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

Optimal. Leaf size=388 \[ -\frac{d \sqrt{a+b x} \left (1715 a^2 b c d^2-1155 a^3 d^3-581 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^6 \sqrt{c+d x}}-\frac{d \sqrt{a+b x} (b c-a d) \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{64 a c^5 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (b c-a d) \left (231 a^2 d^2-156 a b c d+5 b^2 c^2\right )}{64 a c^4 x (c+d x)^{3/2}}+\frac{5 (b c-a d) \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{13/2}}-\frac{\sqrt{a+b x} (59 b c-99 a d) (b c-a d)}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{11 a \sqrt{a+b x} (b c-a d)}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}} \]

[Out]

-(d*(b*c - a*d)*(5*b^2*c^2 - 238*a*b*c*d + 385*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^5*(c + d*x)^(3/2)) - (11*a*(b*c
- a*d)*Sqrt[a + b*x])/(24*c^2*x^3*(c + d*x)^(3/2)) - ((59*b*c - 99*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^
2*(c + d*x)^(3/2)) - ((b*c - a*d)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^4*x*(c + d*x)
^(3/2)) - (a*(a + b*x)^(3/2))/(4*c*x^4*(c + d*x)^(3/2)) - (d*(5*b^3*c^3 - 581*a*b^2*c^2*d + 1715*a^2*b*c*d^2 -
1155*a^3*d^3)*Sqrt[a + b*x])/(64*a*c^6*Sqrt[c + d*x]) + (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*
c*d^2 + 231*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.513979, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {98, 149, 151, 152, 12, 93, 208} \[ -\frac{d \sqrt{a+b x} \left (1715 a^2 b c d^2-1155 a^3 d^3-581 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^6 \sqrt{c+d x}}-\frac{d \sqrt{a+b x} (b c-a d) \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{64 a c^5 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (b c-a d) \left (231 a^2 d^2-156 a b c d+5 b^2 c^2\right )}{64 a c^4 x (c+d x)^{3/2}}+\frac{5 (b c-a d) \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{13/2}}-\frac{\sqrt{a+b x} (59 b c-99 a d) (b c-a d)}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{11 a \sqrt{a+b x} (b c-a d)}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(b*c - a*d)*(5*b^2*c^2 - 238*a*b*c*d + 385*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^5*(c + d*x)^(3/2)) - (11*a*(b*c
- a*d)*Sqrt[a + b*x])/(24*c^2*x^3*(c + d*x)^(3/2)) - ((59*b*c - 99*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^
2*(c + d*x)^(3/2)) - ((b*c - a*d)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^4*x*(c + d*x)
^(3/2)) - (a*(a + b*x)^(3/2))/(4*c*x^4*(c + d*x)^(3/2)) - (d*(5*b^3*c^3 - 581*a*b^2*c^2*d + 1715*a^2*b*c*d^2 -
1155*a^3*d^3)*Sqrt[a + b*x])/(64*a*c^6*Sqrt[c + d*x]) + (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*
c*d^2 + 231*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(13/2))

Rule 98

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

Rule 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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{(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx &=-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{\int \frac{\sqrt{a+b x} \left (-\frac{11}{2} a (b c-a d)-4 b (b c-a d) x\right )}{x^4 (c+d x)^{5/2}} \, dx}{4 c}\\ &=-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{\int \frac{-\frac{1}{4} a (59 b c-99 a d) (b c-a d)-2 b (6 b c-11 a d) (b c-a d) x}{x^3 \sqrt{a+b x} (c+d x)^{5/2}} \, dx}{12 c^2}\\ &=-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}+\frac{\int \frac{\frac{3}{8} a (b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right )-\frac{3}{4} a b d (59 b c-99 a d) (b c-a d) x}{x^2 \sqrt{a+b x} (c+d x)^{5/2}} \, dx}{24 a c^3}\\ &=-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{\int \frac{\frac{15}{16} a (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )+\frac{3}{4} a b d (b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) x}{x \sqrt{a+b x} (c+d x)^{5/2}} \, dx}{24 a^2 c^4}\\ &=-\frac{d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}+\frac{\int \frac{-\frac{45}{32} a (b c-a d)^2 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )-\frac{9}{16} a b d (b c-a d)^2 \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) x}{x \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{36 a^2 c^5 (b c-a d)}\\ &=-\frac{d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt{a+b x}}{64 a c^6 \sqrt{c+d x}}-\frac{\int \frac{45 a (b c-a d)^3 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )}{64 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{18 a^2 c^6 (b c-a d)^2}\\ &=-\frac{d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt{a+b x}}{64 a c^6 \sqrt{c+d x}}-\frac{\left (5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a c^6}\\ &=-\frac{d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt{a+b x}}{64 a c^6 \sqrt{c+d x}}-\frac{\left (5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 a c^6}\\ &=-\frac{d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac{11 a (b c-a d) \sqrt{a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{(59 b c-99 a d) (b c-a d) \sqrt{a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt{a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac{d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt{a+b x}}{64 a c^6 \sqrt{c+d x}}+\frac{5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.660184, size = 255, normalized size = 0.66 \[ \frac{x^2 \left (2 c^{7/2} (a+b x)^{7/2} \left (-99 a^2 d^2+26 a b c d+b^2 c^2\right )+x \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right ) \left (3 c^{5/2} (a+b x)^{5/2}-5 x (b c-a d) \left (\sqrt{c} \sqrt{a+b x} (4 a c+3 a d x+b c x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )\right )-48 a^2 c^{11/2} (a+b x)^{7/2}+8 a c^{9/2} x (a+b x)^{7/2} (11 a d+b c)}{192 a^3 c^{13/2} x^4 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*a^2*c^(11/2)*(a + b*x)^(7/2) + 8*a*c^(9/2)*(b*c + 11*a*d)*x*(a + b*x)^(7/2) + x^2*(2*c^(7/2)*(b^2*c^2 + 2
6*a*b*c*d - 99*a^2*d^2)*(a + b*x)^(7/2) + (b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*x*(3*c^(5
/2)*(a + b*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[c]*Sqrt[a + b*x]*(4*a*c + b*c*x + 3*a*d*x) - 3*a^(3/2)*(c + d*x)^(
3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))))/(192*a^3*c^(13/2)*x^4*(c + d*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.035, size = 1377, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(b*x+a)^(1/2)*(10290*x^5*a^2*b*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-3486*x^5*a*b^2*c^2*d^3*((b*x+a
)*(d*x+c))^(1/2)*(a*c)^(1/2)+14028*x^4*a^2*b*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-4944*x^4*a*b^2*c^3*d^
2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+2322*x^3*a^2*b*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30*x^5*b^3*c^
3*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-15*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^4*c^4*d^2-12600*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*c^2*d^4+6300*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^2*c^3*d^3-600*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^3*c^4*d^2-6300*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*c^3*d^3+3150*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^2*c^4*d^2-300*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^3*c^5*d-966*x^3*a*b^
2*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-632*x^2*a^2*b*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+96*a^3*c^5
*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+3465*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*d^6-15*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^4*c^6+6930*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*c*d^5-30*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^4*c^5*d+3465*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*c^
2*d^4-6930*x^5*a^3*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30*x^3*b^3*c^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-
9240*x^4*a^3*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+60*x^4*b^3*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-13
86*x^3*a^3*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+396*x^2*a^3*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)
+236*x^2*a*b^2*c^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-176*x*a^3*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+272
*x*a^2*b*c^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6300*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*c*d^5+3150*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^2*c^2*d
^4-300*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^3*c^3*d^3)/c^6/a/((b*x+a)*(d*x+
c))^(1/2)/(a*c)^(1/2)/x^4/(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((b*x+a)^(5/2)/x^5/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 166.314, size = 2396, normalized size = 6.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*((b^4*c^4*d^2 + 20*a*b^3*c^3*d^3 - 210*a^2*b^2*c^2*d^4 + 420*a^3*b*c*d^5 - 231*a^4*d^6)*x^6 + 2*(b
^4*c^5*d + 20*a*b^3*c^4*d^2 - 210*a^2*b^2*c^3*d^3 + 420*a^3*b*c^2*d^4 - 231*a^4*c*d^5)*x^5 + (b^4*c^6 + 20*a*b
^3*c^5*d - 210*a^2*b^2*c^4*d^2 + 420*a^3*b*c^3*d^3 - 231*a^4*c^2*d^4)*x^4)*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)*sqrt(d*x + c) + 8*(a*b*c^2 + a
^2*c*d)*x)/x^2) + 4*(48*a^4*c^6 + 3*(5*a*b^3*c^4*d^2 - 581*a^2*b^2*c^3*d^3 + 1715*a^3*b*c^2*d^4 - 1155*a^4*c*d
^5)*x^5 + 6*(5*a*b^3*c^5*d - 412*a^2*b^2*c^4*d^2 + 1169*a^3*b*c^3*d^3 - 770*a^4*c^2*d^4)*x^4 + 3*(5*a*b^3*c^6
- 161*a^2*b^2*c^5*d + 387*a^3*b*c^4*d^2 - 231*a^4*c^3*d^3)*x^3 + 2*(59*a^2*b^2*c^6 - 158*a^3*b*c^5*d + 99*a^4*
c^4*d^2)*x^2 + 8*(17*a^3*b*c^6 - 11*a^4*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^7*d^2*x^6 + 2*a^2*c^8*d*
x^5 + a^2*c^9*x^4), -1/384*(15*((b^4*c^4*d^2 + 20*a*b^3*c^3*d^3 - 210*a^2*b^2*c^2*d^4 + 420*a^3*b*c*d^5 - 231*
a^4*d^6)*x^6 + 2*(b^4*c^5*d + 20*a*b^3*c^4*d^2 - 210*a^2*b^2*c^3*d^3 + 420*a^3*b*c^2*d^4 - 231*a^4*c*d^5)*x^5
+ (b^4*c^6 + 20*a*b^3*c^5*d - 210*a^2*b^2*c^4*d^2 + 420*a^3*b*c^3*d^3 - 231*a^4*c^2*d^4)*x^4)*sqrt(-a*c)*arcta
n(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*(48*a^4*c^6 + 3*(5*a*b^3*c^4*d^2 - 581*a^2*b^2*c^3*d^3 + 1715*a^3*b*c^2*d^4 - 1155*a^4*c*d^5)*x^5
+ 6*(5*a*b^3*c^5*d - 412*a^2*b^2*c^4*d^2 + 1169*a^3*b*c^3*d^3 - 770*a^4*c^2*d^4)*x^4 + 3*(5*a*b^3*c^6 - 161*a^
2*b^2*c^5*d + 387*a^3*b*c^4*d^2 - 231*a^4*c^3*d^3)*x^3 + 2*(59*a^2*b^2*c^6 - 158*a^3*b*c^5*d + 99*a^4*c^4*d^2)
*x^2 + 8*(17*a^3*b*c^6 - 11*a^4*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^7*d^2*x^6 + 2*a^2*c^8*d*x^5 + a^
2*c^9*x^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError