Integrand size = 24, antiderivative size = 174 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=-\frac {2 a}{7 c e (e x)^{7/2} \sqrt {c+d x}}+\frac {16 a d}{35 c^2 e^2 (e x)^{5/2} \sqrt {c+d x}}+\frac {2 \left (35 b c^2+48 a d^2\right )}{35 c^3 e^3 (e x)^{3/2} \sqrt {c+d x}}-\frac {8 \left (35 b c^2+48 a d^2\right ) \sqrt {c+d x}}{105 c^4 e^3 (e x)^{3/2}}+\frac {16 d \left (35 b c^2+48 a d^2\right ) \sqrt {c+d x}}{105 c^5 e^4 \sqrt {e x}} \] Output:
-2/7*a/c/e/(e*x)^(7/2)/(d*x+c)^(1/2)+16/35*a*d/c^2/e^2/(e*x)^(5/2)/(d*x+c) ^(1/2)+2/35*(48*a*d^2+35*b*c^2)/c^3/e^3/(e*x)^(3/2)/(d*x+c)^(1/2)-8/105*(4 8*a*d^2+35*b*c^2)*(d*x+c)^(1/2)/c^4/e^3/(e*x)^(3/2)+16/105*d*(48*a*d^2+35* b*c^2)*(d*x+c)^(1/2)/c^5/e^4/(e*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.58 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=-\frac {2 x \left (15 a c^4-24 a c^3 d x+35 b c^4 x^2+48 a c^2 d^2 x^2-140 b c^3 d x^3-192 a c d^3 x^3-280 b c^2 d^2 x^4-384 a d^4 x^4\right )}{105 c^5 (e x)^{9/2} \sqrt {c+d x}} \] Input:
Integrate[(a + b*x^2)/((e*x)^(9/2)*(c + d*x)^(3/2)),x]
Output:
(-2*x*(15*a*c^4 - 24*a*c^3*d*x + 35*b*c^4*x^2 + 48*a*c^2*d^2*x^2 - 140*b*c ^3*d*x^3 - 192*a*c*d^3*x^3 - 280*b*c^2*d^2*x^4 - 384*a*d^4*x^4))/(105*c^5* (e*x)^(9/2)*Sqrt[c + d*x])
Time = 0.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {519, 27, 87, 55, 55, 48}
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.
\(\displaystyle \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 519 |
\(\displaystyle \frac {2 \left (a+\frac {b c^2}{d^2}\right )}{c e (e x)^{7/2} \sqrt {c+d x}}-\frac {2 \int -\frac {\left (\frac {7 b c^2}{d^2}+8 a\right ) d+b c x}{2 d (e x)^{9/2} \sqrt {c+d x}}dx}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\frac {7 b c^2}{d}+b x c+8 a d}{(e x)^{9/2} \sqrt {c+d x}}dx}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{c e (e x)^{7/2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {-\frac {\left (48 a d^2+35 b c^2\right ) \int \frac {1}{(e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {2 \sqrt {c+d x} \left (8 a d+\frac {7 b c^2}{d}\right )}{7 c e (e x)^{7/2}}}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{c e (e x)^{7/2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {-\frac {\left (48 a d^2+35 b c^2\right ) \left (-\frac {4 d \int \frac {1}{(e x)^{5/2} \sqrt {c+d x}}dx}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c+d x} \left (8 a d+\frac {7 b c^2}{d}\right )}{7 c e (e x)^{7/2}}}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{c e (e x)^{7/2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {-\frac {\left (48 a d^2+35 b c^2\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {1}{(e x)^{3/2} \sqrt {c+d x}}dx}{3 c e}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c+d x} \left (8 a d+\frac {7 b c^2}{d}\right )}{7 c e (e x)^{7/2}}}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{c e (e x)^{7/2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {-\frac {\left (48 a d^2+35 b c^2\right ) \left (-\frac {4 d \left (\frac {4 d \sqrt {c+d x}}{3 c^2 e^2 \sqrt {e x}}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c+d x} \left (8 a d+\frac {7 b c^2}{d}\right )}{7 c e (e x)^{7/2}}}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{c e (e x)^{7/2} \sqrt {c+d x}}\) |
Input:
Int[(a + b*x^2)/((e*x)^(9/2)*(c + d*x)^(3/2)),x]
Output:
(2*(a + (b*c^2)/d^2))/(c*e*(e*x)^(7/2)*Sqrt[c + d*x]) + ((-2*((7*b*c^2)/d + 8*a*d)*Sqrt[c + d*x])/(7*c*e*(e*x)^(7/2)) - ((35*b*c^2 + 48*a*d^2)*((-2* Sqrt[c + d*x])/(5*c*e*(e*x)^(5/2)) - (4*d*((-2*Sqrt[c + d*x])/(3*c*e*(e*x) ^(3/2)) + (4*d*Sqrt[c + d*x])/(3*c^2*e^2*Sqrt[e*x])))/(5*c*e)))/(7*c*e))/( c*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, c + d*x, x], R = PolynomialRemainder[(a + b*x^2)^p, c + d*x, x]}, Simp[(-R)*(e*x)^(m + 1)*( (c + d*x)^(n + 1)/(c*e*(n + 1))), x] + Simp[1/(c*(n + 1)) Int[(e*x)^m*(c + d*x)^(n + 1)*ExpandToSum[c*(n + 1)*Qx + R*(m + n + 2), x], x], x]] /; Fre eQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && LtQ[n, -1] && !IntegerQ[m]
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {2 x \left (-384 a \,d^{4} x^{4}-280 b \,c^{2} d^{2} x^{4}-192 a c \,d^{3} x^{3}-140 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+35 b \,c^{4} x^{2}-24 a \,c^{3} d x +15 a \,c^{4}\right )}{105 \sqrt {d x +c}\, c^{5} \left (e x \right )^{\frac {9}{2}}}\) | \(96\) |
orering | \(-\frac {2 x \left (-384 a \,d^{4} x^{4}-280 b \,c^{2} d^{2} x^{4}-192 a c \,d^{3} x^{3}-140 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+35 b \,c^{4} x^{2}-24 a \,c^{3} d x +15 a \,c^{4}\right )}{105 \sqrt {d x +c}\, c^{5} \left (e x \right )^{\frac {9}{2}}}\) | \(96\) |
default | \(-\frac {2 \left (-384 a \,d^{4} x^{4}-280 b \,c^{2} d^{2} x^{4}-192 a c \,d^{3} x^{3}-140 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+35 b \,c^{4} x^{2}-24 a \,c^{3} d x +15 a \,c^{4}\right )}{105 \sqrt {e x}\, e^{4} \sqrt {d x +c}\, c^{5} x^{3}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {d x +c}\, \left (-279 a \,x^{3} d^{3}-175 b \,c^{2} d \,x^{3}+87 a \,d^{2} x^{2} c +35 b \,c^{3} x^{2}-39 a d x \,c^{2}+15 c^{3} a \right )}{105 c^{5} x^{3} e^{4} \sqrt {e x}}+\frac {2 \left (a \,d^{2}+b \,c^{2}\right ) d^{2} x}{c^{5} e^{4} \sqrt {e x}\, \sqrt {d x +c}}\) | \(113\) |
Input:
int((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/105*x*(-384*a*d^4*x^4-280*b*c^2*d^2*x^4-192*a*c*d^3*x^3-140*b*c^3*d*x^3 +48*a*c^2*d^2*x^2+35*b*c^4*x^2-24*a*c^3*d*x+15*a*c^4)/(d*x+c)^(1/2)/c^5/(e *x)^(9/2)
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (24 \, a c^{3} d x - 15 \, a c^{4} + 8 \, {\left (35 \, b c^{2} d^{2} + 48 \, a d^{4}\right )} x^{4} + 4 \, {\left (35 \, b c^{3} d + 48 \, a c d^{3}\right )} x^{3} - {\left (35 \, b c^{4} + 48 \, a c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{105 \, {\left (c^{5} d e^{5} x^{5} + c^{6} e^{5} x^{4}\right )}} \] Input:
integrate((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(3/2),x, algorithm="fricas")
Output:
2/105*(24*a*c^3*d*x - 15*a*c^4 + 8*(35*b*c^2*d^2 + 48*a*d^4)*x^4 + 4*(35*b *c^3*d + 48*a*c*d^3)*x^3 - (35*b*c^4 + 48*a*c^2*d^2)*x^2)*sqrt(d*x + c)*sq rt(e*x)/(c^5*d*e^5*x^5 + c^6*e^5*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 1144 vs. \(2 (168) = 336\).
Time = 110.56 (sec) , antiderivative size = 1144, normalized size of antiderivative = 6.57 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x**2+a)/(e*x)**(9/2)/(d*x+c)**(3/2),x)
Output:
a*(-10*c**7*d**(33/2)*sqrt(c/(d*x) + 1)/(35*c**9*d**16*e**(9/2)*x**3 + 140 *c**8*d**17*e**(9/2)*x**4 + 210*c**7*d**18*e**(9/2)*x**5 + 140*c**6*d**19* e**(9/2)*x**6 + 35*c**5*d**20*e**(9/2)*x**7) - 14*c**6*d**(35/2)*x*sqrt(c/ (d*x) + 1)/(35*c**9*d**16*e**(9/2)*x**3 + 140*c**8*d**17*e**(9/2)*x**4 + 2 10*c**7*d**18*e**(9/2)*x**5 + 140*c**6*d**19*e**(9/2)*x**6 + 35*c**5*d**20 *e**(9/2)*x**7) - 14*c**5*d**(37/2)*x**2*sqrt(c/(d*x) + 1)/(35*c**9*d**16* e**(9/2)*x**3 + 140*c**8*d**17*e**(9/2)*x**4 + 210*c**7*d**18*e**(9/2)*x** 5 + 140*c**6*d**19*e**(9/2)*x**6 + 35*c**5*d**20*e**(9/2)*x**7) + 70*c**4* d**(39/2)*x**3*sqrt(c/(d*x) + 1)/(35*c**9*d**16*e**(9/2)*x**3 + 140*c**8*d **17*e**(9/2)*x**4 + 210*c**7*d**18*e**(9/2)*x**5 + 140*c**6*d**19*e**(9/2 )*x**6 + 35*c**5*d**20*e**(9/2)*x**7) + 560*c**3*d**(41/2)*x**4*sqrt(c/(d* x) + 1)/(35*c**9*d**16*e**(9/2)*x**3 + 140*c**8*d**17*e**(9/2)*x**4 + 210* c**7*d**18*e**(9/2)*x**5 + 140*c**6*d**19*e**(9/2)*x**6 + 35*c**5*d**20*e* *(9/2)*x**7) + 1120*c**2*d**(43/2)*x**5*sqrt(c/(d*x) + 1)/(35*c**9*d**16*e **(9/2)*x**3 + 140*c**8*d**17*e**(9/2)*x**4 + 210*c**7*d**18*e**(9/2)*x**5 + 140*c**6*d**19*e**(9/2)*x**6 + 35*c**5*d**20*e**(9/2)*x**7) + 896*c*d** (45/2)*x**6*sqrt(c/(d*x) + 1)/(35*c**9*d**16*e**(9/2)*x**3 + 140*c**8*d**1 7*e**(9/2)*x**4 + 210*c**7*d**18*e**(9/2)*x**5 + 140*c**6*d**19*e**(9/2)*x **6 + 35*c**5*d**20*e**(9/2)*x**7) + 256*d**(47/2)*x**7*sqrt(c/(d*x) + 1)/ (35*c**9*d**16*e**(9/2)*x**3 + 140*c**8*d**17*e**(9/2)*x**4 + 210*c**7*...
Exception generated. \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(3/2),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. 364 vs. \(2 (144) = 288\).
Time = 0.20 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.09 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {{\left (175 \, b c^{11} d^{8} e^{3} {\left | d \right |} + 279 \, a c^{9} d^{10} e^{3} {\left | d \right |}\right )} {\left (d x + c\right )}}{c^{14} d^{4} e^{4}} - \frac {28 \, {\left (20 \, b c^{12} d^{8} e^{3} {\left | d \right |} + 33 \, a c^{10} d^{10} e^{3} {\left | d \right |}\right )}}{c^{14} d^{4} e^{4}}\right )} + \frac {35 \, {\left (17 \, b c^{13} d^{8} e^{3} {\left | d \right |} + 30 \, a c^{11} d^{10} e^{3} {\left | d \right |}\right )}}{c^{14} d^{4} e^{4}}\right )} - \frac {210 \, {\left (b c^{14} d^{8} e^{3} {\left | d \right |} + 2 \, a c^{12} d^{10} e^{3} {\left | d \right |}\right )}}{c^{14} d^{4} e^{4}}\right )} \sqrt {d x + c}}{105 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {7}{2}}} + \frac {4 \, {\left (b^{2} c^{4} d^{7} + 2 \, a b c^{2} d^{9} + a^{2} d^{11}\right )}}{{\left (\sqrt {d e} b c^{3} d^{4} e + \sqrt {d e} a c d^{6} e + \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} b c^{2} d^{3} + \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} a d^{5}\right )} c^{4} e^{3} {\left | d \right |}} \] Input:
integrate((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(3/2),x, algorithm="giac")
Output:
2/105*((d*x + c)*((d*x + c)*((175*b*c^11*d^8*e^3*abs(d) + 279*a*c^9*d^10*e ^3*abs(d))*(d*x + c)/(c^14*d^4*e^4) - 28*(20*b*c^12*d^8*e^3*abs(d) + 33*a* c^10*d^10*e^3*abs(d))/(c^14*d^4*e^4)) + 35*(17*b*c^13*d^8*e^3*abs(d) + 30* a*c^11*d^10*e^3*abs(d))/(c^14*d^4*e^4)) - 210*(b*c^14*d^8*e^3*abs(d) + 2*a *c^12*d^10*e^3*abs(d))/(c^14*d^4*e^4))*sqrt(d*x + c)/((d*x + c)*d*e - c*d* e)^(7/2) + 4*(b^2*c^4*d^7 + 2*a*b*c^2*d^9 + a^2*d^11)/((sqrt(d*e)*b*c^3*d^ 4*e + sqrt(d*e)*a*c*d^6*e + sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*b*c^2*d^3 + sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqr t((d*x + c)*d*e - c*d*e))^2*a*d^5)*c^4*e^3*abs(d))
Time = 9.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.83 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {16\,a\,x}{35\,c^2\,e^4}-\frac {2\,a}{7\,c\,d\,e^4}-\frac {x^2\,\left (70\,b\,c^4+96\,a\,c^2\,d^2\right )}{105\,c^5\,d\,e^4}+\frac {x^4\,\left (560\,b\,c^2\,d^2+768\,a\,d^4\right )}{105\,c^5\,d\,e^4}+\frac {x^3\,\left (280\,b\,c^3\,d+384\,a\,c\,d^3\right )}{105\,c^5\,d\,e^4}\right )}{x^4\,\sqrt {e\,x}+\frac {c\,x^3\,\sqrt {e\,x}}{d}} \] Input:
int((a + b*x^2)/((e*x)^(9/2)*(c + d*x)^(3/2)),x)
Output:
((c + d*x)^(1/2)*((16*a*x)/(35*c^2*e^4) - (2*a)/(7*c*d*e^4) - (x^2*(70*b*c ^4 + 96*a*c^2*d^2))/(105*c^5*d*e^4) + (x^4*(768*a*d^4 + 560*b*c^2*d^2))/(1 05*c^5*d*e^4) + (x^3*(384*a*c*d^3 + 280*b*c^3*d))/(105*c^5*d*e^4)))/(x^4*( e*x)^(1/2) + (c*x^3*(e*x)^(1/2))/d)
Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^2}{(e x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {e}\, \left (-384 \sqrt {d}\, \sqrt {d x +c}\, a \,d^{3} x^{4}-280 \sqrt {d}\, \sqrt {d x +c}\, b \,c^{2} d \,x^{4}-15 \sqrt {x}\, a \,c^{4}+24 \sqrt {x}\, a \,c^{3} d x -48 \sqrt {x}\, a \,c^{2} d^{2} x^{2}+192 \sqrt {x}\, a c \,d^{3} x^{3}+384 \sqrt {x}\, a \,d^{4} x^{4}-35 \sqrt {x}\, b \,c^{4} x^{2}+140 \sqrt {x}\, b \,c^{3} d \,x^{3}+280 \sqrt {x}\, b \,c^{2} d^{2} x^{4}\right )}{105 \sqrt {d x +c}\, c^{5} e^{5} x^{4}} \] Input:
int((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(3/2),x)
Output:
(2*sqrt(e)*( - 384*sqrt(d)*sqrt(c + d*x)*a*d**3*x**4 - 280*sqrt(d)*sqrt(c + d*x)*b*c**2*d*x**4 - 15*sqrt(x)*a*c**4 + 24*sqrt(x)*a*c**3*d*x - 48*sqrt (x)*a*c**2*d**2*x**2 + 192*sqrt(x)*a*c*d**3*x**3 + 384*sqrt(x)*a*d**4*x**4 - 35*sqrt(x)*b*c**4*x**2 + 140*sqrt(x)*b*c**3*d*x**3 + 280*sqrt(x)*b*c**2 *d**2*x**4))/(105*sqrt(c + d*x)*c**5*e**5*x**4)