Integrand size = 24, antiderivative size = 599 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=-\frac {e \sqrt {e x} (c-d x)}{4 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}-\frac {e \sqrt {e x} \left (a d^2 (7 c-5 d x)-b c^2 (c-3 d x)\right )}{16 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}+\frac {2 c^{3/2} d^{7/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\left (b c^2+a d^2\right )^3}-\frac {\left (3 b^{5/2} c^5-3 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2-30 a^{3/2} b c^2 d^3-21 a^2 \sqrt {b} c d^4+5 a^{5/2} d^5\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{7/4} b^{3/4} \left (b c^2+a d^2\right )^3}+\frac {\left (3 b^{5/2} c^5-3 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2-30 a^{3/2} b c^2 d^3-21 a^2 \sqrt {b} c d^4+5 a^{5/2} d^5\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{7/4} b^{3/4} \left (b c^2+a d^2\right )^3}+\frac {\left (\sqrt {b} c \left (3 b^2 c^4+14 a b c^2 d^2-21 a^2 d^4\right )+\sqrt {a} d \left (3 b^2 c^4+30 a b c^2 d^2-5 a^2 d^4\right )\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{7/4} b^{3/4} \left (b c^2+a d^2\right )^3} \] Output:
-1/4*e*(e*x)^(1/2)*(-d*x+c)/(a*d^2+b*c^2)/(b*x^2+a)^2-1/16*e*(e*x)^(1/2)*( a*d^2*(-5*d*x+7*c)-b*c^2*(-3*d*x+c))/a/(a*d^2+b*c^2)^2/(b*x^2+a)+2*c^(3/2) *d^(7/2)*e^(3/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/(a*d^2+b*c^2) ^3-1/64*(3*b^(5/2)*c^5-3*a^(1/2)*b^2*c^4*d+14*a*b^(3/2)*c^3*d^2-30*a^(3/2) *b*c^2*d^3-21*a^2*b^(1/2)*c*d^4+5*a^(5/2)*d^5)*e^(3/2)*arctan(1-2^(1/2)*b^ (1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(7/4)/b^(3/4)/(a*d^2+b*c^2)^3 +1/64*(3*b^(5/2)*c^5-3*a^(1/2)*b^2*c^4*d+14*a*b^(3/2)*c^3*d^2-30*a^(3/2)*b *c^2*d^3-21*a^2*b^(1/2)*c*d^4+5*a^(5/2)*d^5)*e^(3/2)*arctan(1+2^(1/2)*b^(1 /4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(7/4)/b^(3/4)/(a*d^2+b*c^2)^3+1 /64*(b^(1/2)*c*(-21*a^2*d^4+14*a*b*c^2*d^2+3*b^2*c^4)+a^(1/2)*d*(-5*a^2*d^ 4+30*a*b*c^2*d^2+3*b^2*c^4))*e^(3/2)*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^(7/4)/b^(3/4)/(a*d^2+b*c^2)^ 3
Time = 1.90 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\frac {(e x)^{3/2} \left (\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} \left (b^2 c^2 x^2 (c-3 d x)+a^2 d^2 (-11 c+9 d x)+a b \left (-3 c^3+c^2 d x-7 c d^2 x^2+5 d^3 x^3\right )\right )}{a \left (a+b x^2\right )^2}+128 c^{3/2} d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )-\frac {\sqrt {2} \left (3 b^{5/2} c^5-3 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2-30 a^{3/2} b c^2 d^3-21 a^2 \sqrt {b} c d^4+5 a^{5/2} d^5\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4} b^{3/4}}+\frac {\sqrt {2} \left (3 b^{5/2} c^5+3 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2+30 a^{3/2} b c^2 d^3-21 a^2 \sqrt {b} c d^4-5 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4} b^{3/4}}\right )}{64 \left (b c^2+a d^2\right )^3 x^{3/2}} \] Input:
Integrate[(e*x)^(3/2)/((c + d*x)*(a + b*x^2)^3),x]
Output:
((e*x)^(3/2)*((4*(b*c^2 + a*d^2)*Sqrt[x]*(b^2*c^2*x^2*(c - 3*d*x) + a^2*d^ 2*(-11*c + 9*d*x) + a*b*(-3*c^3 + c^2*d*x - 7*c*d^2*x^2 + 5*d^3*x^3)))/(a* (a + b*x^2)^2) + 128*c^(3/2)*d^(7/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] - ( Sqrt[2]*(3*b^(5/2)*c^5 - 3*Sqrt[a]*b^2*c^4*d + 14*a*b^(3/2)*c^3*d^2 - 30*a ^(3/2)*b*c^2*d^3 - 21*a^2*Sqrt[b]*c*d^4 + 5*a^(5/2)*d^5)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(7/4)*b^(3/4)) + (Sqrt[ 2]*(3*b^(5/2)*c^5 + 3*Sqrt[a]*b^2*c^4*d + 14*a*b^(3/2)*c^3*d^2 + 30*a^(3/2 )*b*c^2*d^3 - 21*a^2*Sqrt[b]*c*d^4 - 5*a^(5/2)*d^5)*ArcTanh[(Sqrt[2]*a^(1/ 4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(7/4)*b^(3/4))))/(64*(b*c^2 + a*d^2)^3*x^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(1312\) vs. \(2(599)=1198\).
Time = 3.32 (sec) , antiderivative size = 1312, normalized size of antiderivative = 2.19, 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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^3 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (-\frac {b d^2 (e x)^{3/2} (d x-c)}{\left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}+\frac {b (e x)^{3/2} (c-d x)}{\left (a+b x^2\right )^3 \left (a d^2+b c^2\right )}+\frac {d^6 (e x)^{3/2}}{(c+d x) \left (a d^2+b c^2\right )^3}-\frac {b d^4 (e x)^{3/2} (d x-c)}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{a} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} b^{3/4} \left (b c^2+a d^2\right )^3}-\frac {\sqrt [4]{a} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} b^{3/4} \left (b c^2+a d^2\right )^3}+\frac {\sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \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} b^{3/4} \left (b c^2+a d^2\right )^3}-\frac {\sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \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} b^{3/4} \left (b c^2+a d^2\right )^3}+\frac {2 c^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{7/2}}{\left (b c^2+a d^2\right )^3}-\frac {\left (\sqrt {b} c-3 \sqrt {a} d\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{4 \sqrt {2} a^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2}+\frac {\left (\sqrt {b} c-3 \sqrt {a} d\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{4 \sqrt {2} a^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2}-\frac {\left (\sqrt {b} c+3 \sqrt {a} d\right ) e^{3/2} \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^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2}+\frac {\left (\sqrt {b} c+3 \sqrt {a} d\right ) e^{3/2} \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^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2}-\frac {e \sqrt {e x} (c-d x) d^2}{2 \left (b c^2+a d^2\right )^2 \left (b x^2+a\right )}-\frac {3 \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{7/4} b^{3/4} \left (b c^2+a d^2\right )}+\frac {3 \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{7/4} b^{3/4} \left (b c^2+a d^2\right )}-\frac {3 \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \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^{7/4} b^{3/4} \left (b c^2+a d^2\right )}+\frac {3 \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \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^{7/4} b^{3/4} \left (b c^2+a d^2\right )}+\frac {e \sqrt {e x} (c-3 d x)}{16 a \left (b c^2+a d^2\right ) \left (b x^2+a\right )}-\frac {e \sqrt {e x} (c-d x)}{4 \left (b c^2+a d^2\right ) \left (b x^2+a\right )^2}\) |
Input:
Int[(e*x)^(3/2)/((c + d*x)*(a + b*x^2)^3),x]
Output:
-1/4*(e*Sqrt[e*x]*(c - d*x))/((b*c^2 + a*d^2)*(a + b*x^2)^2) + (e*Sqrt[e*x ]*(c - 3*d*x))/(16*a*(b*c^2 + a*d^2)*(a + b*x^2)) - (d^2*e*Sqrt[e*x]*(c - d*x))/(2*(b*c^2 + a*d^2)^2*(a + b*x^2)) + (2*c^(3/2)*d^(7/2)*e^(3/2)*ArcTa n[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^2)^3 + (a^(1/4)*d^4 *(Sqrt[b]*c - Sqrt[a]*d)*e^(3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a ^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(3/4)*(b*c^2 + a*d^2)^3) - (d^2*(Sqrt[b]*c - 3*Sqrt[a]*d)*e^(3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[ e])])/(4*Sqrt[2]*a^(3/4)*b^(3/4)*(b*c^2 + a*d^2)^2) - (3*(Sqrt[b]*c - Sqrt [a]*d)*e^(3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/ (32*Sqrt[2]*a^(7/4)*b^(3/4)*(b*c^2 + a*d^2)) - (a^(1/4)*d^4*(Sqrt[b]*c - S qrt[a]*d)*e^(3/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e]) ])/(Sqrt[2]*b^(3/4)*(b*c^2 + a*d^2)^3) + (d^2*(Sqrt[b]*c - 3*Sqrt[a]*d)*e^ (3/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqrt[2 ]*a^(3/4)*b^(3/4)*(b*c^2 + a*d^2)^2) + (3*(Sqrt[b]*c - Sqrt[a]*d)*e^(3/2)* ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sqrt[2]*a^( 7/4)*b^(3/4)*(b*c^2 + a*d^2)) + (a^(1/4)*d^4*(Sqrt[b]*c + Sqrt[a]*d)*e^(3/ 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]*b^(3/4)*(b*c^2 + a*d^2)^3) - (d^2*(Sqrt[b]*c + 3*Sqrt[a] *d)*e^(3/2)*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^(3/4)*b^(3/4)*(b*c^2 + a*d^2)^2) - (3*(Sqr...
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 = 0.79 (sec) , antiderivative size = 553, normalized size of antiderivative = 0.92
| method | result | size |
| pseudoelliptic | \(\frac {5 \left (-\frac {44 \left (-\frac {32 a \,c^{2} d^{4} e \left (b \,x^{2}+a \right )^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{11}+\left (d^{2} \left (-\frac {9 d x}{11}+c \right ) a^{2}+\frac {3 \left (-\frac {5}{3} d^{3} x^{3}+\frac {7}{3} c \,d^{2} x^{2}-\frac {1}{3} c^{2} d x +c^{3}\right ) b a}{11}-\frac {b^{2} c^{2} x^{2} \left (-3 d x +c \right )}{11}\right ) \sqrt {e x}\, \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\right ) b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a}{5}+\sqrt {d e c}\, \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \left (-\frac {21 \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 )+\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}+\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 {2}{3} b \,c^{2} d^{2} a -\frac {1}{7} b^{2} c^{4}\right ) c \sqrt {\frac {a \,e^{2}}{b}}}{5}+d 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 \left (a^{2} d^{4}-6 b \,c^{2} d^{2} a -\frac {3}{5} b^{2} c^{4}\right )\right )\right ) e}{64 \sqrt {d e c}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3} b \left (b \,x^{2}+a \right )^{2}}\) | \(553\) |
| derivativedivides | \(2 e^{6} \left (-\frac {\frac {-\frac {b d \left (5 a^{2} d^{4}+2 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a}+\frac {b c 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 {d \,e^{2} \left (9 a^{2} d^{4}+10 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{32}+\frac {c \,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 c e \,a^{2} d^{4}-14 a b \,d^{2} e \,c^{3}-3 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 (-5 a^{2} d^{5}+30 d^{3} a \,c^{2} b +3 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}}{\left (a \,d^{2}+b \,c^{2}\right )^{3} e^{4}}+\frac {c^{2} d^{4} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3} \sqrt {d e c}}\right )\) | \(585\) |
| default | \(2 e^{6} \left (-\frac {\frac {-\frac {b d \left (5 a^{2} d^{4}+2 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right ) \left (e x \right )^{\frac {7}{2}}}{32 a}+\frac {b c 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 {d \,e^{2} \left (9 a^{2} d^{4}+10 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{32}+\frac {c \,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 c e \,a^{2} d^{4}-14 a b \,d^{2} e \,c^{3}-3 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 (-5 a^{2} d^{5}+30 d^{3} a \,c^{2} b +3 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}}{\left (a \,d^{2}+b \,c^{2}\right )^{3} e^{4}}+\frac {c^{2} d^{4} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3} \sqrt {d e c}}\right )\) | \(585\) |
Input:
int((e*x)^(3/2)/(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
5/64*(-44/5*(-32/11*a*c^2*d^4*e*(b*x^2+a)^2*arctan(d*(e*x)^(1/2)/(d*e*c)^( 1/2))+(d^2*(-9/11*d*x+c)*a^2+3/11*(-5/3*d^3*x^3+7/3*c*d^2*x^2-1/3*c^2*d*x+ c^3)*b*a-1/11*b^2*c^2*x^2*(-3*d*x+c))*(e*x)^(1/2)*(a*d^2+b*c^2)*(d*e*c)^(1 /2))*b*(a*e^2/b)^(1/4)*a+(d*e*c)^(1/2)*2^(1/2)*(b*x^2+a)^2*(-21/5*(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)))+arctan((2^(1/2)*(e*x)^(1/2)-(a*e^2/b)^(1/4) )/(a*e^2/b)^(1/4)))*b*(a^2*d^4-2/3*b*c^2*d^2*a-1/7*b^2*c^4)*c*(a*e^2/b)^(1 /2)+d*e*(arctan((2^(1/2)*(e*x)^(1/2)-(a*e^2/b)^(1/4))/(a*e^2/b)^(1/4))+arc tan((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^2*d^4-6*b*c^2*d^2*a-3/5*b^2*c^4)) )/(d*e*c)^(1/2)*e/(a*e^2/b)^(1/4)/a^2/(a*d^2+b*c^2)^3/b/(b*x^2+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 8140 vs. \(2 (488) = 976\).
Time = 92.71 (sec) , antiderivative size = 16292, normalized size of antiderivative = 27.20 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(3/2)/(d*x+c)/(b*x**2+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(3/2)/(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. 1201 vs. \(2 (488) = 976\).
Time = 0.23 (sec) , antiderivative size = 1201, normalized size of antiderivative = 2.01 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")
Output:
1/64*(128*c^2*d^4*e^3*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) *b^4*c^5*e^2 + 14*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2*e^2 - 21*(a*b^3*e^2)^(1/ 4)*a^2*b^2*c*d^4*e^2 - 3*(a*b^3*e^2)^(3/4)*b^2*c^4*d*e - 30*(a*b^3*e^2)^(3 /4)*a*b*c^2*d^3*e + 5*(a*b^3*e^2)^(3/4)*a^2*d^5*e)*arctan(1/2*sqrt(2)*(sqr t(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a^2*b^6*c^6 + 3*sqrt(2)*a^3*b^5*c^4*d^2 + 3*sqrt(2)*a^4*b^4*c^2*d^4 + sqrt(2)*a^5*b^3* d^6) + 2*(3*(a*b^3*e^2)^(1/4)*b^4*c^5*e^2 + 14*(a*b^3*e^2)^(1/4)*a*b^3*c^3 *d^2*e^2 - 21*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^4*e^2 - 3*(a*b^3*e^2)^(3/4)*b^ 2*c^4*d*e - 30*(a*b^3*e^2)^(3/4)*a*b*c^2*d^3*e + 5*(a*b^3*e^2)^(3/4)*a^2*d ^5*e)*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^6*c^6 + 3*sqrt(2)*a^3*b^5*c^4*d^2 + 3*sqrt(2)*a^4* b^4*c^2*d^4 + sqrt(2)*a^5*b^3*d^6) + (3*(a*b^3*e^2)^(1/4)*b^4*c^5*e^2 + 14 *(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2*e^2 - 21*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^4* e^2 + 3*(a*b^3*e^2)^(3/4)*b^2*c^4*d*e + 30*(a*b^3*e^2)^(3/4)*a*b*c^2*d^3*e - 5*(a*b^3*e^2)^(3/4)*a^2*d^5*e)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e *x) + sqrt(a*e^2/b))/(sqrt(2)*a^2*b^6*c^6 + 3*sqrt(2)*a^3*b^5*c^4*d^2 + 3* sqrt(2)*a^4*b^4*c^2*d^4 + sqrt(2)*a^5*b^3*d^6) - (3*(a*b^3*e^2)^(1/4)*b^4* c^5*e^2 + 14*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d^2*e^2 - 21*(a*b^3*e^2)^(1/4)*a^ 2*b^2*c*d^4*e^2 + 3*(a*b^3*e^2)^(3/4)*b^2*c^4*d*e + 30*(a*b^3*e^2)^(3/4...
Time = 11.17 (sec) , antiderivative size = 6273, normalized size of antiderivative = 10.47 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int((e*x)^(3/2)/((a + b*x^2)^3*(c + d*x)),x)
Output:
(((e*x)^(3/2)*(9*a*d^3*e^4 + b*c^2*d*e^4))/(16*(a^2*d^4 + b^2*c^4 + 2*a*b* c^2*d^2)) - ((e*x)^(1/2)*(3*b*c^3*e^5 + 11*a*c*d^2*e^5))/(16*(a^2*d^4 + b^ 2*c^4 + 2*a*b*c^2*d^2)) + (b*(e*x)^(5/2)*(b*c^3*e^3 - 7*a*c*d^2*e^3))/(16* a*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) + (b*(e*x)^(7/2)*(5*a*d^3*e^2 - 3*b *c^2*d*e^2))/(16*a*(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) + symsum(log((81*b^6*c^12*d^8*e^18 + 1098*a^2*b^4*c ^8*d^12*e^18 + 16266*a^3*b^3*c^6*d^14*e^18 + 7925*a^4*b^2*c^4*d^16*e^18 + 405*a*b^5*c^10*d^10*e^18 + 625*a^5*b*c^2*d^18*e^18)/(32768*(a^12*d^16 + a^ 4*b^8*c^16 + 8*a^11*b*c^2*d^14 + 8*a^5*b^7*c^14*d^2 + 28*a^6*b^6*c^12*d^4 + 56*a^7*b^5*c^10*d^6 + 70*a^8*b^4*c^8*d^8 + 56*a^9*b^3*c^6*d^10 + 28*a^10 *b^2*c^4*d^12)) - root(15502147584*a^13*b^9*c^12*d^12*g^6 + 1107296256*a^1 7*b^5*c^4*d^20*g^6 + 1107296256*a^9*b^13*c^20*d^4*g^6 + 13287555072*a^14*b ^8*c^10*d^14*g^6 + 13287555072*a^12*b^10*c^14*d^10*g^6 + 201326592*a^18*b^ 4*c^2*d^22*g^6 + 201326592*a^8*b^14*c^22*d^2*g^6 + 8304721920*a^15*b^7*c^8 *d^16*g^6 + 8304721920*a^11*b^11*c^16*d^8*g^6 + 3690987520*a^16*b^6*c^6*d^ 18*g^6 + 3690987520*a^10*b^12*c^18*d^6*g^6 + 16777216*a^19*b^3*d^24*g^6 + 16777216*a^7*b^15*c^24*g^6 + 138067968*a^12*b^4*c^5*d^17*e^3*g^4 - 4076339 2*a^7*b^9*c^15*d^7*e^3*g^4 + 38240256*a^8*b^8*c^13*d^9*e^3*g^4 - 20791296* a^6*b^10*c^17*d^5*e^3*g^4 + 17924096*a^13*b^3*c^3*d^19*e^3*g^4 - 3047424*a ^5*b^11*c^19*d^3*e^3*g^4 - 1720320*a^14*b^2*c*d^21*e^3*g^4 - 147456*a^4...
Time = 0.29 (sec) , antiderivative size = 3333, normalized size of antiderivative = 5.56 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((e*x)^(3/2)/(d*x+c)/(b*x^2+a)^3,x)
Output:
(sqrt(e)*e*( - 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**4*d**5 + 60*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**2*d**3 - 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)*s qrt(2)))*a**3*b*d**5*x**2 + 6*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**4*d + 120*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**2*d**3*x**2 - 10*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*d**5*x**4 + 12*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**4*d*x**2 + 60*b**(1/4)*a**(3/4)*sqrt(2)*at an((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**2*d**3*x**4 + 6*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** 4*d*x**4 + 42*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*d**4 - 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**3*b*c**3*d**2 + 84*b**(3/4)*a**(1/4)*sqrt(2...