^8*e^3 + 4 *a^6*b^3*c^6*d^2*e^3 + 6*a^7*b^2*c^4*d^4*e^3 + 4*a^8*b*c^2*d^6*e^3)))^(1/2 )*((e*x)^(1/2)*((a^2*d^4*(-a^5*b^5)^(1/2) + b^2*c^4*(-a^5*b^5)^(1/2) - 4*a ^3*b^4*c^3*d + 4*a^4*b^3*c*d^3 - 6*a*b*c^2*d^2*(-a^5*b^5)^(1/2))/(4*(a^9*d ^8*e^3 + a^5*b^4*c^8*e^3 + 4*a^6*b^3*c^6*d^2*e^3 + 6*a^7*b^2*c^4*d^4*e^...
\[ \int \frac {1}{(e x)^{3/2} (c+d x)^2 \left (a+b x^2\right )} \, dx=\int \frac {1}{\left (e x \right )^{\frac {3}{2}} \left (d x +c \right )^{2} \left (b \,x^{2}+a \right )}d x \] Input:
int(1/(e*x)^(3/2)/(d*x+c)^2/(b*x^2+a),x)
Output:
int(1/(e*x)^(3/2)/(d*x+c)^2/(b*x^2+a),x)