3.7.48 \(\int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=280 \[ \frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \]

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Rubi [A]  time = 0.18, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {94, 93, 208} \begin {gather*} \frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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

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Mathematica [A]  time = 0.83, size = 268, normalized size = 0.96 \begin {gather*} -\frac {(b c-a d) \left (128 a^{7/2} c^{3/2} (a+b x)^{3/2} (c+d x)^{7/2}+x (b c-a d) \left (x (b c-a d) \left (8 a^{5/2} \sqrt {c} \sqrt {a+b x} (c+d x)^{5/2}-5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )\right )+48 a^{7/2} \sqrt {c} \sqrt {a+b x} (c+d x)^{7/2}\right )\right )}{1536 a^{7/2} c^{7/2} x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

-1/6*((a + b*x)^(5/2)*(c + d*x)^(7/2))/(c*x^6) - ((b*c - a*d)*(128*a^(7/2)*c^(3/2)*(a + b*x)^(3/2)*(c + d*x)^(
7/2) + (b*c - a*d)*x*(48*a^(7/2)*Sqrt[c]*Sqrt[a + b*x]*(c + d*x)^(7/2) + (b*c - a*d)*x*(8*a^(5/2)*Sqrt[c]*Sqrt
[a + b*x]*(c + d*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*
a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))))/(1536*a^(7/2)*c^(7/
2)*x^5)

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IntegrateAlgebraic [A]  time = 0.52, size = 220, normalized size = 0.79 \begin {gather*} \frac {5 (a d-b c)^6 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{512 a^{7/2} c^{7/2}}-\frac {\sqrt {c+d x} (a d-b c)^6 \left (\frac {15 a^5 (c+d x)^5}{(a+b x)^5}-\frac {85 a^4 c (c+d x)^4}{(a+b x)^4}+\frac {198 a^3 c^2 (c+d x)^3}{(a+b x)^3}+\frac {198 a^2 c^3 (c+d x)^2}{(a+b x)^2}-\frac {85 a c^4 (c+d x)}{a+b x}+15 c^5\right )}{1536 a^3 c^3 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

-1/1536*((-(b*c) + a*d)^6*Sqrt[c + d*x]*(15*c^5 - (85*a*c^4*(c + d*x))/(a + b*x) + (198*a^2*c^3*(c + d*x)^2)/(
a + b*x)^2 + (198*a^3*c^2*(c + d*x)^3)/(a + b*x)^3 - (85*a^4*c*(c + d*x)^4)/(a + b*x)^4 + (15*a^5*(c + d*x)^5)
/(a + b*x)^5))/(a^3*c^3*Sqrt[a + b*x]*(-c + (a*(c + d*x))/(a + b*x))^6) + (5*(-(b*c) + a*d)^6*ArcTanh[(Sqrt[a]
*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(512*a^(7/2)*c^(7/2))

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fricas [A]  time = 39.62, size = 908, normalized size = 3.24 \begin {gather*} \left [\frac {15 \, {\left (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}\right )} \sqrt {a c} x^{6} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (256 \, a^{6} c^{6} + {\left (15 \, a b^{5} c^{6} - 85 \, a^{2} b^{4} c^{5} d + 198 \, a^{3} b^{3} c^{4} d^{2} + 198 \, a^{4} b^{2} c^{3} d^{3} - 85 \, a^{5} b c^{2} d^{4} + 15 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (5 \, a^{2} b^{4} c^{6} - 28 \, a^{3} b^{3} c^{5} d - 594 \, a^{4} b^{2} c^{4} d^{2} - 28 \, a^{5} b c^{3} d^{3} + 5 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (a^{3} b^{3} c^{6} + 159 \, a^{4} b^{2} c^{5} d + 159 \, a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (27 \, a^{4} b^{2} c^{6} + 106 \, a^{5} b c^{5} d + 27 \, a^{6} c^{4} d^{2}\right )} x^{2} + 640 \, {\left (a^{5} b c^{6} + a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6144 \, a^{4} c^{4} x^{6}}, -\frac {15 \, {\left (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}\right )} \sqrt {-a c} x^{6} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (256 \, a^{6} c^{6} + {\left (15 \, a b^{5} c^{6} - 85 \, a^{2} b^{4} c^{5} d + 198 \, a^{3} b^{3} c^{4} d^{2} + 198 \, a^{4} b^{2} c^{3} d^{3} - 85 \, a^{5} b c^{2} d^{4} + 15 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (5 \, a^{2} b^{4} c^{6} - 28 \, a^{3} b^{3} c^{5} d - 594 \, a^{4} b^{2} c^{4} d^{2} - 28 \, a^{5} b c^{3} d^{3} + 5 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (a^{3} b^{3} c^{6} + 159 \, a^{4} b^{2} c^{5} d + 159 \, a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (27 \, a^{4} b^{2} c^{6} + 106 \, a^{5} b c^{5} d + 27 \, a^{6} c^{4} d^{2}\right )} x^{2} + 640 \, {\left (a^{5} b c^{6} + a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3072 \, a^{4} c^{4} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,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(a*c)*x^6*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c + a*d)*x
)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(256*a^6*c^6 + (15*a*b^5*c^6 - 85*
a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^
4*c^6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*
a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 16*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*
x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^6), -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(-a*c)*x
^6*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2
+ a^2*c*d)*x)) + 2*(256*a^6*c^6 + (15*a*b^5*c^6 - 85*a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^
3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^4*c^6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*
b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 1
6*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(
d*x + c))/(a^4*c^4*x^6)]

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giac [B]  time = 34.64, size = 8500, normalized size = 30.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="giac")

[Out]

1/1536*(15*(sqrt(b*d)*b^7*c^6*abs(b) - 6*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 15*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b) -
20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b) + 15*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) - 6*sqrt(b*d)*a^5*b^2*c*d^5*abs(b) +
sqrt(b*d)*a^6*b*d^6*abs(b))*arctan(-1/2*(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)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^3*b*c^3) - 2*(15*sqrt(b*d)*b^29*c^17*abs(b) - 265*sqrt(b*
d)*a*b^28*c^16*d*abs(b) + 2208*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 11088*sqrt(b*d)*a^3*b^26*c^14*d^3*abs(b) +
36732*sqrt(b*d)*a^4*b^25*c^13*d^4*abs(b) - 83412*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 129840*sqrt(b*d)*a^6*b^
23*c^11*d^6*abs(b) - 129536*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 55506*sqrt(b*d)*a^8*b^21*c^9*d^8*abs(b) + 555
06*sqrt(b*d)*a^9*b^20*c^8*d^9*abs(b) - 129536*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) + 129840*sqrt(b*d)*a^11*b^18
*c^6*d^11*abs(b) - 83412*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) + 36732*sqrt(b*d)*a^13*b^16*c^4*d^13*abs(b) - 110
88*sqrt(b*d)*a^14*b^15*c^3*d^14*abs(b) + 2208*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) - 265*sqrt(b*d)*a^16*b^13*c*
d^16*abs(b) + 15*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^2*b^27*c^16*abs(b) + 2400*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^2*a*b^26*c^15*d*abs(b) - 16056*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^2*a^2*b^25*c^14*d^2*abs(b) + 62880*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2*a^3*b^24*c^13*d^3*abs(b) - 156876*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2*a^4*b^23*c^12*d^4*abs(b) + 257760*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2*a^5*b^22*c^11*d^5*abs(b) - 282120*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2*a^6*b^21*c^10*d^6*abs(b) + 217632*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2*a^7*b^20*c^9*d^7*abs(b) - 170910*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*a^8*b^19*c^8*d^8*abs(b) + 217632*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*a^9*b^18*c^7*d^9*abs(b) - 282120*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2*a^10*b^17*c^6*d^10*abs(b) + 257760*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*a^11*b^16*c^5*d^11*abs(b) - 156876*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*a^12*b^15*c^4*d^12*abs(b) + 62880*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*a^13*b^14*c^3*d^13*abs(b) - 16056*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*a^14*b^13*c^2*d^14*abs(b) + 2400*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*a^15*b^12*c*d^15*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a
^16*b^11*d^16*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^25*c^
15*abs(b) - 9765*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^24*c^14*d*abs
(b) + 51813*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^23*c^13*d^2*abs(
b) - 147657*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^22*c^12*d^3*abs(
b) + 229845*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^21*c^11*d^4*abs(
b) - 168225*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^20*c^10*d^5*abs(
b) + 6945*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^6*b^19*c^9*d^6*abs(b)
+ 36219*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^7*b^18*c^8*d^7*abs(b) +
36219*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^8*b^17*c^7*d^8*abs(b) + 69
45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^9*b^16*c^6*d^9*abs(b) - 16822
5*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^10*b^15*c^5*d^10*abs(b) + 2298
45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^11*b^14*c^4*d^11*abs(b) - 147
657*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^12*b^13*c^3*d^12*abs(b) + 51
813*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^13*b^12*c^2*d^13*abs(b) - 97
65*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^14*b^11*c*d^14*abs(b) + 825*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^15*b^10*d^15*abs(b) - 2475*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^23*c^14*abs(b) + 23610*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^22*c^13*d*abs(b) - 98505*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^21*c^12*d^2*abs(b) + 183780*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^20*c^11*d^3*abs(b) - 103075*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^19*c^10*d^4*abs(b) - 125850*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^18*c^9*d^5*abs(b) + 224535*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^17*c^8*d^6*abs(b) - 204040*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^16*c^7*d^7*abs(b) + 224535*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^15*c^6*d^8*abs(b) - 125850*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^9*b^14*c^5*d^9*abs(b) - 103075*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^10*b^13*c^4*d^10*abs(b) + 183780*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^11*b^12*c^3*d^11*abs(b) - 98505*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^12*b^11*c^2*d^12*abs(b) + 23610*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^13*b^10*c*d^13*abs(b) - 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^14*b^9*d^14*abs(b) + 4950*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^21*c^13*abs(b) - 37890*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^20*c^12*d*abs(b) + 124020*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^8*a^2*b^19*c^11*d^2*abs(b) - 115500*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^8*a^3*b^18*c^10*d^3*abs(b) - 57390*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*a^4*b^17*c^9*d^4*abs(b) + 89370*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^8*a^5*b^16*c^8*d^5*abs(b) - 7560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^8*a^6*b^15*c^7*d^6*abs(b) - 7560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^7*b^14*c^6*d^7*abs(b) + 89370*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^8*a^8*b^13*c^5*d^8*abs(b) - 57390*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^8*a^9*b^12*c^4*d^9*abs(b) - 115500*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^8*a^10*b^11*c^3*d^10*abs(b) + 124020*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^8*a^11*b^10*c^2*d^11*abs(b) - 37890*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^8*a^12*b^9*c*d^12*abs(b) + 4950*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^8*a^13*b^8*d^13*abs(b) - 6930*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^10*b^19*c^12*abs(b) + 42840*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*
a*b^18*c^11*d*abs(b) - 110628*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2
*b^17*c^10*d^2*abs(b) - 16392*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3
*b^16*c^9*d^3*abs(b) + 60978*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^4*
b^15*c^8*d^4*abs(b) + 65712*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^5*b
^14*c^7*d^5*abs(b) - 71160*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^
13*c^6*d^6*abs(b) + 65712*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^7*b^1
2*c^5*d^7*abs(b) + 60978*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^8*b^11
*c^4*d^8*abs(b) - 16392*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^9*b^10*
c^3*d^9*abs(b) - 110628*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^10*b^9*
c^2*d^10*abs(b) + 42840*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^11*b^8*
c*d^11*abs(b) - 6930*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^12*b^7*d^1
2*abs(b) + 6930*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^17*c^11*abs(b)
- 35490*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b^16*c^10*d*abs(b) + 73
206*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^2*b^15*c^9*d^2*abs(b) + 126
522*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^14*c^8*d^3*abs(b) + 383
56*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^4*b^13*c^7*d^4*abs(b) + 5262
0*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^5*b^12*c^6*d^5*abs(b) + 52620
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^6*b^11*c^5*d^6*abs(b) + 38356*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^7*b^10*c^4*d^7*abs(b) + 126522*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^8*b^9*c^3*d^8*abs(b) + 73206*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^9*b^8*c^2*d^9*abs(b) - 35490*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^10*b^7*c*d^10*abs(b) + 6930*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^11*b^6*d^11*abs(b) - 4950*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*b^15*c^10*abs(b) + 22140*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a*b^14*c^9*d*abs(b) - 37278*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^13*c^8*d^2*abs(b) - 149808*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^3*b^12*c^7*d^3*abs(b) - 131340*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^4*b^11*c^6*d^4*abs(b) - 183960*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^5*b^10*c^5*d^5*abs(b) - 131340*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^6*b^9*c^4*d^6*abs(b) - 149808*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^7*b^8*c^3*d^7*abs(b) - 37278*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^8*b^7*c^2*d^8*abs(b) + 22140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^9*b^6*c*d^9*abs(b) - 4950*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^10*b^5*d^10*abs(b) + 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^16*b^13*c^9*abs(b) - 10485*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^16*a*b^12*c^8*d*abs(b) + 15300*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^16*a^2*b^11*c^7*d^2*abs(b) + 99780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^16*a^3*b^10*c^6*d^3*abs(b) + 138690*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^16*a^4*b^9*c^5*d^4*abs(b) + 138690*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^16*a^5*b^8*c^4*d^5*abs(b) + 99780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^16*a^6*b^7*c^3*d^6*abs(b) + 15300*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^16*a^7*b^6*c^2*d^7*abs(b) - 10485*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^16*a^8*b^5*c*d^8*abs(b) + 2475*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^16*a^9*b^4*d^9*abs(b) - 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^1
8*b^11*c^8*abs(b) + 3640*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a*b^10*c
^7*d*abs(b) - 5340*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^2*b^9*c^6*d^
2*abs(b) - 44280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^3*b^8*c^5*d^3*
abs(b) - 70230*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^4*b^7*c^4*d^4*ab
s(b) - 44280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^5*b^6*c^3*d^5*abs(
b) - 5340*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^6*b^5*c^2*d^6*abs(b)
+ 3640*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^7*b^4*c*d^7*abs(b) - 825
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^18*a^8*b^3*d^8*abs(b) + 165*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*b^9*c^7*abs(b) - 825*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*a*b^8*c^6*d*abs(b) + 1485*sqrt(b*d)*(sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*a^2*b^7*c^5*d^2*abs(b) + 14535*sqrt(b*d)*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*a^3*b^6*c^4*d^3*abs(b) + 14535*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*a^4*b^5*c^3*d^4*abs(b) + 1485*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*a^5*b^4*c^2*d^5*abs(b) - 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^20*a^6*b^3*c*d^6*abs(b) + 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^20*a^7*b^2*d^7*abs(b) - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^22*b^7*c^6*abs(b) + 90*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^22*a*b^6*c^5*d*abs(b) - 225*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^22*a^2*b^5*c^4*d^2*abs(b) - 2772*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22
*a^3*b^4*c^3*d^3*abs(b) - 225*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^4
*b^3*c^2*d^4*abs(b) + 90*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^5*b^2*
c*d^5*abs(b) - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^22*a^6*b*d^6*abs(b
))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2
*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^6*a^3*c^3))/b

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maple [B]  time = 0.02, size = 1271, normalized size = 4.54 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 a^{6} d^{6} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-90 a^{5} b c \,d^{5} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+225 a^{4} b^{2} c^{2} d^{4} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-300 a^{3} b^{3} c^{3} d^{3} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+225 a^{2} b^{4} c^{4} d^{2} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-90 a \,b^{5} c^{5} d \,x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+15 b^{6} c^{6} x^{6} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} d^{5} x^{5}+170 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b c \,d^{4} x^{5}-396 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{2} d^{3} x^{5}-396 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{3} d^{2} x^{5}+170 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{4} d \,x^{5}-30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{5} x^{5}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c \,d^{4} x^{4}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{2} d^{3} x^{4}-2376 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{3} d^{2} x^{4}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{4} d \,x^{4}+20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{5} x^{4}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c^{2} d^{3} x^{3}-2544 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{3} d^{2} x^{3}-2544 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{4} d \,x^{3}-16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{5} x^{3}-864 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c^{3} d^{2} x^{2}-3392 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{4} d \,x^{2}-864 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{5} x^{2}-1280 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} c^{4} d x -1280 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,c^{5} x -512 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{5} c^{5}\right )}{3072 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{3} c^{3} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x)

[Out]

1/3072*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(225*a^2*b^4*c^4*d^2*x^6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2))/x)-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*d^5*x^5+15*a^6*d^6*x^6*ln((a*
d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+15*b^6*c^6*x^6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-512*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*a^5*c^5-30*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c^5*x^5-90*a*b^5*c^5*d*x^6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2))/x)+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c*d^4*x^4+20*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*c^5*x^4-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c^2*d^3*x^3-16*(a*
c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^3*c^5*x^3-864*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c
^3*d^2*x^2-864*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c^5*x^2-1280*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*a^5*c^4*d*x-1280*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c^5*x-90*a^5*b*c*d^5*x^6*ln((
a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+225*a^4*b^2*c^2*d^4*x^6*ln((a*d*x+b*c*x+2*
a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-300*a^3*b^3*c^3*d^3*x^6*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+170*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c*d^4*x^5-396*(a
*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c^2*d^3*x^5-396*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*
a^2*b^3*c^3*d^2*x^5+170*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*c^4*d*x^5-3392*(a*c)^(1/2)*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c^4*d*x^2-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c^2*d^3*x^4-2376*
(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c^3*d^2*x^4-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*a^2*b^3*c^4*d*x^4-2544*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c^3*d^2*x^3-2544*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c^4*d*x^3)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^6/(a*c)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**7,x)

[Out]

Timed out

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