Integrand size = 22, antiderivative size = 316 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\frac {\left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
-1/4*(b*x+a)^(3/2)*(d*x+c)^(5/2)/x^4-1/64*(-5*a^4*d^4+60*a^3*b*c*d^3+90*a^ 2*b^2*c^2*d^2-20*a*b^3*c^3*d+3*b^4*c^4)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1 /2)/(d*x+c)^(1/2))/a^(5/2)/c^(3/2)+2*b^(3/2)*d^(5/2)*arctanh(d^(1/2)*(b*x+ a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))-1/96*(3*b^2*c/a+50*b*d-5*a*d^2/c)*(d*x+c)^ (3/2)*(b*x+a)^(1/2)/x^2-1/24*(5*a*d+3*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/c/x ^3+1/64*(5*a^3*d^3-55*a^2*b*c*d^2-17*a*b^2*c^2*d+3*b^3*c^3)*(b*x+a)^(1/2)* (d*x+c)^(1/2)/a^2/c/x
Time = 0.75 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-9 b^3 c^3 x^3+3 a b^2 c^2 x^2 (2 c+19 d x)+a^2 b c x \left (72 c^2+244 c d x+337 d^2 x^2\right )+a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a^2 c x^4}+\frac {\left (-3 b^4 c^4+20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-60 a^3 b c d^3+5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \]
-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*b^3*c^3*x^3 + 3*a*b^2*c^2*x^2*(2*c + 19*d*x) + a^2*b*c*x*(72*c^2 + 244*c*d*x + 337*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^2*c*x^4) + ((-3*b^4*c^4 + 2 0*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 5*a^4*d^4)*ArcTanh[( Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(64*a^(5/2)*c^(3/2)) + 2* b^(3/2)*d^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])]
Time = 0.48 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {108, 27, 166, 27, 166, 27, 166, 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.
\(\displaystyle \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} (3 b c+5 a d+8 b d x)}{2 x^4}dx-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} (3 b c+5 a d+8 b d x)}{x^4}dx-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {1}{8} \left (\frac {\int \frac {(c+d x)^{3/2} \left (3 c^2 b^2+48 c d x b^2+50 a c d b-5 a^2 d^2\right )}{2 x^3 \sqrt {a+b x}}dx}{3 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {\int \frac {(c+d x)^{3/2} \left (3 c^2 b^2+48 c d x b^2+50 a c d b-5 a^2 d^2\right )}{x^3 \sqrt {a+b x}}dx}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {1}{8} \left (\frac {\frac {\int -\frac {3 \sqrt {c+d x} \left (3 b^3 c^3-17 a b^2 d c^2-55 a^2 b d^2 c-64 a b^2 d^2 x c+5 a^3 d^3\right )}{2 x^2 \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \int \frac {\sqrt {c+d x} \left (3 b^3 c^3-17 a b^2 d c^2-55 a^2 b d^2 c-64 a b^2 d^2 x c+5 a^3 d^3\right )}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (\frac {\int -\frac {3 b^4 c^4-20 a b^3 d c^3+90 a^2 b^2 d^2 c^2+60 a^3 b d^3 c+128 a^2 b^2 d^3 x c-5 a^4 d^4}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {\int \frac {3 b^4 c^4-20 a b^3 d c^3+90 a^2 b^2 d^2 c^2+60 a^3 b d^3 c+128 a^2 b^2 d^3 x c-5 a^4 d^4}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {128 a^2 b^2 c d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {256 a^2 b^2 c d^3 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {256 a^2 b^2 c d^3 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 \left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{8} \left (\frac {-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}-\frac {3 \left (-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}-\frac {256 a^2 b^{3/2} c d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {2 \left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}}{2 a}\right )}{4 a}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\) |
-1/4*((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^4 + (-1/3*((3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(c*x^3) + (-1/2*((3*b^2*c^2 + 50*a*b*c*d - 5*a^2* d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(a*x^2) - (3*(-(((3*b^3*c^3 - 17*a*b^2 *c^2*d - 55*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - ((-2*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt [a]*Sqrt[c]) + 256*a^2*b^(3/2)*c*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/( Sqrt[b]*Sqrt[c + d*x])])/(2*a)))/(4*a))/(6*c))/8
3.7.23.3.1 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[(((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. \(732\) vs. \(2(266)=532\).
Time = 0.55 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.32
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (384 \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}+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}-180 \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}-270 \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}+60 \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}-9 \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}-30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}-674 \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}-114 \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}+18 \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}-488 \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}-12 \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 -144 \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^{2} c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {b d}\, \sqrt {a c}}\) | \(733\) |
1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c*(384*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)+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)-180*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)-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)*a^2*b^2*c^2*d^2*x^4*(b*d)^(1/2)+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)*a*b^3*c^3*d*x ^4*(b*d)^(1/2)-9*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)-30*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c) )^(1/2)*a^3*d^3*x^3-674*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^ 2*b*c*d^2*x^3-114*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^ 2*d*x^3+18*(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^2-488*((b*x+a )*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2-12*((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-144*((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)/x^4/(b*d)^(1/2)/(a*c)^(1/2)
Time = 4.14 (sec) , antiderivative size = 1529, normalized size of antiderivative = 4.84 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Too large to display} \]
[1/768*(384*sqrt(b*d)*a^3*b*c^2*d^2*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) - 3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2* b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*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)*s qrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(48*a^4*c^4 - (9*a*b^3*c^4 - 57*a^2*b^2*c^3*d - 337*a^3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 + 2*(3*a^2*b^2*c^4 + 122*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c ^4 + 17*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^2*x^4), -1/768*( 768*sqrt(-b*d)*a^3*b*c^2*d^2*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)) + 3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^ 3 - 5*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*(48*a^4*c^4 - (9*a*b^3*c^4 - 57*a^2*b^ 2*c^3*d - 337*a^3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 + 2*(3*a^2*b^2*c^4 + 122*a ^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c^4 + 17*a^4*c^3*d)*x)*sqrt( b*x + a)*sqrt(d*x + c))/(a^3*c^2*x^4), 1/384*(192*sqrt(b*d)*a^3*b*c^2*d^2* 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) ...
\[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Exception raised: ValueError} \]
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. 3887 vs. \(2 (266) = 532\).
Time = 1.38 (sec) , antiderivative size = 3887, normalized size of antiderivative = 12.30 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Too large to display} \]
-1/192*(192*sqrt(b*d)*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) + 3*(3*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) + 60*sqrt(b* d)*a^3*b^2*c*d^3*abs(b) - 5*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^2*b*c) - 2*(9*sqrt(b*d)*b^19*c^1 1*abs(b) - 129*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 371*sqrt(b*d)*a^2*b^17*c^9 *d^2*abs(b) + 581*sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b) - 5494*sqrt(b*d)*a^4*b ^15*c^7*d^4*abs(b) + 13958*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 19306*sqrt( b*d)*a^6*b^13*c^5*d^6*abs(b) + 16154*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 8 131*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) + 2219*sqrt(b*d)*a^9*b^10*c^2*d^9*ab s(b) - 217*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 15*sqrt(b*d)*a^11*b^8*d^11*a bs(b) - 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) + 702*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d*abs(b) - 619*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) - 5272*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^14*c^7*d^3*abs(b) + 13362*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^13*c^6*d^ 4*abs(b) - 7372*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x ...
Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x^5} \,d x \]