\(\int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [813]

Optimal result

Integrand size = 22, antiderivative size = 462 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {d (b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^4 (b c-a d)^4 \sqrt {c+d x}}-\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2} c^{9/2}} \]

[Out]

1/12*b*(-21*a^2*d^2-6*a*b*c*d+35*b^2*c^2)/a^3/c^2/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-1/2/a/c/x^2/(b*x+a)^(
3/2)/(d*x+c)^(3/2)+7/4*(a*d+b*c)/a^2/c^2/x/(b*x+a)^(3/2)/(d*x+c)^(3/2)-5/4*(7*a^2*d^2+10*a*b*c*d+7*b^2*c^2)*ar
ctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(9/2)+1/4*b*(7*a^3*d^3-3*a^2*b*c*d^2-55*a*b^2*c^2
*d+35*b^3*c^3)/a^4/c^2/(-a*d+b*c)^2/(d*x+c)^(3/2)/(b*x+a)^(1/2)+1/12*d*(-35*a^4*d^4+48*a^3*b*c*d^3+18*a^2*b^2*
c^2*d^2-200*a*b^3*c^3*d+105*b^4*c^4)*(b*x+a)^(1/2)/a^4/c^3/(-a*d+b*c)^3/(d*x+c)^(3/2)+1/12*d*(a*d+b*c)*(105*a^
4*d^4-340*a^3*b*c*d^3+406*a^2*b^2*c^2*d^2-340*a*b^3*c^3*d+105*b^4*c^4)*(b*x+a)^(1/2)/a^4/c^4/(-a*d+b*c)^4/(d*x
+c)^(1/2)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {105, 156, 157, 12, 95, 214} \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {7 (a d+b c)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2} c^{9/2}}+\frac {b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{12 a^3 c^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac {b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{4 a^4 c^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {d \sqrt {a+b x} \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right ) (a d+b c)}{12 a^4 c^4 \sqrt {c+d x} (b c-a d)^4}+\frac {d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{12 a^4 c^3 (c+d x)^{3/2} (b c-a d)^3}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \]

[In]

Int[1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(b*(35*b^2*c^2 - 6*a*b*c*d - 21*a^2*d^2))/(12*a^3*c^2*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - 1/(2*a*c*
x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (7*(b*c + a*d))/(4*a^2*c^2*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*(35*
b^3*c^3 - 55*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 7*a^3*d^3))/(4*a^4*c^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2))
+ (d*(105*b^4*c^4 - 200*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 48*a^3*b*c*d^3 - 35*a^4*d^4)*Sqrt[a + b*x])/(12*a^
4*c^3*(b*c - a*d)^3*(c + d*x)^(3/2)) + (d*(b*c + a*d)*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^2*b^2*c^2*d^2 - 3
40*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[a + b*x])/(12*a^4*c^4*(b*c - a*d)^4*Sqrt[c + d*x]) - (5*(7*b^2*c^2 + 10*a*b
*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(9/2)*c^(9/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 214

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {\int \frac {\frac {7}{2} (b c+a d)+5 b d x}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{2 a c} \\ & = -\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {\int \frac {\frac {5}{4} \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )+14 b d (b c+a d) x}{x (a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{2 a^2 c^2} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {\int \frac {\frac {15}{8} (b c-a d) \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )+\frac {3}{4} b d \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right ) x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 a^3 c^2 (b c-a d)} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \int \frac {\frac {15}{16} (b c-a d)^2 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )+\frac {3}{4} b d \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 a^4 c^2 (b c-a d)^2} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}-\frac {4 \int \frac {-\frac {45}{32} (b c-a d)^3 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )-\frac {3}{16} b d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 a^4 c^3 (b c-a d)^3} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {d (b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^4 (b c-a d)^4 \sqrt {c+d x}}+\frac {8 \int \frac {45 (b c-a d)^4 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )}{64 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a^4 c^4 (b c-a d)^4} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {d (b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^4 (b c-a d)^4 \sqrt {c+d x}}+\frac {\left (5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^4 c^4} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {d (b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^4 (b c-a d)^4 \sqrt {c+d x}}+\frac {\left (5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^4 c^4} \\ & = \frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {d (b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^4 (b c-a d)^4 \sqrt {c+d x}}-\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2} c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {105 b^7 c^5 x^3 (c+d x)^2+5 a b^6 c^4 x^2 (28 c-47 d x) (c+d x)^2+3 a^2 b^5 c^3 x (c+d x)^2 \left (7 c^2-106 c d x+22 d^2 x^2\right )-3 a^3 b^4 c^2 (c+d x)^2 \left (2 c^3+17 c^2 d x-32 c d^2 x^2-22 d^3 x^3\right )+a^7 d^4 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )+3 a^6 b d^3 \left (8 c^4-21 c^3 d x-92 c^2 d^2 x^2+15 c d^3 x^3+70 d^4 x^4\right )+a^4 b^3 c d \left (24 c^5+42 c^4 d x+168 c^3 d^2 x^2+207 c^2 d^3 x^3-186 c d^4 x^4-235 d^5 x^5\right )-3 a^5 b^2 d^2 \left (12 c^5-14 c^4 d x+4 c^3 d^2 x^2+183 c^2 d^3 x^3+110 c d^4 x^4-35 d^5 x^5\right )}{12 a^4 c^4 (b c-a d)^4 x^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{9/2} c^{9/2}} \]

[In]

Integrate[1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(105*b^7*c^5*x^3*(c + d*x)^2 + 5*a*b^6*c^4*x^2*(28*c - 47*d*x)*(c + d*x)^2 + 3*a^2*b^5*c^3*x*(c + d*x)^2*(7*c^
2 - 106*c*d*x + 22*d^2*x^2) - 3*a^3*b^4*c^2*(c + d*x)^2*(2*c^3 + 17*c^2*d*x - 32*c*d^2*x^2 - 22*d^3*x^3) + a^7
*d^4*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3) + 3*a^6*b*d^3*(8*c^4 - 21*c^3*d*x - 92*c^2*d^2*x^2 +
15*c*d^3*x^3 + 70*d^4*x^4) + a^4*b^3*c*d*(24*c^5 + 42*c^4*d*x + 168*c^3*d^2*x^2 + 207*c^2*d^3*x^3 - 186*c*d^4*
x^4 - 235*d^5*x^5) - 3*a^5*b^2*d^2*(12*c^5 - 14*c^4*d*x + 4*c^3*d^2*x^2 + 183*c^2*d^3*x^3 + 110*c*d^4*x^4 - 35
*d^5*x^5))/(12*a^4*c^4*(b*c - a*d)^4*x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2
*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(4*a^(9/2)*c^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3391\) vs. \(2(412)=824\).

Time = 0.61 (sec) , antiderivative size = 3392, normalized size of antiderivative = 7.34

method	result	size
default	\(\text {Expression too large to display}\)	\(3392\)

[In]

int(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/24/c^4/a^4*(105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^8*d^8*x^4+105*ln((a*d*x+b
*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^8*c^8*x^4-210*a^7*d^7*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)-210*b^7*c^7*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^7*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
12*a^3*b^4*c^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2
*a*c)/x)*a^6*b^2*d^8*x^6+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^8*c^6*d^2*x^6+2
10*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^7*b*d^8*x^5+210*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^8*c^7*d*x^5+210*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
+2*a*c)/x)*a^8*c*d^7*x^3+210*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^7*c^8*x^3+105
*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^8*c^2*d^6*x^2+105*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^6*c^8*x^2+470*a^4*b^3*c*d^6*x^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)-132*a^3*b^4*c^2*d^5*x^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-132*a^2*b^5*c^3*d^4*x^5*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)+470*a*b^6*c^4*d^3*x^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+660*a^5*b^2*c*d^6*x^4*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2)+372*a^4*b^3*c^2*d^5*x^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-456*a^3*b^4*c^3*d^4*x^4*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+372*a^2*b^5*c^4*d^3*x^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+660*a*b^6*c^5*d^2*x^4
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-90*a^6*b*c*d^6*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1098*a^5*b^2*c^2*d
^5*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*a^4*b^3*c^3*d^4*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*a^3
*b^4*c^4*d^3*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1098*a^2*b^5*c^5*d^2*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)-90*a*b^6*c^6*d*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+552*a^6*b*c^2*d^5*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+24*a^5*b^2*c^3*d^4*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-336*a^4*b^3*c^4*d^3*x^2*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+24*a^3*b^4*c^5*d^2*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+552*a^2*b^5*c^6*d*x^2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+126*a^6*b*c^3*d^4*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-84*a^5*b^2*c^4*d^3*x*(a*c)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)-84*a^4*b^3*c^5*d^2*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+126*a^3*b^4*c^6*d*x*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+135*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b^4*c^6*d
^2*x^2-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^5*c^7*d*x^2-210*a^5*b^2*d^7*x
^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*b^7*c^5*d^2*x^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-420*a^6*b*d^7*x
^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-420*b^7*c^6*d*x^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*a^7*c*d^6*x^2
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*a*b^6*c^7*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*a^7*c^2*d^5*x*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*a^2*b^5*c^7*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-48*a^6*b*c^4*d^3*(a*c)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)+72*a^5*b^2*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-48*a^4*b^3*c^6*d*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*b^3*c*d^7*
x^6+135*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b^4*c^2*d^6*x^6+60*ln((a*d*x+b*c*x
+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^5*c^3*d^5*x^6+135*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^6*c^4*d^4*x^6-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c
)/x)*a*b^7*c^5*d^3*x^6-330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^6*b^2*c*d^7*x^5-2
70*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*b^3*c^2*d^6*x^5+390*ln((a*d*x+b*c*x+2*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b^4*c^3*d^5*x^5+390*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)+2*a*c)/x)*a^3*b^5*c^4*d^4*x^5-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)
*a^2*b^6*c^5*d^3*x^5-330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^7*c^6*d^2*x^5+150
*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^7*b*c*d^7*x^4-840*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^6*b^2*c^2*d^6*x^4+330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+2*a*c)/x)*a^5*b^3*c^3*d^5*x^4+510*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b
^4*c^4*d^4*x^4+330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^5*c^5*d^3*x^4-840*ln(
(a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^6*c^6*d^2*x^4+150*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^7*c^7*d*x^4-330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+2*a*c)/x)*a^7*b*c^2*d^6*x^3-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^6*b^2*c^3
*d^5*x^3+390*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*b^3*c^4*d^4*x^3+390*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b^4*c^5*d^3*x^3-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^5*c^6*d^2*x^3-330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
+2*a*c)/x)*a^2*b^6*c^7*d*x^3-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^7*b*c^3*d^5
*x^2+135*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^6*b^2*c^4*d^4*x^2+60*ln((a*d*x+b*c*
x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*b^3*c^5*d^3*x^2)/((b*x+a)*(d*x+c))^(1/2)/x^2/(a*c)^(1/2)
/(a*d-b*c)^4/(b*x+a)^(3/2)/(d*x+c)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (412) = 824\).

Time = 11.15 (sec) , antiderivative size = 2996, normalized size of antiderivative = 6.48 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*((7*b^8*c^6*d^2 - 18*a*b^7*c^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*
a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^6 + 2*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*c^4*d^
4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^8)*x^5 + (7*b^8*c^8 + 10*a*b^7*c^7*d
- 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d^3 + 34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10
*a^7*b*c*d^7 + 7*a^8*d^8)*x^4 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13*a^4*b^4*c^5*d^3 + 1
3*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b*c^2*d^6 + 7*a^8*c*d^7)*x^3 + (7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*
d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*d^3 + 9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6)*x^2)*sqrt(a*
c)*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)*sq
rt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(6*a^4*b^4*c^8 - 24*a^5*b^3*c^7*d + 36*a^6*b^2*c^6*d^2 - 24*a^
7*b*c^5*d^3 + 6*a^8*c^4*d^4 - (105*a*b^7*c^6*d^2 - 235*a^2*b^6*c^5*d^3 + 66*a^3*b^5*c^4*d^4 + 66*a^4*b^4*c^3*d
^5 - 235*a^5*b^3*c^2*d^6 + 105*a^6*b^2*c*d^7)*x^5 - 6*(35*a*b^7*c^7*d - 55*a^2*b^6*c^6*d^2 - 31*a^3*b^5*c^5*d^
3 + 38*a^4*b^4*c^4*d^4 - 31*a^5*b^3*c^3*d^5 - 55*a^6*b^2*c^2*d^6 + 35*a^7*b*c*d^7)*x^4 - 3*(35*a*b^7*c^8 + 15*
a^2*b^6*c^7*d - 183*a^3*b^5*c^6*d^2 + 69*a^4*b^4*c^5*d^3 + 69*a^5*b^3*c^4*d^4 - 183*a^6*b^2*c^3*d^5 + 15*a^7*b
*c^2*d^6 + 35*a^8*c*d^7)*x^3 - 4*(35*a^2*b^6*c^8 - 69*a^3*b^5*c^7*d - 3*a^4*b^4*c^6*d^2 + 42*a^5*b^3*c^5*d^3 -
3*a^6*b^2*c^4*d^4 - 69*a^7*b*c^3*d^5 + 35*a^8*c^2*d^6)*x^2 - 21*(a^3*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^
6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^5*b^6*c^9*d^2 -
4*a^6*b^5*c^8*d^3 + 6*a^7*b^4*c^7*d^4 - 4*a^8*b^3*c^6*d^5 + a^9*b^2*c^5*d^6)*x^6 + 2*(a^5*b^6*c^10*d - 3*a^6*
b^5*c^9*d^2 + 2*a^7*b^4*c^8*d^3 + 2*a^8*b^3*c^7*d^4 - 3*a^9*b^2*c^6*d^5 + a^10*b*c^5*d^6)*x^5 + (a^5*b^6*c^11
- 9*a^7*b^4*c^9*d^2 + 16*a^8*b^3*c^8*d^3 - 9*a^9*b^2*c^7*d^4 + a^11*c^5*d^6)*x^4 + 2*(a^6*b^5*c^11 - 3*a^7*b^4
*c^10*d + 2*a^8*b^3*c^9*d^2 + 2*a^9*b^2*c^8*d^3 - 3*a^10*b*c^7*d^4 + a^11*c^6*d^5)*x^3 + (a^7*b^4*c^11 - 4*a^8
*b^3*c^10*d + 6*a^9*b^2*c^9*d^2 - 4*a^10*b*c^8*d^3 + a^11*c^7*d^4)*x^2), 1/24*(15*((7*b^8*c^6*d^2 - 18*a*b^7*c
^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^6 + 2
*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2
*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^8)*x^5 + (7*b^8*c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5
*d^3 + 34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*a^7*b*c*d^7 + 7*a^8*d^8)*x^4 + 2*(7*a
*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5
- 11*a^7*b*c^2*d^6 + 7*a^8*c*d^7)*x^3 + (7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*
d^3 + 9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6)*x^2)*sqrt(-a*c)*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*(6*a^4*b^4*c^8 -
24*a^5*b^3*c^7*d + 36*a^6*b^2*c^6*d^2 - 24*a^7*b*c^5*d^3 + 6*a^8*c^4*d^4 - (105*a*b^7*c^6*d^2 - 235*a^2*b^6*c^
5*d^3 + 66*a^3*b^5*c^4*d^4 + 66*a^4*b^4*c^3*d^5 - 235*a^5*b^3*c^2*d^6 + 105*a^6*b^2*c*d^7)*x^5 - 6*(35*a*b^7*c
^7*d - 55*a^2*b^6*c^6*d^2 - 31*a^3*b^5*c^5*d^3 + 38*a^4*b^4*c^4*d^4 - 31*a^5*b^3*c^3*d^5 - 55*a^6*b^2*c^2*d^6
+ 35*a^7*b*c*d^7)*x^4 - 3*(35*a*b^7*c^8 + 15*a^2*b^6*c^7*d - 183*a^3*b^5*c^6*d^2 + 69*a^4*b^4*c^5*d^3 + 69*a^5
*b^3*c^4*d^4 - 183*a^6*b^2*c^3*d^5 + 15*a^7*b*c^2*d^6 + 35*a^8*c*d^7)*x^3 - 4*(35*a^2*b^6*c^8 - 69*a^3*b^5*c^7
*d - 3*a^4*b^4*c^6*d^2 + 42*a^5*b^3*c^5*d^3 - 3*a^6*b^2*c^4*d^4 - 69*a^7*b*c^3*d^5 + 35*a^8*c^2*d^6)*x^2 - 21*
(a^3*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^5)*x)*sqr
t(b*x + a)*sqrt(d*x + c))/((a^5*b^6*c^9*d^2 - 4*a^6*b^5*c^8*d^3 + 6*a^7*b^4*c^7*d^4 - 4*a^8*b^3*c^6*d^5 + a^9*
b^2*c^5*d^6)*x^6 + 2*(a^5*b^6*c^10*d - 3*a^6*b^5*c^9*d^2 + 2*a^7*b^4*c^8*d^3 + 2*a^8*b^3*c^7*d^4 - 3*a^9*b^2*c
^6*d^5 + a^10*b*c^5*d^6)*x^5 + (a^5*b^6*c^11 - 9*a^7*b^4*c^9*d^2 + 16*a^8*b^3*c^8*d^3 - 9*a^9*b^2*c^7*d^4 + a^
11*c^5*d^6)*x^4 + 2*(a^6*b^5*c^11 - 3*a^7*b^4*c^10*d + 2*a^8*b^3*c^9*d^2 + 2*a^9*b^2*c^8*d^3 - 3*a^10*b*c^7*d^
4 + a^11*c^6*d^5)*x^3 + (a^7*b^4*c^11 - 4*a^8*b^3*c^10*d + 6*a^9*b^2*c^9*d^2 - 4*a^10*b*c^8*d^3 + a^11*c^7*d^4
)*x^2)]

Sympy [F]

\[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/x**3/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Integral(1/(x**3*(a + b*x)**(5/2)*(c + d*x)**(5/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),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 detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1901 vs. \(2 (412) = 824\).

Time = 14.19 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.11 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

-2/3*sqrt(b*x + a)*((17*b^7*c^8*d^7*abs(b) - 60*a*b^6*c^7*d^8*abs(b) + 78*a^2*b^5*c^6*d^9*abs(b) - 44*a^3*b^4*
c^5*d^10*abs(b) + 9*a^4*b^3*c^4*d^11*abs(b))*(b*x + a)/(b^9*c^15*d - 7*a*b^8*c^14*d^2 + 21*a^2*b^7*c^13*d^3 -
35*a^3*b^6*c^12*d^4 + 35*a^4*b^5*c^11*d^5 - 21*a^5*b^4*c^10*d^6 + 7*a^6*b^3*c^9*d^7 - a^7*b^2*c^8*d^8) + 9*(2*
b^8*c^9*d^6*abs(b) - 9*a*b^7*c^8*d^7*abs(b) + 16*a^2*b^6*c^7*d^8*abs(b) - 14*a^3*b^5*c^6*d^9*abs(b) + 6*a^4*b^
4*c^5*d^10*abs(b) - a^5*b^3*c^4*d^11*abs(b))/(b^9*c^15*d - 7*a*b^8*c^14*d^2 + 21*a^2*b^7*c^13*d^3 - 35*a^3*b^6
*c^12*d^4 + 35*a^4*b^5*c^11*d^5 - 21*a^5*b^4*c^10*d^6 + 7*a^6*b^3*c^9*d^7 - a^7*b^2*c^8*d^8))/(b^2*c + (b*x +
a)*b*d - a*b*d)^(3/2) + 4/3*(9*sqrt(b*d)*b^11*c^3 - 35*sqrt(b*d)*a*b^10*c^2*d + 43*sqrt(b*d)*a^2*b^9*c*d^2 - 1
7*sqrt(b*d)*a^3*b^8*d^3 - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^9*c
^2 + 54*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^8*c*d - 36*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^7*d^2 + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^7*c - 15*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^6*d)/((a^4*b^3*c^3*abs(b) - 3*a^5*b^2*c^2*d*abs(b) + 3*a^6*b*c*d^2*abs(b) - a^7*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)^3) - 5/4*(7*sqrt
(b*d)*b^4*c^2 + 10*sqrt(b*d)*a*b^3*c*d + 7*sqrt(b*d)*a^2*b^2*d^2)*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^4*b*c^4*abs(b)) + 1/
2*(11*sqrt(b*d)*b^10*c^5 - 33*sqrt(b*d)*a*b^9*c^4*d + 22*sqrt(b*d)*a^2*b^8*c^3*d^2 + 22*sqrt(b*d)*a^3*b^7*c^2*
d^3 - 33*sqrt(b*d)*a^4*b^6*c*d^4 + 11*sqrt(b*d)*a^5*b^5*d^5 - 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^2*b^8*c^4 - 4*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^7*c^3*d + 74*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^6*
c^2*d^2 - 4*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^5*c*d^3 - 33*sqr
t(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^4*d^4 + 33*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^3 + 55*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^5*c^2*d + 55*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^4*c*d^2 + 33*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^3*d^3 - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^4*c^2 - 18*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^3*c*d - 11*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^2*d^2)/((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)^2*
a^4*c^4*abs(b))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^3\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

int(1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x)

[Out]

int(1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)), x)