3.769 \(\int \frac{x (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-7 a d)}{3 b^2 (b c-a d)}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-7 a d)}{12 b^3}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-7 a d) (b c-a d)}{8 b^4}+\frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} \sqrt{d}}+\frac{2 a (c+d x)^{7/2}}{b \sqrt{a+b x} (b c-a d)} \]

[Out]

(5*(b*c - 7*a*d)*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^4) + (5*(b*c - 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(
3/2))/(12*b^3) + ((b*c - 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)) + (2*a*(c + d*x)^(7/2))/(b*
(b*c - a*d)*Sqrt[a + b*x]) + (5*(b*c - 7*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(8*b^(9/2)*Sqrt[d])

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Rubi [A]  time = 0.115045, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 50, 63, 217, 206} \[ \frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-7 a d)}{3 b^2 (b c-a d)}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-7 a d)}{12 b^3}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-7 a d) (b c-a d)}{8 b^4}+\frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} \sqrt{d}}+\frac{2 a (c+d x)^{7/2}}{b \sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

(5*(b*c - 7*a*d)*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^4) + (5*(b*c - 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(
3/2))/(12*b^3) + ((b*c - 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)) + (2*a*(c + d*x)^(7/2))/(b*
(b*c - a*d)*Sqrt[a + b*x]) + (5*(b*c - 7*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(8*b^(9/2)*Sqrt[d])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 50

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] + Dist[(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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx &=\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{(b c-7 a d) \int \frac{(c+d x)^{5/2}}{\sqrt{a+b x}} \, dx}{b (b c-a d)}\\ &=\frac{(b c-7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{(5 (b c-7 a d)) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{6 b^2}\\ &=\frac{5 (b c-7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3}+\frac{(b c-7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{(5 (b c-7 a d) (b c-a d)) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{8 b^3}\\ &=\frac{5 (b c-7 a d) (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4}+\frac{5 (b c-7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3}+\frac{(b c-7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{\left (5 (b c-7 a d) (b c-a d)^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b^4}\\ &=\frac{5 (b c-7 a d) (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4}+\frac{5 (b c-7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3}+\frac{(b c-7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{\left (5 (b c-7 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b^5}\\ &=\frac{5 (b c-7 a d) (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4}+\frac{5 (b c-7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3}+\frac{(b c-7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{\left (5 (b c-7 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 b^5}\\ &=\frac{5 (b c-7 a d) (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4}+\frac{5 (b c-7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3}+\frac{(b c-7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt{a+b x}}+\frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.637361, size = 175, normalized size = 0.82 \[ \frac{\sqrt{c+d x} \left (\frac{5 a^2 b d (7 d x-38 c)+105 a^3 d^2+a b^2 \left (81 c^2-68 c d x-14 d^2 x^2\right )+b^3 x \left (33 c^2+26 c d x+8 d^2 x^2\right )}{\sqrt{a+b x}}+\frac{15 (b c-7 a d) (b c-a d)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{d} \sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{24 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(c + d*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

(Sqrt[c + d*x]*((105*a^3*d^2 + 5*a^2*b*d*(-38*c + 7*d*x) + a*b^2*(81*c^2 - 68*c*d*x - 14*d^2*x^2) + b^3*x*(33*
c^2 + 26*c*d*x + 8*d^2*x^2))/Sqrt[a + b*x] + (15*(b*c - 7*a*d)*(b*c - a*d)^(3/2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x
])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(24*b^4)

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Maple [B]  time = 0.023, size = 689, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^(5/2)/(b*x+a)^(3/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(-16*x^3*b^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^3*b*d^3-225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b^2*c*d^2+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b
*d)^(1/2))*x*a*b^3*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^4*
c^3+28*x^2*a*b^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-52*x^2*b^3*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+10
5*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^3-225*ln(1/2*(2*b*d*x+2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c*d^2+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*a*b^3*c^3-70*x*a^2*b*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+136*x*a*b^2*c*d*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)-66*x*b^3*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-210*a^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+380*a^2*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-162*a*b^2*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(1/2)/b^4

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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(x*(d*x+c)^(5/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.60166, size = 1319, normalized size = 6.16 \begin{align*} \left [-\frac{15 \,{\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b d^{3}\right )} x\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (8 \, b^{4} d^{3} x^{3} + 81 \, a b^{3} c^{2} d - 190 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 2 \,{\left (13 \, b^{4} c d^{2} - 7 \, a b^{3} d^{3}\right )} x^{2} +{\left (33 \, b^{4} c^{2} d - 68 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (b^{6} d x + a b^{5} d\right )}}, -\frac{15 \,{\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b d^{3}\right )} x\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \,{\left (8 \, b^{4} d^{3} x^{3} + 81 \, a b^{3} c^{2} d - 190 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 2 \,{\left (13 \, b^{4} c d^{2} - 7 \, a b^{3} d^{3}\right )} x^{2} +{\left (33 \, b^{4} c^{2} d - 68 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (b^{6} d x + a b^{5} d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)^(5/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + (b^4*c^3 - 9*a*b^3*c^2*d + 15*a^2*b^2*c
*d^2 - 7*a^3*b*d^3)*x)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*s
qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(8*b^4*d^3*x^3 + 81*a*b^3*c^2*d - 190*a^2*
b^2*c*d^2 + 105*a^3*b*d^3 + 2*(13*b^4*c*d^2 - 7*a*b^3*d^3)*x^2 + (33*b^4*c^2*d - 68*a*b^3*c*d^2 + 35*a^2*b^2*d
^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d*x + a*b^5*d), -1/48*(15*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d
^2 - 7*a^4*d^3 + (b^4*c^3 - 9*a*b^3*c^2*d + 15*a^2*b^2*c*d^2 - 7*a^3*b*d^3)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x
+ b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*(8*b^
4*d^3*x^3 + 81*a*b^3*c^2*d - 190*a^2*b^2*c*d^2 + 105*a^3*b*d^3 + 2*(13*b^4*c*d^2 - 7*a*b^3*d^3)*x^2 + (33*b^4*
c^2*d - 68*a*b^3*c*d^2 + 35*a^2*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d*x + a*b^5*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (c + d x\right )^{\frac{5}{2}}}{\left (a + b x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)**(5/2)/(b*x+a)**(3/2),x)

[Out]

Integral(x*(c + d*x)**(5/2)/(a + b*x)**(3/2), x)

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Giac [B]  time = 3.45441, size = 500, normalized size = 2.34 \begin{align*} \frac{1}{24} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{6}} + \frac{13 \, b^{18} c d^{5}{\left | b \right |} - 19 \, a b^{17} d^{6}{\left | b \right |}}{b^{23} d^{4}}\right )} + \frac{3 \,{\left (11 \, b^{19} c^{2} d^{4}{\left | b \right |} - 40 \, a b^{18} c d^{5}{\left | b \right |} + 29 \, a^{2} b^{17} d^{6}{\left | b \right |}\right )}}{b^{23} d^{4}}\right )} + \frac{4 \,{\left (\sqrt{b d} a b^{3} c^{3}{\left | b \right |} - 3 \, \sqrt{b d} a^{2} b^{2} c^{2} d{\left | b \right |} + 3 \, \sqrt{b d} a^{3} b c d^{2}{\left | b \right |} - \sqrt{b d} a^{4} d^{3}{\left | b \right |}\right )}}{{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} b^{5}} - \frac{5 \,{\left (\sqrt{b d} b^{3} c^{3}{\left | b \right |} - 9 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} + 15 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} - 7 \, \sqrt{b d} a^{3} d^{3}{\left | b \right |}\right )} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{16 \, b^{6} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)^(5/2)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*d^2*abs(b)/b^6 + (13*b^18*c*d
^5*abs(b) - 19*a*b^17*d^6*abs(b))/(b^23*d^4)) + 3*(11*b^19*c^2*d^4*abs(b) - 40*a*b^18*c*d^5*abs(b) + 29*a^2*b^
17*d^6*abs(b))/(b^23*d^4)) + 4*(sqrt(b*d)*a*b^3*c^3*abs(b) - 3*sqrt(b*d)*a^2*b^2*c^2*d*abs(b) + 3*sqrt(b*d)*a^
3*b*c*d^2*abs(b) - sqrt(b*d)*a^4*d^3*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2)*b^5) - 5/16*(sqrt(b*d)*b^3*c^3*abs(b) - 9*sqrt(b*d)*a*b^2*c^2*d*abs(b) + 15*sqrt(b*d)*a^2*
b*c*d^2*abs(b) - 7*sqrt(b*d)*a^3*d^3*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2)/(b^6*d)