### 3.370 $$\int \frac{(d+e x)^{7/2}}{(b x+c x^2)^2} \, dx$$

Optimal. Leaf size=200 $\frac{e \sqrt{d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}+\frac{d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac{e (d+e x)^{3/2} (2 c d-b e)}{b^2 c}$

[Out]

(e*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2)*Sqrt[d + e*x])/(b^2*c^2) + (e*(2*c*d - b*e)*(d + e*x)^(3/2))/(b^2*c) -
((d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)) + (d^(5/2)*(4*c*d - 7*b*e)*ArcTanh[Sqrt[d + e*x]
/Sqrt[d]])/b^3 - ((c*d - b*e)^(5/2)*(4*c*d + 3*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(
5/2))

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Rubi [A]  time = 0.395153, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.238, Rules used = {738, 824, 826, 1166, 208} $\frac{e \sqrt{d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}+\frac{d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac{e (d+e x)^{3/2} (2 c d-b e)}{b^2 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2)*Sqrt[d + e*x])/(b^2*c^2) + (e*(2*c*d - b*e)*(d + e*x)^(3/2))/(b^2*c) -
((d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)) + (d^(5/2)*(4*c*d - 7*b*e)*ArcTanh[Sqrt[d + e*x]
/Sqrt[d]])/b^3 - ((c*d - b*e)^(5/2)*(4*c*d + 3*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(
5/2))

Rule 738

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 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 0]

Rule 826

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 1166

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]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.311376, size = 167, normalized size = 0.84 $\frac{\frac{b \sqrt{d+e x} \left (b^2 c e^2 x (2 e x-3 d)+3 b^3 e^3 x-b c^2 d^2 (d-3 e x)-2 c^3 d^3 x\right )}{c^2 x (b+c x)}-\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{5/2}}+d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b*Sqrt[d + e*x]*(-2*c^3*d^3*x + 3*b^3*e^3*x - b*c^2*d^2*(d - 3*e*x) + b^2*c*e^2*x*(-3*d + 2*e*x)))/(c^2*x*(b
+ c*x)) + d^(5/2)*(4*c*d - 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - ((c*d - b*e)^(5/2)*(4*c*d + 3*b*e)*ArcTanh
[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(5/2))/b^3

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Maple [B]  time = 0.229, size = 403, normalized size = 2. \begin{align*} 2\,{\frac{{e}^{3}\sqrt{ex+d}}{{c}^{2}}}+{\frac{{e}^{4}b}{{c}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{{e}^{3}\sqrt{ex+d}d}{c \left ( cex+be \right ) }}+3\,{\frac{{e}^{2}\sqrt{ex+d}{d}^{2}}{b \left ( cex+be \right ) }}-{\frac{ce{d}^{3}}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{{e}^{4}b}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+5\,{\frac{{e}^{3}d}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+3\,{\frac{{d}^{2}{e}^{2}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-9\,{\frac{ce{d}^{3}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{4}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{{d}^{3}}{{b}^{2}x}\sqrt{ex+d}}-7\,{\frac{e{d}^{5/2}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{7/2}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3/c^2*(e*x+d)^(1/2)+e^4/c^2*b*(e*x+d)^(1/2)/(c*e*x+b*e)-3*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*d+3*e^2/b*(e*x+d
)^(1/2)/(c*e*x+b*e)*d^2-e*c/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*d^3-3*e^4/c^2*b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(
1/2)*c/((b*e-c*d)*c)^(1/2))+5*e^3/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d+3*e^2/b/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d^2-9*e*c/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+
d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d^3+4*c^2/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*
d^4-d^3/b^2*(e*x+d)^(1/2)/x-7*e*d^(5/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))+4*d^(7/2)/b^3*arctanh((e*x+d)^(1/2)
/d^(1/2))*c

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.24772, size = 2628, normalized size = 13.14 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^
3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c)
)/(c*x + b)) - ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt
(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^
2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -1/2*(2*((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*
e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(-(c*d - b*e)/c)*a
rctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7
*b^2*c^2*d^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (
2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -1/2*(
2*((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)
/d) - ((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^
3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c)
)/(c*x + b)) - 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - 3*b^4*e^3)*
x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -(((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^
2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*
c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sq
rt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*
b^3*c*d*e^2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.38429, size = 464, normalized size = 2.32 \begin{align*} \frac{2 \, \sqrt{x e + d} e^{3}}{c^{2}} - \frac{{\left (4 \, c d^{4} - 7 \, b d^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (4 \, c^{4} d^{4} - 9 \, b c^{3} d^{3} e + 3 \, b^{2} c^{2} d^{2} e^{2} + 5 \, b^{3} c d e^{3} - 3 \, b^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c^{2}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e - 2 \, \sqrt{x e + d} c^{3} d^{4} e - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{2} + 4 \, \sqrt{x e + d} b c^{2} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{3} - 3 \, \sqrt{x e + d} b^{2} c d^{2} e^{3} -{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{4} + \sqrt{x e + d} b^{3} d e^{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 )} b^{2} c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x*e + d)*e^3/c^2 - (4*c*d^4 - 7*b*d^3*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^4*d^4 - 9
*b*c^3*d^3*e + 3*b^2*c^2*d^2*e^2 + 5*b^3*c*d*e^3 - 3*b^4*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sq
rt(-c^2*d + b*c*e)*b^3*c^2) - (2*(x*e + d)^(3/2)*c^3*d^3*e - 2*sqrt(x*e + d)*c^3*d^4*e - 3*(x*e + d)^(3/2)*b*c
^2*d^2*e^2 + 4*sqrt(x*e + d)*b*c^2*d^3*e^2 + 3*(x*e + d)^(3/2)*b^2*c*d*e^3 - 3*sqrt(x*e + d)*b^2*c*d^2*e^3 - (
x*e + d)^(3/2)*b^3*e^4 + sqrt(x*e + d)*b^3*d*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e -
b*d*e)*b^2*c^2)