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

Optimal. Leaf size=227 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{64 b^2 d^2}-\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{80 b^2 d}+\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b} \]

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^2*d^3) - ((b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt[c + d*x])/(6
4*b^2*d^2) + ((b*c - a*d)^2*(a + b*x)^(5/2)*Sqrt[c + d*x])/(80*b^2*d) + (3*(b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c
+ d*x])/(40*b^2) + ((a + b*x)^(7/2)*(c + d*x)^(3/2))/(5*b) - (3*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/
(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(7/2))

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Rubi [A]  time = 0.126493, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 217, 206} \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{64 b^2 d^2}-\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{80 b^2 d}+\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^2*d^3) - ((b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt[c + d*x])/(6
4*b^2*d^2) + ((b*c - a*d)^2*(a + b*x)^(5/2)*Sqrt[c + d*x])/(80*b^2*d) + (3*(b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c
+ d*x])/(40*b^2) + ((a + b*x)^(7/2)*(c + d*x)^(3/2))/(5*b) - (3*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/
(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(7/2))

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 (a+b x)^{5/2} (c+d x)^{3/2} \, dx &=\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}+\frac{(3 (b c-a d)) \int (a+b x)^{5/2} \sqrt{c+d x} \, dx}{10 b}\\ &=\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}+\frac{\left (3 (b c-a d)^2\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{80 b^2}\\ &=\frac{(b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{80 b^2 d}+\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}-\frac{(b c-a d)^3 \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{32 b^2 d}\\ &=-\frac{(b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{64 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{80 b^2 d}+\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}+\frac{\left (3 (b c-a d)^4\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{128 b^2 d^2}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^3}-\frac{(b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{64 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{80 b^2 d}+\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}-\frac{\left (3 (b c-a d)^5\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^2 d^3}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^3}-\frac{(b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{64 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{80 b^2 d}+\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}-\frac{\left (3 (b c-a d)^5\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 )}{128 b^3 d^3}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^3}-\frac{(b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{64 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{80 b^2 d}+\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}-\frac{\left (3 (b c-a d)^5\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 )}{128 b^3 d^3}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b^2 d^3}-\frac{(b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{64 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{80 b^2 d}+\frac{3 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}-\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.77056, size = 187, normalized size = 0.82 \[ \frac{(a+b x)^{7/2} \sqrt{c+d x} \left (\frac{15 (b c-a d)^4}{d^3 (a+b x)^3}+\frac{10 (a d-b c)^3}{d^2 (a+b x)^2}-\frac{15 (b c-a d)^{9/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{d^{7/2} (a+b x)^{7/2} \sqrt{\frac{b (c+d x)}{b c-a d}}}+\frac{8 (b c-a d)^2}{d (a+b x)}+48 (b c-a d)+128 b (c+d x)\right )}{640 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(7/2)*Sqrt[c + d*x]*(48*(b*c - a*d) + (15*(b*c - a*d)^4)/(d^3*(a + b*x)^3) + (10*(-(b*c) + a*d)^3)/
(d^2*(a + b*x)^2) + (8*(b*c - a*d)^2)/(d*(a + b*x)) + 128*b*(c + d*x) - (15*(b*c - a*d)^(9/2)*ArcSinh[(Sqrt[d]
*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(d^(7/2)*(a + b*x)^(7/2)*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(640*b^2)

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Maple [B]  time = 0.007, size = 853, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/d*(b*x+a)^(5/2)*(d*x+c)^(5/2)-3/128/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^4*b^2-3/128*d/b^2*(d*x+c)^(1/2)*(b*x
+a)^(1/2)*a^4-9/64/d*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2*c^2-3/64/d*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^2*c-1/64/d^3*(d*
x+c)^(3/2)*(b*x+a)^(1/2)*c^3*b^2-1/8/d^2*(b*x+a)^(3/2)*(d*x+c)^(5/2)*b*c+1/16/d^3*(b*x+a)^(1/2)*(d*x+c)^(5/2)*
b^2*c^2+1/64/b*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^3+1/8/d*(b*x+a)^(3/2)*(d*x+c)^(5/2)*a+1/16/d*(b*x+a)^(1/2)*(d*x+c
)^(5/2)*a^2-15/256*d/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1
/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^4*c+15/128*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1
/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^3*c^2+15/256/d^2*((b
*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a
*c)^(1/2))/(b*d)^(1/2)*a*c^4*b^2-1/8/d^2*(b*x+a)^(1/2)*(d*x+c)^(5/2)*a*b*c+3/256*d^2/b^2*((b*x+a)*(d*x+c))^(1/
2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(
1/2)*a^5-15/128/d*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(
d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^2*c^3*b-3/256/d^3*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^
(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*c^5*b^3+3/64/d^2*(d*
x+c)^(3/2)*(b*x+a)^(1/2)*a*c^2*b+3/32/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3*c+3/32/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)
*a*c^3*b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.81882, size = 1547, normalized size = 6.81 \begin{align*} \left [-\frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\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 (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 70 \, a b^{4} c^{3} d^{2} + 128 \, a^{2} b^{3} c^{2} d^{3} + 70 \, a^{3} b^{2} c d^{4} - 15 \, a^{4} b d^{5} + 16 \,{\left (11 \, b^{5} c d^{4} + 21 \, a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (b^{5} c^{2} d^{3} + 64 \, a b^{4} c d^{4} + 31 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \,{\left (5 \, b^{5} c^{3} d^{2} - 23 \, a b^{4} c^{2} d^{3} - 233 \, a^{2} b^{3} c d^{4} - 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, b^{3} d^{4}}, \frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\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 (128 \, b^{5} d^{5} x^{4} + 15 \, b^{5} c^{4} d - 70 \, a b^{4} c^{3} d^{2} + 128 \, a^{2} b^{3} c^{2} d^{3} + 70 \, a^{3} b^{2} c d^{4} - 15 \, a^{4} b d^{5} + 16 \,{\left (11 \, b^{5} c d^{4} + 21 \, a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (b^{5} c^{2} d^{3} + 64 \, a b^{4} c d^{4} + 31 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \,{\left (5 \, b^{5} c^{3} d^{2} - 23 \, a b^{4} c^{2} d^{3} - 233 \, a^{2} b^{3} c d^{4} - 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, b^{3} d^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2560*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqr
t(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)*sqrt(b*d)*sqrt(b*x + a)*sqr
t(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(128*b^5*d^5*x^4 + 15*b^5*c^4*d - 70*a*b^4*c^3*d^2 + 128*a^2*b^3*c^2
*d^3 + 70*a^3*b^2*c*d^4 - 15*a^4*b*d^5 + 16*(11*b^5*c*d^4 + 21*a*b^4*d^5)*x^3 + 8*(b^5*c^2*d^3 + 64*a*b^4*c*d^
4 + 31*a^2*b^3*d^5)*x^2 - 2*(5*b^5*c^3*d^2 - 23*a*b^4*c^2*d^3 - 233*a^2*b^3*c*d^4 - 5*a^3*b^2*d^5)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(b^3*d^4), 1/1280*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3
+ 5*a^4*b*c*d^4 - a^5*d^5)*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*(128*b^5*d^5*x^4 + 15*b^5*c^4*d - 70*a*b^4*c^3*d^2 + 128*
a^2*b^3*c^2*d^3 + 70*a^3*b^2*c*d^4 - 15*a^4*b*d^5 + 16*(11*b^5*c*d^4 + 21*a*b^4*d^5)*x^3 + 8*(b^5*c^2*d^3 + 64
*a*b^4*c*d^4 + 31*a^2*b^3*d^5)*x^2 - 2*(5*b^5*c^3*d^2 - 23*a*b^4*c^2*d^3 - 233*a^2*b^3*c*d^4 - 5*a^3*b^2*d^5)*
x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.89817, size = 1987, normalized size = 8.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*(10*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*c*d^5 - 17*a
*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*
c^2*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c
^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(
sqrt(b*d)*b*d^3))*c*abs(b) + 20*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b^4*d^2) + (b
*c*d - a*d^2)/(b^4*d^4)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*
x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^3))*a^2*c*abs(b)/b^2 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x
+ a)*(6*(b*x + a)*(8*(b*x + a)/b^3 + (b^13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*
d^7 - 263*a^2*b^12*d^8)/(b^15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*d^7 - 121*a^3*b^12*d
^8)/(b^15*d^8))*(b*x + a) - 15*(7*b^16*c^4*d^4 + 2*a*b^15*c^3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d
^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7
*a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^4))*d*abs(
b) + 20*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*c*d^5 - 17*a*b^
6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*c^2
*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*
d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqr
t(b*d)*b*d^3))*a*d*abs(b)/b + 2*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/(
b^6*d^2) + (b*c*d^3 - 7*a*d^4)/(b^6*d^6)) - 3*(b^2*c^2*d^2 - a^2*d^4)/(b^6*d^6)) - 3*(b^3*c^3 - a*b^2*c^2*d -
a^2*b*c*d^2 + a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5
*d^4))*a*c*abs(b)/b^2 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/(b^6*d^2)
+ (b*c*d^3 - 7*a*d^4)/(b^6*d^6)) - 3*(b^2*c^2*d^2 - a^2*d^4)/(b^6*d^6)) - 3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*
d^2 + a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5*d^4))*a
^2*d*abs(b)/b^3)/b