Integrand size = 22, antiderivative size = 333 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 a^4 c^3 x}-\frac {(b c-a d)^3 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{9/2} c^{7/2}} \]
1/60*(5*a*d+7*b*c)*(b*x+a)^(3/2)*(d*x+c)^(7/2)/a/c^2/x^5-1/6*(b*x+a)^(5/2) *(d*x+c)^(7/2)/a/c/x^6-1/512*(-a*d+b*c)^5*(5*a*d+7*b*c)*arctanh(c^(1/2)*(b *x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(7/2)-1/768*(-a*d+b*c)^3*(5*a *d+7*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a^3/c^3/x^2+1/960*(-a*d+b*c)^2*(5*a* d+7*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a^2/c^3/x^3+1/160*(-a*d+b*c)*(5*a*d+7 *b*c)*(d*x+c)^(7/2)*(b*x+a)^(1/2)/a/c^3/x^4+1/512*(-a*d+b*c)^4*(5*a*d+7*b* c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3/x
Time = 0.69 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\frac {(-b c+a d)^5 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (-105 b^5 c^5 x^5+5 a b^4 c^4 x^4 (14 c+83 d x)-2 a^2 b^3 c^3 x^3 \left (28 c^2+136 c d x+273 d^2 x^2\right )+6 a^3 b^2 c^2 x^2 \left (8 c^3+36 c^2 d x+58 c d^2 x^2+25 d^3 x^3\right )+a^4 b c x \left (1664 c^4+4448 c^3 d x+3384 c^2 d^2 x^2+160 c d^3 x^3-245 d^4 x^4\right )+5 a^5 \left (256 c^5+640 c^4 d x+432 c^3 d^2 x^2+8 c^2 d^3 x^3-10 c d^4 x^4+15 d^5 x^5\right )\right )}{(b c-a d)^5 x^6}+15 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{7680 a^{9/2} c^{7/2}} \]
((-(b*c) + a*d)^5*((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^5* c^5*x^5 + 5*a*b^4*c^4*x^4*(14*c + 83*d*x) - 2*a^2*b^3*c^3*x^3*(28*c^2 + 13 6*c*d*x + 273*d^2*x^2) + 6*a^3*b^2*c^2*x^2*(8*c^3 + 36*c^2*d*x + 58*c*d^2* x^2 + 25*d^3*x^3) + a^4*b*c*x*(1664*c^4 + 4448*c^3*d*x + 3384*c^2*d^2*x^2 + 160*c*d^3*x^3 - 245*d^4*x^4) + 5*a^5*(256*c^5 + 640*c^4*d*x + 432*c^3*d^ 2*x^2 + 8*c^2*d^3*x^3 - 10*c*d^4*x^4 + 15*d^5*x^5)))/((b*c - a*d)^5*x^6) + 15*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x] )]))/(7680*a^(9/2)*c^(7/2))
Time = 0.33 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {107, 105, 105, 105, 105, 105, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(5 a d+7 b c) \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6}dx}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5}dx}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}}dx}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 a c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}\) |
-1/6*((a + b*x)^(5/2)*(c + d*x)^(7/2))/(a*c*x^6) - ((7*b*c + 5*a*d)*(-1/5* ((a + b*x)^(3/2)*(c + d*x)^(7/2))/(c*x^5) + (3*(b*c - a*d)*(-1/4*(Sqrt[a + b*x]*(c + d*x)^(7/2))/(c*x^4) + ((b*c - a*d)*(-1/3*(Sqrt[a + b*x]*(c + d* x)^(5/2))/(a*x^3) - (5*(b*c - a*d)*(-1/2*(Sqrt[a + b*x]*(c + d*x)^(3/2))/( a*x^2) - (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sq rt[c])))/(4*a)))/(6*a)))/(8*c)))/(10*c)))/(12*a*c)
3.7.25.3.1 Defintions of rubi rules used
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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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. \(1067\) vs. \(2(283)=566\).
Time = 0.55 (sec) , antiderivative size = 1068, normalized size of antiderivative = 3.21
1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3*(100*((b*x+a)*(d*x+c))^(1/2)*( a*c)^(1/2)*a^5*c*d^4*x^4-3328*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^ 5*x-4320*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^3*d^2*x^2-96*((b*x+a)*( d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c^5*x^2-140*((b*x+a)*(d*x+c))^(1/2)*(a*c )^(1/2)*a*b^4*c^5*x^4-80*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^2*d^3*x ^3-6400*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^4*d*x+544*((b*x+a)*(d*x+ c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^4*d*x^4-6768*((b*x+a)*(d*x+c))^(1/2)*(a*c) ^(1/2)*a^4*b*c^3*d^2*x^3-432*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c ^4*d*x^3-8896*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^4*d*x^2-320*((b* x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^2*d^3*x^4-696*((b*x+a)*(d*x+c))^(1 /2)*(a*c)^(1/2)*a^3*b^2*c^3*d^2*x^4-270*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^5*b*c*d^5*x^6+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^4*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^3*b^3*c^3*d^3*x^6- 675*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^ 4*c^4*d^2*x^6+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^5*c^5*d*x^6+112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^ 5*x^3-150*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*d^5*x^5+210*((b*x+a)*(d* x+c))^(1/2)*(a*c)^(1/2)*b^5*c^5*x^5-2560*((b*x+a)*(d*x+c))^(1/2)*a^5*c^5*( a*c)^(1/2)+490*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c*d^4*x^5-300*...
Time = 6.24 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.77 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\right )} \sqrt {a c} x^{6} \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 (1280 \, a^{6} c^{6} - {\left (105 \, a b^{5} c^{6} - 415 \, a^{2} b^{4} c^{5} d + 546 \, a^{3} b^{3} c^{4} d^{2} - 150 \, a^{4} b^{2} c^{3} d^{3} + 245 \, a^{5} b c^{2} d^{4} - 75 \, a^{6} c d^{5}\right )} x^{5} + 2 \, {\left (35 \, a^{2} b^{4} c^{6} - 136 \, a^{3} b^{3} c^{5} d + 174 \, a^{4} b^{2} c^{4} d^{2} + 80 \, a^{5} b c^{3} d^{3} - 25 \, a^{6} c^{2} d^{4}\right )} x^{4} - 8 \, {\left (7 \, a^{3} b^{3} c^{6} - 27 \, a^{4} b^{2} c^{5} d - 423 \, a^{5} b c^{4} d^{2} - 5 \, a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (3 \, a^{4} b^{2} c^{6} + 278 \, a^{5} b c^{5} d + 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (13 \, a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, a^{5} c^{4} x^{6}}, \frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\right )} \sqrt {-a c} x^{6} \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 (1280 \, a^{6} c^{6} - {\left (105 \, a b^{5} c^{6} - 415 \, a^{2} b^{4} c^{5} d + 546 \, a^{3} b^{3} c^{4} d^{2} - 150 \, a^{4} b^{2} c^{3} d^{3} + 245 \, a^{5} b c^{2} d^{4} - 75 \, a^{6} c d^{5}\right )} x^{5} + 2 \, {\left (35 \, a^{2} b^{4} c^{6} - 136 \, a^{3} b^{3} c^{5} d + 174 \, a^{4} b^{2} c^{4} d^{2} + 80 \, a^{5} b c^{3} d^{3} - 25 \, a^{6} c^{2} d^{4}\right )} x^{4} - 8 \, {\left (7 \, a^{3} b^{3} c^{6} - 27 \, a^{4} b^{2} c^{5} d - 423 \, a^{5} b c^{4} d^{2} - 5 \, a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (3 \, a^{4} b^{2} c^{6} + 278 \, a^{5} b c^{5} d + 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (13 \, a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, a^{5} c^{4} x^{6}}\right ] \]
[-1/30720*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^ 3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*a^6*d^6)*sqrt(a*c)*x^6 *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*(1280*a^6*c^6 - (105*a*b^5*c^6 - 415*a^2*b^4*c^5*d + 546*a^3*b^3*c ^4*d^2 - 150*a^4*b^2*c^3*d^3 + 245*a^5*b*c^2*d^4 - 75*a^6*c*d^5)*x^5 + 2*( 35*a^2*b^4*c^6 - 136*a^3*b^3*c^5*d + 174*a^4*b^2*c^4*d^2 + 80*a^5*b*c^3*d^ 3 - 25*a^6*c^2*d^4)*x^4 - 8*(7*a^3*b^3*c^6 - 27*a^4*b^2*c^5*d - 423*a^5*b* c^4*d^2 - 5*a^6*c^3*d^3)*x^3 + 16*(3*a^4*b^2*c^6 + 278*a^5*b*c^5*d + 135*a ^6*c^4*d^2)*x^2 + 128*(13*a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt( d*x + c))/(a^5*c^4*x^6), 1/15360*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2* b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5 *a^6*d^6)*sqrt(-a*c)*x^6*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqr t(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*(1280*a^6*c^6 - (105*a*b^5*c^6 - 415*a^2*b^4*c^5*d + 546*a^3*b^3*c^4*d ^2 - 150*a^4*b^2*c^3*d^3 + 245*a^5*b*c^2*d^4 - 75*a^6*c*d^5)*x^5 + 2*(35*a ^2*b^4*c^6 - 136*a^3*b^3*c^5*d + 174*a^4*b^2*c^4*d^2 + 80*a^5*b*c^3*d^3 - 25*a^6*c^2*d^4)*x^4 - 8*(7*a^3*b^3*c^6 - 27*a^4*b^2*c^5*d - 423*a^5*b*c^4* d^2 - 5*a^6*c^3*d^3)*x^3 + 16*(3*a^4*b^2*c^6 + 278*a^5*b*c^5*d + 135*a^6*c ^4*d^2)*x^2 + 128*(13*a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d...
\[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{7}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\text {Exception raised: ValueError} \]
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. 8502 vs. \(2 (283) = 566\).
Time = 3.44 (sec) , antiderivative size = 8502, normalized size of antiderivative = 25.53 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\text {Too large to display} \]
-1/7680*(15*(7*sqrt(b*d)*b^7*c^6*abs(b) - 30*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 45*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b) - 20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b ) - 15*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) + 18*sqrt(b*d)*a^5*b^2*c*d^5*abs(b ) - 5*sqrt(b*d)*a^6*b*d^6*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^4*b*c^3) - 2*(105*sqrt(b*d)*b^29*c^17*abs(b) - 1675*sq rt(b*d)*a*b^28*c^16*d*abs(b) + 12456*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 57192*sqrt(b*d)*a^3*b^26*c^14*d^3*abs(b) + 181356*sqrt(b*d)*a^4*b^25*c^13* d^4*abs(b) - 421620*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 746040*sqrt(b*d)* a^6*b^23*c^11*d^6*abs(b) - 1032152*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 11 41734*sqrt(b*d)*a^8*b^21*c^9*d^8*abs(b) - 1030722*sqrt(b*d)*a^9*b^20*c^8*d ^9*abs(b) + 773080*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) - 486360*sqrt(b*d)* a^11*b^18*c^6*d^11*abs(b) + 254796*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) - 1 07892*sqrt(b*d)*a^13*b^16*c^4*d^13*abs(b) + 35016*sqrt(b*d)*a^14*b^15*c^3* d^14*abs(b) - 8040*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) + 1145*sqrt(b*d)*a^ 16*b^13*c*d^16*abs(b) - 75*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 1155*sqrt(b*d )*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^27*c ^16*abs(b) + 14820*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^26*c^15*d*abs(b) - 85572*sqrt(b*d)*(sqrt(b*d)*sqr t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^25*c^14*d^2*a...
Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \]