3.1481 \(\int (a+b x)^{5/2} (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=262 \[ \frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 b^3 d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}{768 b^3 d^2}-\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}{192 b^3 d}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}{32 b^3}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b} \]

[Out]

(5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^3) - (5*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c + d*x])/
(768*b^3*d^2) + ((b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x])/(192*b^3*d) + ((b*c - a*d)^2*(a + b*x)^(7/2)*Sqr
t[c + d*x])/(32*b^3) + ((b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(12*b^2) + ((a + b*x)^(7/2)*(c + d*x)^(5/
2))/(6*b) - (5*(b*c - a*d)^6*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(7/2))

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Rubi [A]  time = 0.1493, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 217, 206} \[ \frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 b^3 d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}{768 b^3 d^2}-\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}{192 b^3 d}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}{32 b^3}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^3) - (5*(b*c - a*d)^4*(a + b*x)^(3/2)*Sqrt[c + d*x])/
(768*b^3*d^2) + ((b*c - a*d)^3*(a + b*x)^(5/2)*Sqrt[c + d*x])/(192*b^3*d) + ((b*c - a*d)^2*(a + b*x)^(7/2)*Sqr
t[c + d*x])/(32*b^3) + ((b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(12*b^2) + ((a + b*x)^(7/2)*(c + d*x)^(5/
2))/(6*b) - (5*(b*c - a*d)^6*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*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)^{5/2} \, dx &=\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac{(5 (b c-a d)) \int (a+b x)^{5/2} (c+d x)^{3/2} \, dx}{12 b}\\ &=\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac{(b c-a d)^2 \int (a+b x)^{5/2} \sqrt{c+d x} \, dx}{8 b^2}\\ &=\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac{(b c-a d)^3 \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{64 b^3}\\ &=\frac{(b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}{192 b^3 d}+\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac{\left (5 (b c-a d)^4\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{384 b^3 d}\\ &=-\frac{5 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}{768 b^3 d^2}+\frac{(b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}{192 b^3 d}+\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}+\frac{\left (5 (b c-a d)^5\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{512 b^3 d^2}\\ &=\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 b^3 d^3}-\frac{5 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}{768 b^3 d^2}+\frac{(b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}{192 b^3 d}+\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac{\left (5 (b c-a d)^6\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{1024 b^3 d^3}\\ &=\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 b^3 d^3}-\frac{5 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}{768 b^3 d^2}+\frac{(b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}{192 b^3 d}+\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac{\left (5 (b c-a d)^6\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 )}{512 b^4 d^3}\\ &=\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 b^3 d^3}-\frac{5 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}{768 b^3 d^2}+\frac{(b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}{192 b^3 d}+\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac{\left (5 (b c-a d)^6\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 )}{512 b^4 d^3}\\ &=\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 b^3 d^3}-\frac{5 (b c-a d)^4 (a+b x)^{3/2} \sqrt{c+d x}}{768 b^3 d^2}+\frac{(b c-a d)^3 (a+b x)^{5/2} \sqrt{c+d x}}{192 b^3 d}+\frac{(b c-a d)^2 (a+b x)^{7/2} \sqrt{c+d x}}{32 b^3}+\frac{(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{7/2}}\\ \end{align*}

Mathematica [A]  time = 2.5292, size = 209, normalized size = 0.8 \[ \frac{(a+b x)^{7/2} \sqrt{c+d x} \left (\frac{15 (b c-a d)^5}{d^3 (a+b x)^3}-\frac{10 (b c-a d)^4}{d^2 (a+b x)^2}-\frac{15 (b c-a d)^{11/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{d^{7/2} (a+b x)^{7/2} \sqrt{\frac{b (c+d x)}{b c-a d}}}+\frac{8 (b c-a d)^3}{d (a+b x)}+128 b (c+d x) (b c-a d)+48 (b c-a d)^2+256 b^2 (c+d x)^2\right )}{1536 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(7/2)*Sqrt[c + d*x]*(48*(b*c - a*d)^2 + (15*(b*c - a*d)^5)/(d^3*(a + b*x)^3) - (10*(b*c - a*d)^4)/(
d^2*(a + b*x)^2) + (8*(b*c - a*d)^3)/(d*(a + b*x)) + 128*b*(b*c - a*d)*(c + d*x) + 256*b^2*(c + d*x)^2 - (15*(
b*c - a*d)^(11/2)*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)])))/(1536*b^3)

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Maple [B]  time = 0.006, size = 1089, normalized size = 4.2 \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)^(5/2),x)

[Out]

5/512*d^2/b^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^5-5/512/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^5*b^2-1/64/d*(d*x+c)^(5/
2)*(b*x+a)^(1/2)*a^2*c-1/192/d^3*(d*x+c)^(5/2)*(b*x+a)^(1/2)*c^3*b^2-5/768*d/b^2*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a
^4-5/768/d^3*(d*x+c)^(3/2)*(b*x+a)^(1/2)*c^4*b^2-1/12/d^2*(b*x+a)^(3/2)*(d*x+c)^(7/2)*b*c-25/256/d*(d*x+c)^(1/
2)*(b*x+a)^(1/2)*a^2*c^3-5/128/d*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^2*c^2+1/32/d^3*(b*x+a)^(1/2)*(d*x+c)^(7/2)*b^2*
c^2+1/32/d*(b*x+a)^(1/2)*(d*x+c)^(7/2)*a^2+1/192/b*(d*x+c)^(5/2)*(b*x+a)^(1/2)*a^3+1/12/d*(b*x+a)^(3/2)*(d*x+c
)^(7/2)*a+25/256/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3*c^2+5/192/b*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^3*c+25/256*((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^3+1/6/d*(b*x+a)^(5/2)*(d*x+c)^(7/2)-25/512*d/b^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^4*c-1
/16/d^2*(b*x+a)^(1/2)*(d*x+c)^(7/2)*a*b*c+1/64/d^2*(d*x+c)^(5/2)*(b*x+a)^(1/2)*a*c^2*b+5/192/d^2*(d*x+c)^(3/2)
*(b*x+a)^(1/2)*a*c^3*b+25/512/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^4*b-5/1024/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^6
*b^3-5/1024*d^3/b^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)*a^6+15/512/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^5*b^2-75/1024*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^2-75/1024/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^4*b+15/512*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*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((b*x+a)^(5/2)*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.50246, size = 1924, normalized size = 7.34 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*
c*d^5 + a^6*d^6)*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)*sqrt(b*
d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(256*b^6*d^6*x^5 + 15*b^6*c^5*d - 85*a*b^5*c^4*d
^2 + 198*a^2*b^4*c^3*d^3 + 198*a^3*b^3*c^2*d^4 - 85*a^4*b^2*c*d^5 + 15*a^5*b*d^6 + 640*(b^6*c*d^5 + a*b^5*d^6)
*x^4 + 16*(27*b^6*c^2*d^4 + 106*a*b^5*c*d^5 + 27*a^2*b^4*d^6)*x^3 + 8*(b^6*c^3*d^3 + 159*a*b^5*c^2*d^4 + 159*a
^2*b^4*c*d^5 + a^3*b^3*d^6)*x^2 - 2*(5*b^6*c^4*d^2 - 28*a*b^5*c^3*d^3 - 594*a^2*b^4*c^2*d^4 - 28*a^3*b^3*c*d^5
+ 5*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^4), 1/3072*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*
c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*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*(256*b^
6*d^6*x^5 + 15*b^6*c^5*d - 85*a*b^5*c^4*d^2 + 198*a^2*b^4*c^3*d^3 + 198*a^3*b^3*c^2*d^4 - 85*a^4*b^2*c*d^5 + 1
5*a^5*b*d^6 + 640*(b^6*c*d^5 + a*b^5*d^6)*x^4 + 16*(27*b^6*c^2*d^4 + 106*a*b^5*c*d^5 + 27*a^2*b^4*d^6)*x^3 + 8
*(b^6*c^3*d^3 + 159*a*b^5*c^2*d^4 + 159*a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^2 - 2*(5*b^6*c^4*d^2 - 28*a*b^5*c^3*d^3
- 594*a^2*b^4*c^2*d^4 - 28*a^3*b^3*c*d^5 + 5*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^4)]

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

[Out]

Timed out

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Giac [B]  time = 1.65402, size = 3542, normalized size = 13.52 \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)^(5/2),x, algorithm="giac")

[Out]

1/7680*(40*(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^2*abs(b) + 80*(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^2*abs(b)/b^2 + 8*(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))*
c*d*abs(b) + 160*(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))*a*c*d*abs(b)/b + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*
(10*(b*x + a)/b^4 + (b^21*c*d^9 - 49*a*b^20*d^10)/(b^24*d^10)) - 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2
*b^20*d^10)/(b^24*d^10)) + (21*b^23*c^3*d^7 + 49*a*b^22*c^2*d^8 + 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^2
4*d^10))*(b*x + a) - 5*(21*b^24*c^4*d^6 + 28*a*b^23*c^3*d^7 + 30*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^
4*b^20*d^10)/(b^24*d^10))*(b*x + a) + 15*(21*b^25*c^5*d^5 + 7*a*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22
*c^2*d^8 - 7*a^4*b^21*c*d^9 - 21*a^5*b^20*d^10)/(b^24*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d -
5*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 14*a^5*b*c*d^5 + 21*a^6*d^6)*log(abs(-sqrt(b*d)*s
qrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^5))*d^2*abs(b) + 40*(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))*a^2*d^2*abs(b
)/b^2 + 8*(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 - 3
1*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))*a*d^2*abs(b)/b + 8*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sq
rt(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^2*abs(b)/b^2 + 8*(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*c*d*abs(b)/b^3)/b