3.4.79 \(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^3} \, dx\) [379]

Optimal. Leaf size=248 \[ -\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}} \]

[Out]

-1/2*(e*x+d)^(5/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^2-1/4*d^(3/2)*(35*b^2*e^2-84*b*c*d*e+48*c^2*d^2)*arcta
nh((e*x+d)^(1/2)/d^(1/2))/b^5+1/4*(-b*e+c*d)^(3/2)*(-b^2*e^2-12*b*c*d*e+48*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1
/2)/(-b*e+c*d)^(1/2))/b^5/c^(3/2)+1/4*(b*c*d^2*(-11*b*e+12*c*d)+(-b*e+2*c*d)*(b^2*e^2-12*b*c*d*e+12*c^2*d^2)*x
)*(e*x+d)^(1/2)/b^4/c/(c*x^2+b*x)

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {752, 832, 840, 1180, 214} \begin {gather*} -\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {d^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}+\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{4 b^4 c \left (b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(b*x + c*x^2)^3,x]

[Out]

-1/2*((d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b*c*d^2*(12*c*d - 11*b*
e) + (2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*x))/(4*b^4*c*(b*x + c*x^2)) - (d^(3/2)*(48*c^2*d^2 - 84
*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*e)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e -
b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(3/2))

Rule 214

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]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (12 c d-11 b e)+\frac {1}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{4} c d^2 \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} d e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right )-\frac {1}{4} c d^2 e \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 c}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\left ((c d-b e)^2 \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 c}+\frac {\left (c d^2 \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.28, size = 250, normalized size = 1.01 \begin {gather*} -\frac {-\frac {b \sqrt {d+e x} \left (-b^4 e^3 x^2+24 c^4 d^3 x^3+36 b c^3 d^2 x^2 (d-e x)+b^2 c^2 d x \left (8 d^2-55 d e x+10 e^2 x^2\right )+b^3 c \left (-2 d^3-13 d^2 e x+16 d e^2 x^2+e^3 x^3\right )\right )}{c x^2 (b+c x)^2}+\frac {(-c d+b e)^{3/2} \left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{3/2}}+d^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^3,x]

[Out]

-1/4*(-((b*Sqrt[d + e*x]*(-(b^4*e^3*x^2) + 24*c^4*d^3*x^3 + 36*b*c^3*d^2*x^2*(d - e*x) + b^2*c^2*d*x*(8*d^2 -
55*d*e*x + 10*e^2*x^2) + b^3*c*(-2*d^3 - 13*d^2*e*x + 16*d*e^2*x^2 + e^3*x^3)))/(c*x^2*(b + c*x)^2)) + ((-(c*d
) + b*e)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(3/2)
+ d^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^5

________________________________________________________________________________________

Maple [A]
time = 0.70, size = 273, normalized size = 1.10

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {d^{2} \left (\frac {\left (\frac {13}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}+\frac {\left (b e -c d \right )^{2} \left (\frac {\left (\frac {1}{8} b^{2} e^{2}+\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (b^{2} e^{2}-13 b c d e +12 d^{2} c^{2}\right ) \sqrt {e x +d}}{8 c}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (b^{2} e^{2}+12 b c d e -48 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}\right )\)	\(273\)
default	\(2 e^{5} \left (-\frac {d^{2} \left (\frac {\left (\frac {13}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 b^{2} e^{2}-84 b c d e +48 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}+\frac {\left (b e -c d \right )^{2} \left (\frac {\left (\frac {1}{8} b^{2} e^{2}+\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}-\frac {b e \left (b^{2} e^{2}-13 b c d e +12 d^{2} c^{2}\right ) \sqrt {e x +d}}{8 c}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (b^{2} e^{2}+12 b c d e -48 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}\right )\)	\(273\)
risch	\(-\frac {d^{2} \sqrt {e x +d}\, \left (13 b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2}}+\frac {e^{4} \left (e x +d \right )^{\frac {3}{2}}}{4 b \left (c e x +b e \right )^{2}}+\frac {5 e^{3} \left (e x +d \right )^{\frac {3}{2}} c d}{2 b^{2} \left (c e x +b e \right )^{2}}-\frac {23 e^{2} \left (e x +d \right )^{\frac {3}{2}} c^{2} d^{2}}{4 b^{3} \left (c e x +b e \right )^{2}}+\frac {3 e \left (e x +d \right )^{\frac {3}{2}} c^{3} d^{3}}{b^{4} \left (c e x +b e \right )^{2}}-\frac {e^{5} \sqrt {e x +d}}{4 \left (c e x +b e \right )^{2} c}+\frac {15 e^{4} \sqrt {e x +d}\, d}{4 b \left (c e x +b e \right )^{2}}-\frac {39 e^{3} c \sqrt {e x +d}\, d^{2}}{4 b^{2} \left (c e x +b e \right )^{2}}+\frac {37 e^{2} c^{2} \sqrt {e x +d}\, d^{3}}{4 b^{3} \left (c e x +b e \right )^{2}}-\frac {3 e \,c^{3} \sqrt {e x +d}\, d^{4}}{b^{4} \left (c e x +b e \right )^{2}}+\frac {e^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 b c \sqrt {\left (b e -c d \right ) c}}+\frac {5 e^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d}{2 b^{2} \sqrt {\left (b e -c d \right ) c}}-\frac {71 e^{2} c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{2}}{4 b^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {27 e \,c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{3}}{b^{4} \sqrt {\left (b e -c d \right ) c}}-\frac {12 c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{4}}{b^{5} \sqrt {\left (b e -c d \right ) c}}-\frac {35 e^{2} d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3}}+\frac {21 e \,d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{4}}-\frac {12 d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c^{2}}{b^{5}}\)	\(580\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(-d^2/b^5/e^5*(((13/8*b^2*e^2-3/2*b*c*d*e)*(e*x+d)^(3/2)+(-11/8*b^2*d*e^2+3/2*b*c*d^2*e)*(e*x+d)^(1/2))/
e^2/x^2+1/8*(35*b^2*e^2-84*b*c*d*e+48*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))+(b*e-c*d)^2/b^5/e^5*(((
1/8*b^2*e^2+3/2*b*c*d*e)*(e*x+d)^(3/2)-1/8*b*e*(b^2*e^2-13*b*c*d*e+12*c^2*d^2)/c*(e*x+d)^(1/2))/(c*(e*x+d)+b*e
-c*d)^2+1/8*(b^2*e^2+12*b*c*d*e-48*c^2*d^2)/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))
)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-%e*b>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (229) = 458\).
time = 3.26, size = 2150, normalized size = 8.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/8*((48*c^5*d^3*x^4 + 96*b*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + (b^3*c^2*x^4 + 2*b^4*c*x^3 + b^5*x^2)*e^3 + 11
*(b^2*c^3*d*x^4 + 2*b^3*c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 60*(b*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2
)*e)*sqrt((c*d - b*e)/c)*log((2*c*d + 2*sqrt(x*e + d)*c*sqrt((c*d - b*e)/c) + (c*x - b)*e)/(c*x + b)) + (48*c^
5*d^3*x^4 + 96*b*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + 35*(b^2*c^3*d*x^4 + 2*b^3*c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 8
4*(b*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x
) + 2*(24*b*c^4*d^3*x^3 + 36*b^2*c^3*d^3*x^2 + 8*b^3*c^2*d^3*x - 2*b^4*c*d^3 + (b^4*c*x^3 - b^5*x^2)*e^3 + 2*(
5*b^3*c^2*d*x^3 + 8*b^4*c*d*x^2)*e^2 - (36*b^2*c^3*d^2*x^3 + 55*b^3*c^2*d^2*x^2 + 13*b^4*c*d^2*x)*e)*sqrt(x*e
+ d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*(48*c^5*d^3*x^4 + 96*b*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^
2 + (b^3*c^2*x^4 + 2*b^4*c*x^3 + b^5*x^2)*e^3 + 11*(b^2*c^3*d*x^4 + 2*b^3*c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 60*(b
*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(x*e + d)*c*sqrt(-(c*d
- b*e)/c)/(c*d - b*e)) + (48*c^5*d^3*x^4 + 96*b*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + 35*(b^2*c^3*d*x^4 + 2*b^3*
c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 84*(b*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2)*e)*sqrt(d)*log((x*e -
2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^4*d^3*x^3 + 36*b^2*c^3*d^3*x^2 + 8*b^3*c^2*d^3*x - 2*b^4*c*d^3 +
(b^4*c*x^3 - b^5*x^2)*e^3 + 2*(5*b^3*c^2*d*x^3 + 8*b^4*c*d*x^2)*e^2 - (36*b^2*c^3*d^2*x^3 + 55*b^3*c^2*d^2*x^
2 + 13*b^4*c*d^2*x)*e)*sqrt(x*e + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*(48*c^5*d^3*x^4 + 96*b
*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + 35*(b^2*c^3*d*x^4 + 2*b^3*c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 84*(b*c^4*d^2*x^4
+ 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (48*c^5*d^3*x^4 + 96*b*
c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + (b^3*c^2*x^4 + 2*b^4*c*x^3 + b^5*x^2)*e^3 + 11*(b^2*c^3*d*x^4 + 2*b^3*c^2*d
*x^3 + b^4*c*d*x^2)*e^2 - 60*(b*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2)*e)*sqrt((c*d - b*e)/c)*log(
(2*c*d + 2*sqrt(x*e + d)*c*sqrt((c*d - b*e)/c) + (c*x - b)*e)/(c*x + b)) + 2*(24*b*c^4*d^3*x^3 + 36*b^2*c^3*d^
3*x^2 + 8*b^3*c^2*d^3*x - 2*b^4*c*d^3 + (b^4*c*x^3 - b^5*x^2)*e^3 + 2*(5*b^3*c^2*d*x^3 + 8*b^4*c*d*x^2)*e^2 -
(36*b^2*c^3*d^2*x^3 + 55*b^3*c^2*d^2*x^2 + 13*b^4*c*d^2*x)*e)*sqrt(x*e + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^
7*c*x^2), 1/4*((48*c^5*d^3*x^4 + 96*b*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + (b^3*c^2*x^4 + 2*b^4*c*x^3 + b^5*x^2)
*e^3 + 11*(b^2*c^3*d*x^4 + 2*b^3*c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 60*(b*c^4*d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^
2*d^2*x^2)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(x*e + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + (48*c^5*d^3*x^4
+ 96*b*c^4*d^3*x^3 + 48*b^2*c^3*d^3*x^2 + 35*(b^2*c^3*d*x^4 + 2*b^3*c^2*d*x^3 + b^4*c*d*x^2)*e^2 - 84*(b*c^4*
d^2*x^4 + 2*b^2*c^3*d^2*x^3 + b^3*c^2*d^2*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (24*b*c^4*d^3*x^
3 + 36*b^2*c^3*d^3*x^2 + 8*b^3*c^2*d^3*x - 2*b^4*c*d^3 + (b^4*c*x^3 - b^5*x^2)*e^3 + 2*(5*b^3*c^2*d*x^3 + 8*b^
4*c*d*x^2)*e^2 - (36*b^2*c^3*d^2*x^3 + 55*b^3*c^2*d^2*x^2 + 13*b^4*c*d^2*x)*e)*sqrt(x*e + d))/(b^5*c^3*x^4 + 2
*b^6*c^2*x^3 + b^7*c*x^2)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (229) = 458\).
time = 2.10, size = 552, normalized size = 2.23 \begin {gather*} \frac {{\left (48 \, c^{2} d^{4} - 84 \, b c d^{3} e + 35 \, b^{2} d^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} - \frac {{\left (48 \, c^{4} d^{4} - 108 \, b c^{3} d^{3} e + 71 \, b^{2} c^{2} d^{2} e^{2} - 10 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5} c} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{4} d^{3} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{4} d^{4} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{5} e - 24 \, \sqrt {x e + d} c^{4} d^{6} e - 36 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{3} d^{2} e^{2} + 144 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{3} d^{3} e^{2} - 180 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{4} e^{2} + 72 \, \sqrt {x e + d} b c^{3} d^{5} e^{2} + 10 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{2} d e^{3} - 85 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} e^{3} + 148 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{3} e^{3} - 73 \, \sqrt {x e + d} b^{2} c^{2} d^{4} e^{3} + {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c e^{4} + 13 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c d e^{4} - 42 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c d^{2} e^{4} + 26 \, \sqrt {x e + d} b^{3} c d^{3} e^{4} - {\left (x e + d\right )}^{\frac {5}{2}} b^{4} e^{5} + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d e^{5} - \sqrt {x e + d} b^{4} d^{2} e^{5}}{4 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2} b^{4} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/4*(48*c^2*d^4 - 84*b*c*d^3*e + 35*b^2*d^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) - 1/4*(48*c^4*d
^4 - 108*b*c^3*d^3*e + 71*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - b^4*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c
*e))/(sqrt(-c^2*d + b*c*e)*b^5*c) + 1/4*(24*(x*e + d)^(7/2)*c^4*d^3*e - 72*(x*e + d)^(5/2)*c^4*d^4*e + 72*(x*e
+ d)^(3/2)*c^4*d^5*e - 24*sqrt(x*e + d)*c^4*d^6*e - 36*(x*e + d)^(7/2)*b*c^3*d^2*e^2 + 144*(x*e + d)^(5/2)*b*
c^3*d^3*e^2 - 180*(x*e + d)^(3/2)*b*c^3*d^4*e^2 + 72*sqrt(x*e + d)*b*c^3*d^5*e^2 + 10*(x*e + d)^(7/2)*b^2*c^2*
d*e^3 - 85*(x*e + d)^(5/2)*b^2*c^2*d^2*e^3 + 148*(x*e + d)^(3/2)*b^2*c^2*d^3*e^3 - 73*sqrt(x*e + d)*b^2*c^2*d^
4*e^3 + (x*e + d)^(7/2)*b^3*c*e^4 + 13*(x*e + d)^(5/2)*b^3*c*d*e^4 - 42*(x*e + d)^(3/2)*b^3*c*d^2*e^4 + 26*sqr
t(x*e + d)*b^3*c*d^3*e^4 - (x*e + d)^(5/2)*b^4*e^5 + 2*(x*e + d)^(3/2)*b^4*d*e^5 - sqrt(x*e + d)*b^4*d^2*e^5)/
(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c)

________________________________________________________________________________________

Mupad [B]
time = 0.73, size = 1792, normalized size = 7.23 \begin {gather*} -\frac {\frac {\sqrt {d+e\,x}\,\left (b^4\,d^2\,e^5-26\,b^3\,c\,d^3\,e^4+73\,b^2\,c^2\,d^4\,e^3-72\,b\,c^3\,d^5\,e^2+24\,c^4\,d^6\,e\right )}{4\,b^4\,c}-\frac {e\,{\left (d+e\,x\right )}^{7/2}\,\left (b^3\,e^3+10\,b^2\,c\,d\,e^2-36\,b\,c^2\,d^2\,e+24\,c^3\,d^3\right )}{4\,b^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (b^4\,d\,e^5-21\,b^3\,c\,d^2\,e^4+74\,b^2\,c^2\,d^3\,e^3-90\,b\,c^3\,d^4\,e^2+36\,c^4\,d^5\,e\right )}{2\,b^4\,c}+\frac {e\,{\left (d+e\,x\right )}^{5/2}\,\left (b^4\,e^4-13\,b^3\,c\,d\,e^3+85\,b^2\,c^2\,d^2\,e^2-144\,b\,c^3\,d^3\,e+72\,c^4\,d^4\right )}{4\,b^4\,c}}{c^2\,{\left (d+e\,x\right )}^4-\left (d+e\,x\right )\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )-\left (4\,c^2\,d-2\,b\,c\,e\right )\,{\left (d+e\,x\right )}^3+{\left (d+e\,x\right )}^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )+c^2\,d^4+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e}-\frac {\mathrm {atanh}\left (\frac {35\,e^{12}\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,\left (\frac {35\,d^2\,e^{12}}{32}+\frac {77\,c\,d^3\,e^{11}}{4\,b}-\frac {1551\,c^2\,d^4\,e^{10}}{16\,b^2}+\frac {5223\,c^3\,d^5\,e^9}{32\,b^3}-\frac {945\,c^4\,d^6\,e^8}{8\,b^4}+\frac {63\,c^5\,d^7\,e^7}{2\,b^5}\right )}+\frac {77\,d\,e^{11}\,\sqrt {d^3}\,\sqrt {d+e\,x}}{4\,\left (\frac {77\,d^3\,e^{11}}{4}+\frac {35\,b\,d^2\,e^{12}}{32\,c}-\frac {1551\,c\,d^4\,e^{10}}{16\,b}+\frac {5223\,c^2\,d^5\,e^9}{32\,b^2}-\frac {945\,c^3\,d^6\,e^8}{8\,b^3}+\frac {63\,c^4\,d^7\,e^7}{2\,b^4}\right )}+\frac {5223\,c^2\,d^3\,e^9\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,\left (\frac {77\,b^2\,d^3\,e^{11}}{4}+\frac {5223\,c^2\,d^5\,e^9}{32}-\frac {945\,c^3\,d^6\,e^8}{8\,b}+\frac {35\,b^3\,d^2\,e^{12}}{32\,c}+\frac {63\,c^4\,d^7\,e^7}{2\,b^2}-\frac {1551\,b\,c\,d^4\,e^{10}}{16}\right )}-\frac {945\,c^3\,d^4\,e^8\,\sqrt {d^3}\,\sqrt {d+e\,x}}{8\,\left (\frac {77\,b^3\,d^3\,e^{11}}{4}-\frac {945\,c^3\,d^6\,e^8}{8}+\frac {5223\,b\,c^2\,d^5\,e^9}{32}-\frac {1551\,b^2\,c\,d^4\,e^{10}}{16}+\frac {63\,c^4\,d^7\,e^7}{2\,b}+\frac {35\,b^4\,d^2\,e^{12}}{32\,c}\right )}+\frac {63\,c^4\,d^5\,e^7\,\sqrt {d^3}\,\sqrt {d+e\,x}}{2\,\left (\frac {77\,b^4\,d^3\,e^{11}}{4}+\frac {63\,c^4\,d^7\,e^7}{2}-\frac {945\,b\,c^3\,d^6\,e^8}{8}-\frac {1551\,b^3\,c\,d^4\,e^{10}}{16}+\frac {5223\,b^2\,c^2\,d^5\,e^9}{32}+\frac {35\,b^5\,d^2\,e^{12}}{32\,c}\right )}-\frac {1551\,c\,d^2\,e^{10}\,\sqrt {d^3}\,\sqrt {d+e\,x}}{16\,\left (\frac {77\,b\,d^3\,e^{11}}{4}-\frac {1551\,c\,d^4\,e^{10}}{16}+\frac {5223\,c^2\,d^5\,e^9}{32\,b}+\frac {35\,b^2\,d^2\,e^{12}}{32\,c}-\frac {945\,c^3\,d^6\,e^8}{8\,b^2}+\frac {63\,c^4\,d^7\,e^7}{2\,b^3}\right )}\right )\,\sqrt {d^3}\,\left (35\,b^2\,e^2-84\,b\,c\,d\,e+48\,c^2\,d^2\right )}{4\,b^5}-\frac {\mathrm {atanh}\left (\frac {183\,d^3\,e^9\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{32\,\left (\frac {31\,b^3\,d^2\,e^{12}}{32}+\frac {3711\,c^3\,d^5\,e^9}{32}-\frac {1593\,b\,c^2\,d^4\,e^{10}}{32}+\frac {59\,b^2\,c\,d^3\,e^{11}}{16}+\frac {b^4\,d\,e^{13}}{32\,c}-\frac {819\,c^4\,d^6\,e^8}{8\,b}+\frac {63\,c^5\,d^7\,e^7}{2\,b^2}\right )}-\frac {315\,d^4\,e^8\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{8\,\left (\frac {59\,b^3\,d^3\,e^{11}}{16}-\frac {819\,c^3\,d^6\,e^8}{8}+\frac {3711\,b\,c^2\,d^5\,e^9}{32}-\frac {1593\,b^2\,c\,d^4\,e^{10}}{32}+\frac {b^5\,d\,e^{13}}{32\,c^2}+\frac {63\,c^4\,d^7\,e^7}{2\,b}+\frac {31\,b^4\,d^2\,e^{12}}{32\,c}\right )}+\frac {33\,d^2\,e^{10}\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{32\,\left (\frac {b^3\,d\,e^{13}}{32}-\frac {1593\,c^3\,d^4\,e^{10}}{32}+\frac {59\,b\,c^2\,d^3\,e^{11}}{16}+\frac {31\,b^2\,c\,d^2\,e^{12}}{32}+\frac {3711\,c^4\,d^5\,e^9}{32\,b}-\frac {819\,c^5\,d^6\,e^8}{8\,b^2}+\frac {63\,c^6\,d^7\,e^7}{2\,b^3}\right )}+\frac {d\,e^{11}\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{32\,\left (\frac {59\,c^3\,d^3\,e^{11}}{16}+\frac {31\,b\,c^2\,d^2\,e^{12}}{32}-\frac {1593\,c^4\,d^4\,e^{10}}{32\,b}+\frac {3711\,c^5\,d^5\,e^9}{32\,b^2}-\frac {819\,c^6\,d^6\,e^8}{8\,b^3}+\frac {63\,c^7\,d^7\,e^7}{2\,b^4}+\frac {b^2\,c\,d\,e^{13}}{32}\right )}+\frac {63\,c\,d^5\,e^7\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{2\,\left (\frac {59\,b^4\,d^3\,e^{11}}{16}+\frac {63\,c^4\,d^7\,e^7}{2}-\frac {819\,b\,c^3\,d^6\,e^8}{8}-\frac {1593\,b^3\,c\,d^4\,e^{10}}{32}+\frac {b^6\,d\,e^{13}}{32\,c^2}+\frac {3711\,b^2\,c^2\,d^5\,e^9}{32}+\frac {31\,b^5\,d^2\,e^{12}}{32\,c}\right )}\right )\,\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (b^2\,e^2+12\,b\,c\,d\,e-48\,c^2\,d^2\right )}{4\,b^5\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(b*x + c*x^2)^3,x)

[Out]

- (((d + e*x)^(1/2)*(24*c^4*d^6*e + b^4*d^2*e^5 - 72*b*c^3*d^5*e^2 - 26*b^3*c*d^3*e^4 + 73*b^2*c^2*d^4*e^3))/(
4*b^4*c) - (e*(d + e*x)^(7/2)*(b^3*e^3 + 24*c^3*d^3 - 36*b*c^2*d^2*e + 10*b^2*c*d*e^2))/(4*b^4) - ((d + e*x)^(
3/2)*(b^4*d*e^5 + 36*c^4*d^5*e - 90*b*c^3*d^4*e^2 - 21*b^3*c*d^2*e^4 + 74*b^2*c^2*d^3*e^3))/(2*b^4*c) + (e*(d
+ e*x)^(5/2)*(b^4*e^4 + 72*c^4*d^4 + 85*b^2*c^2*d^2*e^2 - 144*b*c^3*d^3*e - 13*b^3*c*d*e^3))/(4*b^4*c))/(c^2*(
d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2
*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atanh((35*e^12*(d^3)^(1/2)*(d + e
*x)^(1/2))/(32*((35*d^2*e^12)/32 + (77*c*d^3*e^11)/(4*b) - (1551*c^2*d^4*e^10)/(16*b^2) + (5223*c^3*d^5*e^9)/(
32*b^3) - (945*c^4*d^6*e^8)/(8*b^4) + (63*c^5*d^7*e^7)/(2*b^5))) + (77*d*e^11*(d^3)^(1/2)*(d + e*x)^(1/2))/(4*
((77*d^3*e^11)/4 + (35*b*d^2*e^12)/(32*c) - (1551*c*d^4*e^10)/(16*b) + (5223*c^2*d^5*e^9)/(32*b^2) - (945*c^3*
d^6*e^8)/(8*b^3) + (63*c^4*d^7*e^7)/(2*b^4))) + (5223*c^2*d^3*e^9*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*((77*b^2*d^
3*e^11)/4 + (5223*c^2*d^5*e^9)/32 - (945*c^3*d^6*e^8)/(8*b) + (35*b^3*d^2*e^12)/(32*c) + (63*c^4*d^7*e^7)/(2*b
^2) - (1551*b*c*d^4*e^10)/16)) - (945*c^3*d^4*e^8*(d^3)^(1/2)*(d + e*x)^(1/2))/(8*((77*b^3*d^3*e^11)/4 - (945*
c^3*d^6*e^8)/8 + (5223*b*c^2*d^5*e^9)/32 - (1551*b^2*c*d^4*e^10)/16 + (63*c^4*d^7*e^7)/(2*b) + (35*b^4*d^2*e^1
2)/(32*c))) + (63*c^4*d^5*e^7*(d^3)^(1/2)*(d + e*x)^(1/2))/(2*((77*b^4*d^3*e^11)/4 + (63*c^4*d^7*e^7)/2 - (945
*b*c^3*d^6*e^8)/8 - (1551*b^3*c*d^4*e^10)/16 + (5223*b^2*c^2*d^5*e^9)/32 + (35*b^5*d^2*e^12)/(32*c))) - (1551*
c*d^2*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(16*((77*b*d^3*e^11)/4 - (1551*c*d^4*e^10)/16 + (5223*c^2*d^5*e^9)/(32
*b) + (35*b^2*d^2*e^12)/(32*c) - (945*c^3*d^6*e^8)/(8*b^2) + (63*c^4*d^7*e^7)/(2*b^3))))*(d^3)^(1/2)*(35*b^2*e
^2 + 48*c^2*d^2 - 84*b*c*d*e))/(4*b^5) - (atanh((183*d^3*e^9*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^
4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(32*((31*b^3*d^2*e^12)/32 + (3711*c^3*d^5*e^9)/32 - (1593*b*c^2*d^4*e^10)/32 +
(59*b^2*c*d^3*e^11)/16 + (b^4*d*e^13)/(32*c) - (819*c^4*d^6*e^8)/(8*b) + (63*c^5*d^7*e^7)/(2*b^2))) - (315*d^
4*e^8*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(8*((59*b^3*d^3*e^11)/1
6 - (819*c^3*d^6*e^8)/8 + (3711*b*c^2*d^5*e^9)/32 - (1593*b^2*c*d^4*e^10)/32 + (b^5*d*e^13)/(32*c^2) + (63*c^4
*d^7*e^7)/(2*b) + (31*b^4*d^2*e^12)/(32*c))) + (33*d^2*e^10*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4
*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(32*((b^3*d*e^13)/32 - (1593*c^3*d^4*e^10)/32 + (59*b*c^2*d^3*e^11)/16 + (31*b^
2*c*d^2*e^12)/32 + (3711*c^4*d^5*e^9)/(32*b) - (819*c^5*d^6*e^8)/(8*b^2) + (63*c^6*d^7*e^7)/(2*b^3))) + (d*e^1
1*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(32*((59*c^3*d^3*e^11)/16 +
(31*b*c^2*d^2*e^12)/32 - (1593*c^4*d^4*e^10)/(32*b) + (3711*c^5*d^5*e^9)/(32*b^2) - (819*c^6*d^6*e^8)/(8*b^3)
+ (63*c^7*d^7*e^7)/(2*b^4) + (b^2*c*d*e^13)/32)) + (63*c*d^5*e^7*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b
^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(2*((59*b^4*d^3*e^11)/16 + (63*c^4*d^7*e^7)/2 - (819*b*c^3*d^6*e^8)/8 - (
1593*b^3*c*d^4*e^10)/32 + (b^6*d*e^13)/(32*c^2) + (3711*b^2*c^2*d^5*e^9)/32 + (31*b^5*d^2*e^12)/(32*c))))*(-c^
3*(b*e - c*d)^3)^(1/2)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(4*b^5*c^3)

________________________________________________________________________________________