Integrand size = 22, antiderivative size = 278 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=-\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{24 c^5 \sqrt {c+d x}}-\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{11/2}} \]
-1/3*a*(b*x+a)^(3/2)/c/x^3/(d*x+c)^(3/2)-5/8*(-a*d+b*c)*(21*a^2*d^2-14*a*b *c*d+b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(11/2 )/a^(1/2)-7/24*d*(-15*a*d+7*b*c)*(-a*d+b*c)*(b*x+a)^(1/2)/c^4/(d*x+c)^(3/2 )-3/4*a*(-a*d+b*c)*(b*x+a)^(1/2)/c^2/x^2/(d*x+c)^(3/2)-1/8*(-21*a*d+11*b*c )*(-a*d+b*c)*(b*x+a)^(1/2)/c^3/x/(d*x+c)^(3/2)-1/24*d*(315*a^2*d^2-420*a*b *c*d+113*b^2*c^2)*(b*x+a)^(1/2)/c^5/(d*x+c)^(1/2)
Time = 10.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\frac {-8 a c^{9/2} (a+b x)^{7/2}-2 c^{7/2} (b c-9 a d) x (a+b x)^{7/2}-\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) x^2 \left (3 c^{5/2} (a+b x)^{5/2}-5 (b c-a d) x \left (\sqrt {c} \sqrt {a+b x} (4 a c+b c x+3 a d x)-3 a^{3/2} (c+d x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{24 a^2 c^{11/2} x^3 (c+d x)^{3/2}} \]
(-8*a*c^(9/2)*(a + b*x)^(7/2) - 2*c^(7/2)*(b*c - 9*a*d)*x*(a + b*x)^(7/2) - (b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*x^2*(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])])) )/(24*a^2*c^(11/2)*x^3*(c + d*x)^(3/2))
Time = 0.45 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {109, 27, 27, 166, 27, 168, 27, 169, 27, 169, 27, 104, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int -\frac {3 \sqrt {a+b x} (3 a (b c-a d)+2 b x (b c-a d))}{2 x^3 (c+d x)^{5/2}}dx}{3 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(b c-a d) \sqrt {a+b x} (3 a+2 b x)}{x^3 (c+d x)^{5/2}}dx}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {a+b x} (3 a+2 b x)}{x^3 (c+d x)^{5/2}}dx}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {a (11 b c-21 a d)+2 b (4 b c-9 a d) x}{2 x^2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {a (11 b c-21 a d)+2 b (4 b c-9 a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {-\frac {\int -\frac {a \left (5 \left (b^2 c^2-14 a b d c+21 a^2 d^2\right )-4 b d (11 b c-21 a d) x\right )}{2 x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\int \frac {5 \left (b^2 c^2-14 a b d c+21 a^2 d^2\right )-4 b d (11 b c-21 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {-\frac {2 \int -\frac {(b c-a d) \left (15 \left (b^2 c^2-14 a b d c+21 a^2 d^2\right )-14 b d (7 b c-15 a d) x\right )}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}-\frac {14 d \sqrt {a+b x} (7 b c-15 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\frac {\int \frac {15 \left (b^2 c^2-14 a b d c+21 a^2 d^2\right )-14 b d (7 b c-15 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c}-\frac {14 d \sqrt {a+b x} (7 b c-15 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\frac {-\frac {2 \int -\frac {15 (b c-a d) \left (b^2 c^2-14 a b d c+21 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}-\frac {2 d \sqrt {a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {14 d \sqrt {a+b x} (7 b c-15 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\frac {\frac {15 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}-\frac {2 d \sqrt {a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {14 d \sqrt {a+b x} (7 b c-15 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\frac {\frac {30 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}-\frac {2 d \sqrt {a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {14 d \sqrt {a+b x} (7 b c-15 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {\frac {-\frac {30 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {2 d \sqrt {a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {14 d \sqrt {a+b x} (7 b c-15 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (11 b c-21 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {3 a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\right )}{2 c}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}\) |
-1/3*(a*(a + b*x)^(3/2))/(c*x^3*(c + d*x)^(3/2)) + ((b*c - a*d)*((-3*a*Sqr t[a + b*x])/(2*c*x^2*(c + d*x)^(3/2)) + (-(((11*b*c - 21*a*d)*Sqrt[a + b*x ])/(c*x*(c + d*x)^(3/2))) + ((-14*d*(7*b*c - 15*a*d)*Sqrt[a + b*x])/(3*c*( c + d*x)^(3/2)) + ((-2*d*(113*b^2*c^2 - 420*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (30*(b^2*c^2 - 14*a*b*c*d + 21*a^2 *d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c ^(3/2)))/(3*c))/(2*c))/(4*c)))/(2*c)
3.7.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[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] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> 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] + Simp[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])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] - Simp[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] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1008\) vs. \(2(234)=468\).
Time = 0.57 (sec) , antiderivative size = 1009, normalized size of antiderivative = 3.63
1/48*(b*x+a)^(1/2)*(315*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*d^5*x^5-525*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*c*d^4*x^5+225*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^2*c^2*d^3*x^5-15*ln((a*d*x+b*c*x+2*(a*c) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*d^2*x^5+630*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*c*d^4*x^4-1050*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*c^2*d^3* x^4+450*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^2*c^3*d^2*x^4-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2 *a*c)/x)*b^3*c^4*d*x^4+315*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*c^2*d^3*x^3-525*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*c^3*d^2*x^3+225*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^2*c^4*d*x^3-15*ln((a*d*x+b*c*x+ 2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^5*x^3-630*(a*c)^(1/2 )*((b*x+a)*(d*x+c))^(1/2)*a^2*d^4*x^4+840*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 /2)*a*b*c*d^3*x^4-226*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*d^2*x^4- 840*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d^3*x^3+1148*(a*c)^(1/2)*((b *x+a)*(d*x+c))^(1/2)*a*b*c^2*d^2*x^3-324*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/ 2)*b^2*c^3*d*x^3-126*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*d^2*x^2+1 92*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^3*d*x^2-66*(a*c)^(1/2)*((b...
Time = 3.90 (sec) , antiderivative size = 842, normalized size of antiderivative = 3.03 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 35 \, a^{2} b c d^{4} - 21 \, a^{3} d^{5}\right )} x^{5} + 2 \, {\left (b^{3} c^{4} d - 15 \, a b^{2} c^{3} d^{2} + 35 \, a^{2} b c^{2} d^{3} - 21 \, a^{3} c d^{4}\right )} x^{4} + {\left (b^{3} c^{5} - 15 \, a b^{2} c^{4} d + 35 \, a^{2} b c^{3} d^{2} - 21 \, a^{3} c^{2} d^{3}\right )} x^{3}\right )} \sqrt {a c} \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 (8 \, a^{3} c^{5} + {\left (113 \, a b^{2} c^{3} d^{2} - 420 \, a^{2} b c^{2} d^{3} + 315 \, a^{3} c d^{4}\right )} x^{4} + 2 \, {\left (81 \, a b^{2} c^{4} d - 287 \, a^{2} b c^{3} d^{2} + 210 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a b^{2} c^{5} - 32 \, a^{2} b c^{4} d + 21 \, a^{3} c^{3} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{5} - 9 \, a^{3} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a c^{6} d^{2} x^{5} + 2 \, a c^{7} d x^{4} + a c^{8} x^{3}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 35 \, a^{2} b c d^{4} - 21 \, a^{3} d^{5}\right )} x^{5} + 2 \, {\left (b^{3} c^{4} d - 15 \, a b^{2} c^{3} d^{2} + 35 \, a^{2} b c^{2} d^{3} - 21 \, a^{3} c d^{4}\right )} x^{4} + {\left (b^{3} c^{5} - 15 \, a b^{2} c^{4} d + 35 \, a^{2} b c^{3} d^{2} - 21 \, a^{3} c^{2} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \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 (8 \, a^{3} c^{5} + {\left (113 \, a b^{2} c^{3} d^{2} - 420 \, a^{2} b c^{2} d^{3} + 315 \, a^{3} c d^{4}\right )} x^{4} + 2 \, {\left (81 \, a b^{2} c^{4} d - 287 \, a^{2} b c^{3} d^{2} + 210 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a b^{2} c^{5} - 32 \, a^{2} b c^{4} d + 21 \, a^{3} c^{3} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{5} - 9 \, a^{3} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{6} d^{2} x^{5} + 2 \, a c^{7} d x^{4} + a c^{8} x^{3}\right )}}\right ] \]
[-1/96*(15*((b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 35*a^2*b*c*d^4 - 21*a^3*d^5) *x^5 + 2*(b^3*c^4*d - 15*a*b^2*c^3*d^2 + 35*a^2*b*c^2*d^3 - 21*a^3*c*d^4)* x^4 + (b^3*c^5 - 15*a*b^2*c^4*d + 35*a^2*b*c^3*d^2 - 21*a^3*c^2*d^3)*x^3)* 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*(8*a^3*c^5 + (113*a*b^2*c^3*d^2 - 420*a^2*b*c^2*d^3 + 315 *a^3*c*d^4)*x^4 + 2*(81*a*b^2*c^4*d - 287*a^2*b*c^3*d^2 + 210*a^3*c^2*d^3) *x^3 + 3*(11*a*b^2*c^5 - 32*a^2*b*c^4*d + 21*a^3*c^3*d^2)*x^2 + 2*(13*a^2* b*c^5 - 9*a^3*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^6*d^2*x^5 + 2*a* c^7*d*x^4 + a*c^8*x^3), 1/48*(15*((b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 35*a^2 *b*c*d^4 - 21*a^3*d^5)*x^5 + 2*(b^3*c^4*d - 15*a*b^2*c^3*d^2 + 35*a^2*b*c^ 2*d^3 - 21*a^3*c*d^4)*x^4 + (b^3*c^5 - 15*a*b^2*c^4*d + 35*a^2*b*c^3*d^2 - 21*a^3*c^2*d^3)*x^3)*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*(8*a^3*c^5 + (113*a*b^2*c^3*d^2 - 420*a^2*b*c^2*d^3 + 315*a^3* c*d^4)*x^4 + 2*(81*a*b^2*c^4*d - 287*a^2*b*c^3*d^2 + 210*a^3*c^2*d^3)*x^3 + 3*(11*a*b^2*c^5 - 32*a^2*b*c^4*d + 21*a^3*c^3*d^2)*x^2 + 2*(13*a^2*b*c^5 - 9*a^3*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^6*d^2*x^5 + 2*a*c^7*d *x^4 + a*c^8*x^3)]
\[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{4} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 2354 vs. \(2 (234) = 468\).
Time = 5.39 (sec) , antiderivative size = 2354, normalized size of antiderivative = 8.47 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
-2/3*sqrt(b*x + a)*((5*b^6*c^8*d^3*abs(b) - 22*a*b^5*c^7*d^4*abs(b) + 29*a ^2*b^4*c^6*d^5*abs(b) - 12*a^3*b^3*c^5*d^6*abs(b))*(b*x + a)/(b^3*c^11*d - a*b^2*c^10*d^2) + 6*(b^7*c^9*d^2*abs(b) - 5*a*b^6*c^8*d^3*abs(b) + 9*a^2* b^5*c^7*d^4*abs(b) - 7*a^3*b^4*c^6*d^5*abs(b) + 2*a^4*b^3*c^5*d^6*abs(b))/ (b^3*c^11*d - a*b^2*c^10*d^2))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5/8 *(sqrt(b*d)*b^5*c^3 - 15*sqrt(b*d)*a*b^4*c^2*d + 35*sqrt(b*d)*a^2*b^3*c*d^ 2 - 21*sqrt(b*d)*a^3*b^2*d^3)*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))/(s qrt(-a*b*c*d)*b*c^5*abs(b)) - 1/12*(33*sqrt(b*d)*b^15*c^8 - 346*sqrt(b*d)* a*b^14*c^7*d + 1506*sqrt(b*d)*a^2*b^13*c^6*d^2 - 3618*sqrt(b*d)*a^3*b^12*c ^5*d^3 + 5300*sqrt(b*d)*a^4*b^11*c^4*d^4 - 4878*sqrt(b*d)*a^5*b^10*c^3*d^5 + 2766*sqrt(b*d)*a^6*b^9*c^2*d^6 - 886*sqrt(b*d)*a^7*b^8*c*d^7 + 123*sqrt (b*d)*a^8*b^7*d^8 - 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^13*c^7 + 1287*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^12*c^6*d - 3441*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^11 *c^5*d^2 + 3579*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^10*c^4*d^3 + 129*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^9*c^3*d^4 - 3291*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...
Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^4\,{\left (c+d\,x\right )}^{5/2}} \,d x \]