3.1460 \(\int (a+b x)^{7/2} \sqrt{c+d x} \, dx\)

Optimal. Leaf size=230 \[ \frac{7 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac{7 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b d^4}+\frac{7 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{192 b d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{240 b d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b} \]

[Out]

(-7*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b*d^4) + (7*(b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt[c + d*x])/(
192*b*d^3) - (7*(b*c - a*d)^2*(a + b*x)^(5/2)*Sqrt[c + d*x])/(240*b*d^2) + ((b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c
+ d*x])/(40*b*d) + ((a + b*x)^(9/2)*Sqrt[c + d*x])/(5*b) + (7*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(
Sqrt[b]*Sqrt[c + d*x])])/(128*b^(3/2)*d^(9/2))

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Rubi [A]  time = 0.160316, antiderivative size = 230, 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} \[ \frac{7 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac{7 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 b d^4}+\frac{7 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}{192 b d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}{240 b d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/2)*Sqrt[c + d*x],x]

[Out]

(-7*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b*d^4) + (7*(b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt[c + d*x])/(
192*b*d^3) - (7*(b*c - a*d)^2*(a + b*x)^(5/2)*Sqrt[c + d*x])/(240*b*d^2) + ((b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c
+ d*x])/(40*b*d) + ((a + b*x)^(9/2)*Sqrt[c + d*x])/(5*b) + (7*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(
Sqrt[b]*Sqrt[c + d*x])])/(128*b^(3/2)*d^(9/2))

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)^{7/2} \sqrt{c+d x} \, dx &=\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}+\frac{(b c-a d) \int \frac{(a+b x)^{7/2}}{\sqrt{c+d x}} \, dx}{10 b}\\ &=\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}-\frac{\left (7 (b c-a d)^2\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{80 b d}\\ &=-\frac{7 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{240 b d^2}+\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}+\frac{\left (7 (b c-a d)^3\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{96 b d^2}\\ &=\frac{7 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{192 b d^3}-\frac{7 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{240 b d^2}+\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}-\frac{\left (7 (b c-a d)^4\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{128 b d^3}\\ &=-\frac{7 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b d^4}+\frac{7 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{192 b d^3}-\frac{7 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{240 b d^2}+\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}+\frac{\left (7 (b c-a d)^5\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b d^4}\\ &=-\frac{7 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b d^4}+\frac{7 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{192 b d^3}-\frac{7 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{240 b d^2}+\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}+\frac{\left (7 (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^2 d^4}\\ &=-\frac{7 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b d^4}+\frac{7 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{192 b d^3}-\frac{7 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{240 b d^2}+\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}+\frac{\left (7 (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^2 d^4}\\ &=-\frac{7 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 b d^4}+\frac{7 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}{192 b d^3}-\frac{7 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}{240 b d^2}+\frac{(b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}{40 b d}+\frac{(a+b x)^{9/2} \sqrt{c+d x}}{5 b}+\frac{7 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{3/2} d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.51989, size = 194, normalized size = 0.84 \[ \frac{(a+b x)^{9/2} \sqrt{c+d x} \left (-\frac{70 (b c-a d)^4}{d^4 (a+b x)^4}+\frac{140 (b c-a d)^3}{3 d^3 (a+b x)^3}-\frac{112 (b c-a d)^2}{3 d^2 (a+b x)^2}+\frac{70 (b c-a d)^{9/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{d^{9/2} (a+b x)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}}}+\frac{32 b c-32 a d}{a d+b d x}+256\right )}{1280 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(9/2)*Sqrt[c + d*x]*(256 - (70*(b*c - a*d)^4)/(d^4*(a + b*x)^4) + (140*(b*c - a*d)^3)/(3*d^3*(a + b
*x)^3) - (112*(b*c - a*d)^2)/(3*d^2*(a + b*x)^2) + (32*b*c - 32*a*d)/(a*d + b*d*x) + (70*(b*c - a*d)^(9/2)*Arc
Sinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(d^(9/2)*(a + b*x)^(9/2)*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(12
80*b)

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Maple [B]  time = 0.008, size = 858, normalized size = 3.7 \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)^(7/2)*(d*x+c)^(1/2),x)

[Out]

1/5/d*(b*x+a)^(7/2)*(d*x+c)^(3/2)-7/32/d*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3*c+7/128/d^4*(d*x+c)^(1/2)*(b*x+a)^(1/
2)*c^4*b^3+7/128/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^4-7/40/d^2*(b*x+a)^(5/2)*(d*x+c)^(3/2)*b*c+7/48/d^3*(b*x+a)^(
3/2)*(d*x+c)^(3/2)*b^2*c^2-7/64/d^4*(b*x+a)^(1/2)*(d*x+c)^(3/2)*b^3*c^3-7/32/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a
*c^3*b^2-7/24/d^2*(b*x+a)^(3/2)*(d*x+c)^(3/2)*a*b*c-21/64/d^2*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a^2*b*c+21/64/d^3*(b
*x+a)^(1/2)*(d*x+c)^(3/2)*a*b^2*c^2+21/64/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2*c^2*b+7/40/d*(b*x+a)^(5/2)*(d*x+
c)^(3/2)*a+7/48/d*(b*x+a)^(3/2)*(d*x+c)^(3/2)*a^2+7/64/d*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a^3+35/128/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^2*c^3*b^2-35/256/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)*a*c^4*b^3+35/256*((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^
4*c-35/128/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^3*c^2*b+7/256/d^4*((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^4-7/256*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^5

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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)^(7/2)*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.60415, size = 1573, normalized size = 6.84 \begin{align*} \left [-\frac{105 \,{\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 (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 490 \, a b^{4} c^{3} d^{2} - 896 \, a^{2} b^{3} c^{2} d^{3} + 790 \, a^{3} b^{2} c d^{4} + 105 \, a^{4} b d^{5} + 48 \,{\left (b^{5} c d^{4} + 31 \, a b^{4} d^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{5} c^{2} d^{3} - 32 \, a b^{4} c d^{4} - 263 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (35 \, b^{5} c^{3} d^{2} - 161 \, a b^{4} c^{2} d^{3} + 289 \, a^{2} b^{3} c d^{4} + 605 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, b^{2} d^{5}}, -\frac{105 \,{\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 (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 490 \, a b^{4} c^{3} d^{2} - 896 \, a^{2} b^{3} c^{2} d^{3} + 790 \, a^{3} b^{2} c d^{4} + 105 \, a^{4} b d^{5} + 48 \,{\left (b^{5} c d^{4} + 31 \, a b^{4} d^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{5} c^{2} d^{3} - 32 \, a b^{4} c d^{4} - 263 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (35 \, b^{5} c^{3} d^{2} - 161 \, a b^{4} c^{2} d^{3} + 289 \, a^{2} b^{3} c d^{4} + 605 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, b^{2} d^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/2)*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(105*(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)*sq
rt(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)*sq
rt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(384*b^5*d^5*x^4 - 105*b^5*c^4*d + 490*a*b^4*c^3*d^2 - 896*a^2*b^3*
c^2*d^3 + 790*a^3*b^2*c*d^4 + 105*a^4*b*d^5 + 48*(b^5*c*d^4 + 31*a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 32*a*b^4*
c*d^4 - 263*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 161*a*b^4*c^2*d^3 + 289*a^2*b^3*c*d^4 + 605*a^3*b^2*d^5)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^5), -1/3840*(105*(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)*sqr
t(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*(384*b^5*d^5*x^4 - 105*b^5*c^4*d + 490*a*b^4*c
^3*d^2 - 896*a^2*b^3*c^2*d^3 + 790*a^3*b^2*c*d^4 + 105*a^4*b*d^5 + 48*(b^5*c*d^4 + 31*a*b^4*d^5)*x^3 - 8*(7*b^
5*c^2*d^3 - 32*a*b^4*c*d^4 - 263*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 161*a*b^4*c^2*d^3 + 289*a^2*b^3*c*d^4
+ 605*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^5)]

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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)**(7/2)*(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.38353, size = 1335, normalized size = 5.8 \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)^(7/2)*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

1/1920*(30*(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*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^3*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))*b*abs(b)
+ 3*(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*abs(b)/b^2)/b