Integrand size = 24, antiderivative size = 202 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {3 (b d-a e) (5 b B d-4 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}} \]
3/4*(-a*e+b*d)*(-4*A*b*e-B*a*e+5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1 /2)/(e*x+d)^(1/2))/e^(7/2)/b^(1/2)-2*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d) /(e*x+d)^(1/2)+1/2*(-4*A*b*e-B*a*e+5*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2)/e^ 2/(-a*e+b*d)-3/4*(-4*A*b*e-B*a*e+5*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/e^3
Time = 1.74 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {\frac {\sqrt {e} \sqrt {a+b x} \left (4 A b e (3 d+e x)+a e (13 B d-8 A e+5 B e x)+b B \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{\sqrt {d+e x}}-\frac {6 (b d-a e) (5 b B d-4 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {a-\frac {b d}{e}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b}}}{4 e^{7/2}} \]
((Sqrt[e]*Sqrt[a + b*x]*(4*A*b*e*(3*d + e*x) + a*e*(13*B*d - 8*A*e + 5*B*e *x) + b*B*(-15*d^2 - 5*d*e*x + 2*e^2*x^2)))/Sqrt[d + e*x] - (6*(b*d - a*e) *(5*b*B*d - 4*A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*(Sqr t[a - (b*d)/e] - Sqrt[a + b*x]))])/Sqrt[b])/(4*e^(7/2))
Time = 0.26 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 60, 60, 66, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-a B e-4 A b e+5 b B d) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-a B e-4 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-a B e-4 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(-a B e-4 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-a B e-4 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)}\) |
(-2*(B*d - A*e)*(a + b*x)^(5/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + ((5*b*B*d - 4*A*b*e - a*B*e)*(((a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e) - (3*(b*d - a*e )*((Sqrt[a + b*x]*Sqrt[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/(e*(b*d - a* e))
3.23.15.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(739\) vs. \(2(172)=344\).
Time = 1.09 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.66
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \left (12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,e^{3} x -12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d \,e^{2} x +3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{3} x -18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{2} x +15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e x +4 B b \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{2}-12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e +8 A b \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} d \,e^{2}-18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,d^{2} e +15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{3}+10 B a \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-10 B b d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-16 A a \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+24 A b d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+26 B a d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-30 B b \,d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \sqrt {e x +d}\, e^{3}}\) | \(740\) |
1/8*(b*x+a)^(1/2)*(12*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1 /2)+a*e+b*d)/(b*e)^(1/2))*a*b*e^3*x-12*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d ))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d*e^2*x+3*B*ln(1/2*(2*b*e*x +2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*e^3*x-18* B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/ 2))*a*b*d*e^2*x+15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2) +a*e+b*d)/(b*e)^(1/2))*b^2*d^2*e*x+4*B*b*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*( b*e)^(1/2)+12*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+ b*d)/(b*e)^(1/2))*a*b*d*e^2-12*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2) *(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^2*e+8*A*b*e^2*x*((b*x+a)*(e*x+d)) ^(1/2)*(b*e)^(1/2)+3*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/ 2)+a*e+b*d)/(b*e)^(1/2))*a^2*d*e^2-18*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d) )^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d^2*e+15*B*ln(1/2*(2*b*e*x+2 *((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^3+10*B*a* e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-10*B*b*d*e*x*((b*x+a)*(e*x+d))^( 1/2)*(b*e)^(1/2)-16*A*a*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+24*A*b*d*e *((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+26*B*a*d*e*((b*x+a)*(e*x+d))^(1/2)*(b *e)^(1/2)-30*B*b*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d) )^(1/2)/(b*e)^(1/2)/(e*x+d)^(1/2)/e^3
Time = 0.57 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\left [\frac {3 \, {\left (5 \, B b^{2} d^{3} - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e + {\left (B a^{2} + 4 \, A a b\right )} d e^{2} + {\left (5 \, B b^{2} d^{2} e - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} + {\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (2 \, B b^{2} e^{3} x^{2} - 15 \, B b^{2} d^{2} e - 8 \, A a b e^{3} + {\left (13 \, B a b + 12 \, A b^{2}\right )} d e^{2} - {\left (5 \, B b^{2} d e^{2} - {\left (5 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{16 \, {\left (b e^{5} x + b d e^{4}\right )}}, -\frac {3 \, {\left (5 \, B b^{2} d^{3} - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e + {\left (B a^{2} + 4 \, A a b\right )} d e^{2} + {\left (5 \, B b^{2} d^{2} e - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} + {\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, B b^{2} e^{3} x^{2} - 15 \, B b^{2} d^{2} e - 8 \, A a b e^{3} + {\left (13 \, B a b + 12 \, A b^{2}\right )} d e^{2} - {\left (5 \, B b^{2} d e^{2} - {\left (5 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{8 \, {\left (b e^{5} x + b d e^{4}\right )}}\right ] \]
[1/16*(3*(5*B*b^2*d^3 - 2*(3*B*a*b + 2*A*b^2)*d^2*e + (B*a^2 + 4*A*a*b)*d* e^2 + (5*B*b^2*d^2*e - 2*(3*B*a*b + 2*A*b^2)*d*e^2 + (B*a^2 + 4*A*a*b)*e^3 )*x)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b* e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b* e^2)*x) + 4*(2*B*b^2*e^3*x^2 - 15*B*b^2*d^2*e - 8*A*a*b*e^3 + (13*B*a*b + 12*A*b^2)*d*e^2 - (5*B*b^2*d*e^2 - (5*B*a*b + 4*A*b^2)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*e^5*x + b*d*e^4), -1/8*(3*(5*B*b^2*d^3 - 2*(3*B*a*b + 2*A*b^2)*d^2*e + (B*a^2 + 4*A*a*b)*d*e^2 + (5*B*b^2*d^2*e - 2*(3*B*a*b + 2*A*b^2)*d*e^2 + (B*a^2 + 4*A*a*b)*e^3)*x)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(2*B*b^2*e^3*x^2 - 15*B*b^2*d^2*e - 8*A*a*b *e^3 + (13*B*a*b + 12*A*b^2)*d*e^2 - (5*B*b^2*d*e^2 - (5*B*a*b + 4*A*b^2)* e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*e^5*x + b*d*e^4)]
\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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*(a*e-b*d)>0)', see `assume?` f or more de
Time = 0.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |}}{b e} - \frac {5 \, B b^{2} d e^{3} {\left | b \right |} - B a b e^{4} {\left | b \right |} - 4 \, A b^{2} e^{4} {\left | b \right |}}{b^{2} e^{5}}\right )} - \frac {3 \, {\left (5 \, B b^{3} d^{2} e^{2} {\left | b \right |} - 6 \, B a b^{2} d e^{3} {\left | b \right |} - 4 \, A b^{3} d e^{3} {\left | b \right |} + B a^{2} b e^{4} {\left | b \right |} + 4 \, A a b^{2} e^{4} {\left | b \right |}\right )}}{b^{2} e^{5}}\right )}}{4 \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} - \frac {3 \, {\left (5 \, B b^{2} d^{2} {\left | b \right |} - 6 \, B a b d e {\left | b \right |} - 4 \, A b^{2} d e {\left | b \right |} + B a^{2} e^{2} {\left | b \right |} + 4 \, A a b e^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{4 \, \sqrt {b e} b e^{3}} \]
1/4*sqrt(b*x + a)*((b*x + a)*(2*(b*x + a)*B*abs(b)/(b*e) - (5*B*b^2*d*e^3* abs(b) - B*a*b*e^4*abs(b) - 4*A*b^2*e^4*abs(b))/(b^2*e^5)) - 3*(5*B*b^3*d^ 2*e^2*abs(b) - 6*B*a*b^2*d*e^3*abs(b) - 4*A*b^3*d*e^3*abs(b) + B*a^2*b*e^4 *abs(b) + 4*A*a*b^2*e^4*abs(b))/(b^2*e^5))/sqrt(b^2*d + (b*x + a)*b*e - a* b*e) - 3/4*(5*B*b^2*d^2*abs(b) - 6*B*a*b*d*e*abs(b) - 4*A*b^2*d*e*abs(b) + B*a^2*e^2*abs(b) + 4*A*a*b*e^2*abs(b))*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b*e^3)
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]