Integrand size = 22, antiderivative size = 388 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=-\frac {d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt {a+b x}}{64 a c^6 \sqrt {c+d x}}+\frac {5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{13/2}} \]
-1/4*a*(b*x+a)^(3/2)/c/x^4/(d*x+c)^(3/2)+5/64*(-a*d+b*c)*(231*a^3*d^3-189* a^2*b*c*d^2+21*a*b^2*c^2*d+b^3*c^3)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/ (d*x+c)^(1/2))/a^(3/2)/c^(13/2)-1/64*d*(-a*d+b*c)*(385*a^2*d^2-238*a*b*c*d +5*b^2*c^2)*(b*x+a)^(1/2)/a/c^5/(d*x+c)^(3/2)-11/24*a*(-a*d+b*c)*(b*x+a)^( 1/2)/c^2/x^3/(d*x+c)^(3/2)-1/96*(-99*a*d+59*b*c)*(-a*d+b*c)*(b*x+a)^(1/2)/ c^3/x^2/(d*x+c)^(3/2)-1/64*(-a*d+b*c)*(231*a^2*d^2-156*a*b*c*d+5*b^2*c^2)* (b*x+a)^(1/2)/a/c^4/x/(d*x+c)^(3/2)-1/64*d*(-1155*a^3*d^3+1715*a^2*b*c*d^2 -581*a*b^2*c^2*d+5*b^3*c^3)*(b*x+a)^(1/2)/a/c^6/(d*x+c)^(1/2)
Time = 10.48 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.66 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=\frac {-48 a^2 c^{11/2} (a+b x)^{7/2}+8 a c^{9/2} (b c+11 a d) x (a+b x)^{7/2}+x^2 \left (2 c^{7/2} \left (b^2 c^2+26 a b c d-99 a^2 d^2\right ) (a+b x)^{7/2}+\left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) x \left (3 c^{5/2} (a+b x)^{5/2}-5 (b c-a d) x \left (\sqrt {c} \sqrt {a+b x} (4 a c+b c x+3 a d x)-3 a^{3/2} (c+d x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )\right )}{192 a^3 c^{13/2} x^4 (c+d x)^{3/2}} \]
(-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 + 26*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])/(Sq rt[a]*Sqrt[c + d*x])]))))/(192*a^3*c^(13/2)*x^4*(c + d*x)^(3/2))
Time = 0.55 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {109, 27, 27, 166, 27, 168, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {a+b x} (11 a (b c-a d)+8 b x (b c-a d))}{2 x^4 (c+d x)^{5/2}}dx}{4 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(b c-a d) \sqrt {a+b x} (11 a+8 b x)}{x^4 (c+d x)^{5/2}}dx}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {a+b x} (11 a+8 b x)}{x^4 (c+d x)^{5/2}}dx}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {a (59 b c-99 a d)+8 b (6 b c-11 a d) x}{2 x^3 \sqrt {a+b x} (c+d x)^{5/2}}dx}{3 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {a (59 b c-99 a d)+8 b (6 b c-11 a d) x}{x^3 \sqrt {a+b x} (c+d x)^{5/2}}dx}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {-\frac {\int -\frac {3 a \left (5 b^2 c^2-156 a b d c+231 a^2 d^2-2 b d (59 b c-99 a d) x\right )}{2 x^2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 a c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \int \frac {5 b^2 c^2-156 a b d c+231 a^2 d^2-2 b d (59 b c-99 a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\int \frac {5 \left (b^3 c^3+21 a b^2 d c^2-189 a^2 b d^2 c+231 a^3 d^3\right )+4 b d \left (5 b^2 c^2-156 a b d c+231 a^2 d^2\right ) x}{2 x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\int \frac {5 \left (b^3 c^3+21 a b^2 d c^2-189 a^2 b d^2 c+231 a^3 d^3\right )+4 b d \left (5 b^2 c^2-156 a b d c+231 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\frac {2 d \sqrt {a+b x} \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{c (c+d x)^{3/2}}-\frac {2 \int -\frac {3 (b c-a d) \left (5 \left (b^3 c^3+21 a b^2 d c^2-189 a^2 b d^2 c+231 a^3 d^3\right )+2 b d \left (5 b^2 c^2-238 a b d c+385 a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\frac {\int \frac {5 \left (b^3 c^3+21 a b^2 d c^2-189 a^2 b d^2 c+231 a^3 d^3\right )+2 b d \left (5 b^2 c^2-238 a b d c+385 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{c}+\frac {2 d \sqrt {a+b x} \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{c (c+d x)^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\frac {\frac {2 d \sqrt {a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {5 (b c-a d) \left (b^3 c^3+21 a b^2 d c^2-189 a^2 b d^2 c+231 a^3 d^3\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{c}+\frac {2 d \sqrt {a+b x} \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{c (c+d x)^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\frac {\frac {5 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}}{c}+\frac {2 d \sqrt {a+b x} \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{c (c+d x)^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\frac {\frac {10 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}}{c}+\frac {2 d \sqrt {a+b x} \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{c (c+d x)^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {\frac {3 \left (-\frac {\frac {2 d \sqrt {a+b x} \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{c (c+d x)^{3/2}}+\frac {\frac {2 d \sqrt {a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {10 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{c}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {5 b^2 c}{a}+\frac {231 a d^2}{c}-156 b d\right )}{x (c+d x)^{3/2}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-99 a d)}{2 c x^2 (c+d x)^{3/2}}}{6 c}-\frac {11 a \sqrt {a+b x}}{3 c x^3 (c+d x)^{3/2}}\right )}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}\) |
-1/4*(a*(a + b*x)^(3/2))/(c*x^4*(c + d*x)^(3/2)) + ((b*c - a*d)*((-11*a*Sq rt[a + b*x])/(3*c*x^3*(c + d*x)^(3/2)) + (-1/2*((59*b*c - 99*a*d)*Sqrt[a + b*x])/(c*x^2*(c + d*x)^(3/2)) + (3*(-((((5*b^2*c)/a - 156*b*d + (231*a*d^ 2)/c)*Sqrt[a + b*x])/(x*(c + d*x)^(3/2))) - ((2*d*(5*b^2*c^2 - 238*a*b*c*d + 385*a^2*d^2)*Sqrt[a + b*x])/(c*(c + d*x)^(3/2)) + ((2*d*(5*b^3*c^3 - 58 1*a*b^2*c^2*d + 1715*a^2*b*c*d^2 - 1155*a^3*d^3)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (10*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 23 1*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt [a]*c^(3/2)))/c)/(2*a*c)))/(4*c))/(6*c)))/(8*c)
3.7.98.3.1 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. \(1376\) vs. \(2(338)=676\).
Time = 0.56 (sec) , antiderivative size = 1377, normalized size of antiderivative = 3.55
-1/384*(b*x+a)^(1/2)*(236*a*b^2*c^5*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2 )-176*a^3*c^4*d*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+272*a^2*b*c^5*x*(a*c )^(1/2)*((b*x+a)*(d*x+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)*a^3*b*c*d^5*x^6+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)*a^2*b^2*c^2*d^4*x^6-300*ln((a*d*x+b* c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^3*c^3*d^3*x^6+60*( a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^4*d*x^4-1386*(a*c)^(1/2)*((b*x+a) *(d*x+c))^(1/2)*a^3*c^2*d^3*x^3+96*a^3*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+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)* a^4*d^6*x^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)*b^4*c^6*x^4-15*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^4*c^4*d^2*x^6-30*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^4*c^5*d*x^5+10290*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2) *a^2*b*c*d^4*x^5-3486*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d^3*x^ 5+14028*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c^2*d^3*x^4-4944*(a*c)^( 1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^3*d^2*x^4+2322*(a*c)^(1/2)*((b*x+a)*( d*x+c))^(1/2)*a^2*b*c^3*d^2*x^3-966*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a* b^2*c^4*d*x^3-632*a^2*b*c^4*d*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+30*( a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^3*d^2*x^5-9240*(a*c)^(1/2)*((b*x+ a)*(d*x+c))^(1/2)*a^3*c*d^4*x^4-12600*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b...
Time = 9.92 (sec) , antiderivative size = 1100, normalized size of antiderivative = 2.84 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (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}\right )} x^{6} + 2 \, {\left (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}\right )} x^{5} + {\left (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}\right )} x^{4}\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 (48 \, a^{4} c^{6} + 3 \, {\left (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}\right )} x^{5} + 6 \, {\left (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}\right )} x^{4} + 3 \, {\left (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}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{6} - 158 \, a^{3} b c^{5} d + 99 \, a^{4} c^{4} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{6} - 11 \, a^{4} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{2} c^{7} d^{2} x^{6} + 2 \, a^{2} c^{8} d x^{5} + a^{2} c^{9} x^{4}\right )}}, -\frac {15 \, {\left ({\left (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}\right )} x^{6} + 2 \, {\left (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}\right )} x^{5} + {\left (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}\right )} x^{4}\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 (48 \, a^{4} c^{6} + 3 \, {\left (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}\right )} x^{5} + 6 \, {\left (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}\right )} x^{4} + 3 \, {\left (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}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{6} - 158 \, a^{3} b c^{5} d + 99 \, a^{4} c^{4} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{6} - 11 \, a^{4} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{2} c^{7} d^{2} x^{6} + 2 \, a^{2} c^{8} d x^{5} + a^{2} c^{9} x^{4}\right )}}\right ] \]
[-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)*s qrt(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 + 1 715*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)*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* (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...
\[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{5} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 3915 vs. \(2 (338) = 676\).
Time = 4.35 (sec) , antiderivative size = 3915, normalized size of antiderivative = 10.09 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
2/3*sqrt(b*x + a)*((8*b^6*c^9*d^4*abs(b) - 31*a*b^5*c^8*d^5*abs(b) + 38*a^ 2*b^4*c^7*d^6*abs(b) - 15*a^3*b^3*c^6*d^7*abs(b))*(b*x + a)/(b^3*c^13*d - a*b^2*c^12*d^2) + 3*(3*b^7*c^10*d^3*abs(b) - 14*a*b^6*c^9*d^4*abs(b) + 24* a^2*b^5*c^8*d^5*abs(b) - 18*a^3*b^4*c^7*d^6*abs(b) + 5*a^4*b^3*c^6*d^7*abs (b))/(b^3*c^13*d - a*b^2*c^12*d^2))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 5/64*(sqrt(b*d)*b^6*c^4 + 20*sqrt(b*d)*a*b^5*c^3*d - 210*sqrt(b*d)*a^2*b ^4*c^2*d^2 + 420*sqrt(b*d)*a^3*b^3*c*d^3 - 231*sqrt(b*d)*a^4*b^2*d^4)*arct an(-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*b*c^6*abs(b)) - 1/ 96*(15*sqrt(b*d)*b^20*c^11 - 839*sqrt(b*d)*a*b^19*c^10*d + 8373*sqrt(b*d)* a^2*b^18*c^9*d^2 - 40125*sqrt(b*d)*a^3*b^17*c^8*d^3 + 115302*sqrt(b*d)*a^4 *b^16*c^7*d^4 - 217686*sqrt(b*d)*a^5*b^15*c^6*d^5 + 281274*sqrt(b*d)*a^6*b ^14*c^5*d^6 - 251658*sqrt(b*d)*a^7*b^13*c^4*d^7 + 153915*sqrt(b*d)*a^8*b^1 2*c^3*d^8 - 61587*sqrt(b*d)*a^9*b^11*c^2*d^9 + 14561*sqrt(b*d)*a^10*b^10*c *d^10 - 1545*sqrt(b*d)*a^11*b^9*d^11 - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^18*c^10 + 5794*sqrt(b*d)*(s qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^17*c^9 *d - 44109*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^16*c^8*d^2 + 145304*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^15*c^7*d^3 - 245954*sqrt...
Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^5\,{\left (c+d\,x\right )}^{5/2}} \,d x \]