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

Optimal. Leaf size=462 \[ \frac{d \sqrt{a+b x} \left (406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4-340 a b^3 c^3 d+105 b^4 c^4\right ) (a d+b c)}{12 a^4 c^4 \sqrt{c+d x} (b c-a d)^4}+\frac{d \sqrt{a+b x} \left (18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4-200 a b^3 c^3 d+105 b^4 c^4\right )}{12 a^4 c^3 (c+d x)^{3/2} (b c-a d)^3}+\frac{b \left (-3 a^2 b c d^2+7 a^3 d^3-55 a b^2 c^2 d+35 b^3 c^3\right )}{4 a^4 c^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{12 a^3 c^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2} c^{9/2}}+\frac{7 (a d+b c)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.55599, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {103, 151, 152, 12, 93, 208} \[ \frac{d \sqrt{a+b x} \left (406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4-340 a b^3 c^3 d+105 b^4 c^4\right ) (a d+b c)}{12 a^4 c^4 \sqrt{c+d x} (b c-a d)^4}+\frac{d \sqrt{a+b x} \left (18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4-200 a b^3 c^3 d+105 b^4 c^4\right )}{12 a^4 c^3 (c+d x)^{3/2} (b c-a d)^3}+\frac{b \left (-3 a^2 b c d^2+7 a^3 d^3-55 a b^2 c^2 d+35 b^3 c^3\right )}{4 a^4 c^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{12 a^3 c^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2} c^{9/2}}+\frac{7 (a d+b c)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[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 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

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] && IntegerQ[m]

Rule 152

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 12

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

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 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{1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx &=-\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 ) \operatorname{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]  time = 1.31014, size = 504, normalized size = 1.09 \[ \frac{-x^2 \left ((a+b x) \left (-3 \sqrt{a} c^{3/2} d (a+b x) \left (-18 a^2 b^2 c^2 d^2-48 a^3 b c d^3+35 a^4 d^4+200 a b^3 c^3 d-105 b^4 c^4\right ) (a d-b c)+\sqrt{a+b x} (c+d x) \left (45 \sqrt{c+d x} (b c-a d)^4 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-3 \sqrt{a} \sqrt{c} d \sqrt{a+b x} \left (66 a^2 b^3 c^3 d^2+66 a^3 b^2 c^2 d^3-235 a^4 b c d^4+105 a^5 d^5-235 a b^4 c^4 d+105 b^5 c^5\right )\right )+45 \sqrt{a} b c^{5/2} \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) (a d-b c)^3-18 a^{3/2} b c^{5/2} d (b c-a d)^2 \left (21 a^2 d^2+6 a b c d-35 b^2 c^2\right )\right )-3 a^{3/2} b c^{5/2} (a d-b c)^3 \left (21 a^2 d^2+6 a b c d-35 b^2 c^2\right )\right )+63 a^{5/2} c^{5/2} x (a d+b c) (b c-a d)^4-18 a^{7/2} c^{7/2} (b c-a d)^4}{36 a^{9/2} c^{9/2} x^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*a^(7/2)*c^(7/2)*(b*c - a*d)^4 + 63*a^(5/2)*c^(5/2)*(b*c - a*d)^4*(b*c + a*d)*x - x^2*(-3*a^(3/2)*b*c^(5/2
)*(-(b*c) + a*d)^3*(-35*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2) + (a + b*x)*(45*Sqrt[a]*b*c^(5/2)*(-(b*c) + a*d)^3*(
7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2) - 18*a^(3/2)*b*c^(5/2)*d*(b*c - a*d)^2*(-35*b^2*c^2 + 6*a*b*c*d + 21*a^2*d
^2) - 3*Sqrt[a]*c^(3/2)*d*(-(b*c) + a*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)*(a + b*x) + Sqrt[a + b*x]*(c + d*x)*(-3*Sqrt[a]*Sqrt[c]*d*(105*b^5*c^5 - 235*a*b^4*c^4*d + 66*a
^2*b^3*c^3*d^2 + 66*a^3*b^2*c^2*d^3 - 235*a^4*b*c*d^4 + 105*a^5*d^5)*Sqrt[a + b*x] + 45*(b*c - a*d)^4*(7*b^2*c
^2 + 10*a*b*c*d + 7*a^2*d^2)*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))))/(36*a^
(9/2)*c^(9/2)*(b*c - a*d)^4*x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2))

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Maple [B]  time = 0.063, size = 3392, normalized size = 7.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24/a^4/c^4*(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)*x^3*a^5*b^3*c^4*d^4+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)*x^3*a^4*b^4*c^5*d^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)*x^3*a^3*b^5*c^6*d^2-330*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+2*a*c)/x)*x^3*a^2*b^6*c^7*d-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)*x^2*a^7
*b*c^3*d^5+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)*x^2*a^6*b^2*c^4*d^4+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)*x^2*a^5*b^3*c^5*d^3+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)*x^2*a^4*b^4*c^6*d^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)*x^2*a^3*b^5*c^7*d-210*x^5*a^5*b^2*d^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^5*b^7*c^5*d^2*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-420*x^4*a^6*b*d^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-420*x^4*b^7*c^6*d*(a*c)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*x^2*a^7*c*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*x^2*a*b^6*c^7*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)-42*x*a^7*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*x*a^2*b^5*c^7*(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)*x^6*a^5*b^3*c*d^7+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)*x^6*a^4*b^4*c^2*d^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)*x^6*a^3*b^5*c^
3*d^5+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)*x^6*a^2*b^6*c^4*d^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)*x^4*a^8*d^8+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)*x^4*b^8*c^8-84*x*a^4*b^3*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+126*x*a^3*b^4*c^6*d
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+470*x^5*a^4*b^3*c*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-132*x^5*a^3*b^4
*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^3*a^7*d^7*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^3*b^7*c
^7*(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)*x^6*a^6*b^2
*d^8+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)*x^6*b^8*c^6*d^2+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)*x^5*a^7*b*d^8+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)*x^5*b^8*c^7*d+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)*x^3*a^8*c
*d^7+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)*x^3*a*b^7*c^8+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)*x^2*a^8*c^2*d^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)*x^5*a^6*b^2*c*d^7-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a
^5*b^3*c^2*d^6+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)*x^5*a^4*b^4*c^3*d^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)*x^5*a^3*b^5*c^4*d^4-270*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^2*b^6*c^5*d^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)*x^5*a*b^7*c^6*d^2+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)*x^4*a^7
*b*c*d^7-840*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^6*b^2*c^2*d^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)*x^4*a^5*b^3*c^3*d^5+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)*x^4*a^4*b^4*c^4*d^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)*x^4*a^3*b^5*c^5*d^3-840*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b^6
*c^6*d^2+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)*x^4*a*b^7*c^7*d-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)*x^3*a^7*b*c^2*d^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)*x^3*a^6*b^2*c^3*d^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)*x^6*a*b^7*c^5*d^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)*x^2*a^2*b^6*c^8-132
*x^5*a^2*b^5*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+470*x^5*a*b^6*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(
1/2)+660*x^4*a^5*b^2*c*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+372*x^4*a^4*b^3*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)-456*x^4*a^3*b^4*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+372*x^4*a^2*b^5*c^4*d^3*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+660*x^4*a*b^6*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-90*x^3*a^6*b*c*d^6*(a*c)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)+1098*x^3*a^5*b^2*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*x^3*a^4*b^3*c^3*d^4
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*x^3*a^3*b^4*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1098*x^3*a^2*
b^5*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-90*x^3*a*b^6*c^6*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+552*x^2
*a^6*b*c^2*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*x^2*a^5*b^2*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-
336*x^2*a^4*b^3*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*x^2*a^3*b^4*c^5*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+552*x^2*a^2*b^5*c^6*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+126*x*a^6*b*c^3*d^4*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)-84*x*a^5*b^2*c^4*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(a*d-b*c)^4/(a*
c)^(1/2)/x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2)

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError