Integrand size = 22, antiderivative size = 380 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=\frac {5 d \left (b^3 c^3+45 a b^2 c^2 d+19 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2}-\frac {5 (3 b c+a d) \left (b^2 c^2+24 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 a c^2 x}-\frac {5 \left (3 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{96 c^2 x^2}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}+\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
-5/24*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/c/x^3-1/4*(b*x+a)^(5/2)*(d*x+c )^(5/2)/x^4+5/64*(a^4*d^4-20*a^3*b*c*d^3-90*a^2*b^2*c^2*d^2-20*a*b^3*c^3*d +b^4*c^4)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2)/c^( 3/2)+5*b^(3/2)*d^(3/2)*(a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d* x+c)^(1/2))-5/192*(a*d+3*b*c)*(-a^2*d^2+24*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)* (b*x+a)^(1/2)/a/c^2/x-5/96*(-a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(d*x+c)^(5/2)*( b*x+a)^(1/2)/c^2/x^2+5/64*d*(-a^3*d^3+19*a^2*b*c*d^2+45*a*b^2*c^2*d+b^3*c^ 3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^2
Time = 1.18 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3+a b^2 c x^2 \left (118 c^2+601 c d x-192 d^2 x^2\right )+a^2 b c x \left (136 c^2+452 c d x+601 d^2 x^2\right )+a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a c x^4}+\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \]
-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 + a*b^2*c*x^2*(118*c^2 + 601*c*d*x - 192*d^2*x^2) + a^2*b*c*x*(136*c^2 + 452*c*d*x + 601*d^2*x^2 ) + a^3*(48*c^3 + 136*c^2*d*x + 118*c*d^2*x^2 + 15*d^3*x^3)))/(a*c*x^4) + (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d ^4)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(64*a^(3/2)* c^(3/2)) + 5*b^(3/2)*d^(3/2)*(b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/( Sqrt[d]*Sqrt[a + b*x])]
Time = 0.60 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 171, 27, 175, 66, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{4} \int \frac {5 (a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{2 x^4}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{x^4}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {5}{8} \left (\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b d c-a^2 d^2+2 b d (7 b c+a d) x\right )}{2 x^3}dx}{3 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} \left (\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b d c-a^2 d^2+2 b d (7 b c+a d) x\right )}{x^3}dx}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {\int \frac {(c+d x)^{3/2} \left ((3 b c+a d) \left (b^2 c^2+24 a b d c-a^2 d^2\right )+2 b d \left (31 b^2 c^2+18 a b d c-a^2 d^2\right ) x\right )}{2 x^2 \sqrt {a+b x}}dx}{2 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {\int \frac {(c+d x)^{3/2} \left ((3 b c+a d) \left (b^2 c^2+24 a b d c-a^2 d^2\right )+2 b d \left (31 b^2 c^2+18 a b d c-a^2 d^2\right ) x\right )}{x^2 \sqrt {a+b x}}dx}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {\frac {\int -\frac {3 \sqrt {c+d x} \left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+a^4 d^4-2 b d \left (b^3 c^3+45 a b^2 d c^2+19 a^2 b d^2 c-a^3 d^3\right ) x\right )}{2 x \sqrt {a+b x}}dx}{a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {3 \int \frac {\sqrt {c+d x} \left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+a^4 d^4-2 b d \left (b^3 c^3+45 a b^2 d c^2+19 a^2 b d^2 c-a^3 d^3\right ) x\right )}{x \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {3 \left (\frac {\int \frac {b c \left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c-64 a b^2 d^2 (b c+a d) x c+a^4 d^4\right )}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}-2 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )\right )}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {3 \left (c \int \frac {b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c-64 a b^2 d^2 (b c+a d) x c+a^4 d^4}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )\right )}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {3 \left (c \left (\left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-64 a b^2 c d^2 (a d+b c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx\right )-2 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )\right )}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {3 \left (c \left (\left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-128 a b^2 c d^2 (a d+b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-2 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )\right )}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {3 \left (c \left (2 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-128 a b^2 c d^2 (a d+b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-2 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )\right )}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5}{8} \left (\frac {\frac {-\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{a x}-\frac {3 \left (c \left (-\frac {2 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}-128 a b^{3/2} c d^{3/2} (a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\right )-2 d \sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )\right )}{2 a}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{2 c x^2}}{6 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}\) |
-1/4*((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^4 + (5*(-1/3*((b*c + a*d)*(a + b* x)^(3/2)*(c + d*x)^(5/2))/(c*x^3) + (-1/2*((3*b^2*c^2 + 14*a*b*c*d - a^2*d ^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(c*x^2) + (-(((3*b*c + a*d)*(b^2*c^2 + 24*a*b*c*d - a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(a*x)) - (3*(-2*d*(b^ 3*c^3 + 45*a*b^2*c^2*d + 19*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x] + c*((-2*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c* d^3 + a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/( Sqrt[a]*Sqrt[c]) - 128*a*b^(3/2)*c*d^(3/2)*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sq rt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])))/(2*a))/(4*c))/(6*c)))/8
3.7.71.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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. \(828\) vs. \(2(324)=648\).
Time = 0.64 (sec) , antiderivative size = 829, normalized size of antiderivative = 2.18
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (960 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c \,d^{3} x^{4} \sqrt {a c}+960 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{2} d^{2} x^{4} \sqrt {a c}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4} \sqrt {b d}-300 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4} \sqrt {b d}-1350 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4} \sqrt {b d}-300 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4} \sqrt {b d}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4} \sqrt {b d}+384 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c \,d^{2} x^{4}-30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}-1202 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-1202 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-904 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{2} d x -272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 a c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {b d}\, \sqrt {a c}}\) | \(829\) |
1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c*(960*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c*d^3*x^4*(a*c)^(1/2) +960*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^ (1/2))*a*b^3*c^2*d^2*x^4*(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)*a^4*d^4*x^4*(b*d)^(1/2)-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)*a^3*b*c*d^3*x^4*(b*d)^(1/2 )-1350*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^2*x^4*(b*d)^(1/2)-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)*a*b^3*c^3*d*x^4*(b*d)^(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)*b^4*c^4*x^4*(b*d)^(1/2)+384*(b *d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c*d^2*x^4-30*(b*d)^(1/ 2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3*x^3-1202*(b*d)^(1/2)*(a*c)^ (1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3-1202*(b*d)^(1/2)*(a*c)^(1/2) *((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d*x^3-30*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+ a)*(d*x+c))^(1/2)*b^3*c^3*x^3-236*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c )^(1/2)*a^3*c*d^2*x^2-904*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)* a^2*b*c^2*d*x^2-236*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2* c^3*x^2-272*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x-27 2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*c^3*x-96*((b*x+a)* (d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c^3)/((b*x+a)*(d*x+c))^(1/2)...
Time = 4.71 (sec) , antiderivative size = 1633, normalized size of antiderivative = 4.30 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Too large to display} \]
[1/768*(960*(a^2*b^2*c^3*d + a^3*b*c^2*d^2)*sqrt(b*d)*x^4*log(8*b^2*d^2*x^ 2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt (b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 15*(b^4*c^4 - 20*a*b^ 3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*c)*x^4*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*(192*a^2*b^2*c^2*d^2*x^4 - 48*a^4*c^4 - (15*a*b^3*c^4 + 601*a^2*b^2*c^ 3*d + 601*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(59*a^2*b^2*c^4 + 226*a^3* b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 136*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^2*x^4), -1/768*(1920*(a^2*b^2*c^3*d + a^3*b*c^2*d ^2)*sqrt(-b*d)*x^4*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 15*(b^ 4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sq rt(a*c)*x^4*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*(192*a^2*b^2*c^2*d^2*x^4 - 48*a^4*c^4 - (15*a*b^3*c^4 + 601*a^2*b^2*c^3*d + 601*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(59*a^2*b^2 *c^4 + 226*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 136*(a^3*b*c^4 + a^4*c^3*d) *x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^2*x^4), -1/384*(15*(b^4*c^4 - 20*a *b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a*c)*...
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Exception raised: ValueError} \]
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. 3937 vs. \(2 (324) = 648\).
Time = 1.79 (sec) , antiderivative size = 3937, normalized size of antiderivative = 10.36 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Too large to display} \]
1/192*(192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*b*d^2*abs(b) - 480*(sqrt(b*d)*b^2*c*d*abs(b) + sqrt(b*d)*a*b*d^2*abs(b))*log((sqrt(b*d) *sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2) + 15*(sqrt(b*d)*b ^5*c^4*abs(b) - 20*sqrt(b*d)*a*b^4*c^3*d*abs(b) - 90*sqrt(b*d)*a^2*b^3*c^2 *d^2*abs(b) - 20*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) + sqrt(b*d)*a^4*b*d^4*abs( b))*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*b*c) - 2*( 15*sqrt(b*d)*b^19*c^11*abs(b) + 481*sqrt(b*d)*a*b^18*c^10*d*abs(b) - 3787* sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) + 11195*sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b ) - 15898*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) + 7994*sqrt(b*d)*a^5*b^14*c^6* d^5*abs(b) + 7994*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) - 15898*sqrt(b*d)*a^7* b^12*c^4*d^7*abs(b) + 11195*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) - 3787*sqrt( b*d)*a^9*b^10*c^2*d^9*abs(b) + 481*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) + 15*s qrt(b*d)*a^11*b^8*d^11*abs(b) - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) - 3446*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^16*c^9 *d*abs(b) + 15371*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^15*c^8*d^2*abs(b) - 18056*sqrt(b*d)*(sqrt(b*d)*s qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^14*c^7*d^3*abs (b) - 8354*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*...
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^5} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^5} \,d x \]