Integrand size = 22, antiderivative size = 280 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=-\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 a^3 c^3 x}+\frac {5 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {5 (b c-a d)^6 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{7/2}} \]
-1/12*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(7/2)/c^2/x^5-1/6*(b*x+a)^(5/2)*(d* x+c)^(7/2)/c/x^6+5/512*(-a*d+b*c)^6*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/ (d*x+c)^(1/2))/a^(7/2)/c^(7/2)+5/768*(-a*d+b*c)^4*(d*x+c)^(3/2)*(b*x+a)^(1 /2)/a^2/c^3/x^2-1/192*(-a*d+b*c)^3*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a/c^3/x^3-1 /32*(-a*d+b*c)^2*(d*x+c)^(7/2)*(b*x+a)^(1/2)/c^3/x^4-5/512*(-a*d+b*c)^5*(b *x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3/x
Time = 0.70 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=\frac {(b c-a d)^6 \left (-\frac {\sqrt {a} \sqrt {c} (a+b x)^{11/2} \sqrt {c+d x} \left (15 c^5-\frac {85 a c^4 (c+d x)}{a+b x}+\frac {198 a^2 c^3 (c+d x)^2}{(a+b x)^2}+\frac {198 a^3 c^2 (c+d x)^3}{(a+b x)^3}-\frac {85 a^4 c (c+d x)^4}{(a+b x)^4}+\frac {15 a^5 (c+d x)^5}{(a+b x)^5}\right )}{(b c x-a d x)^6}+15 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{1536 a^{7/2} c^{7/2}} \]
((b*c - a*d)^6*(-((Sqrt[a]*Sqrt[c]*(a + b*x)^(11/2)*Sqrt[c + d*x]*(15*c^5 - (85*a*c^4*(c + d*x))/(a + b*x) + (198*a^2*c^3*(c + d*x)^2)/(a + b*x)^2 + (198*a^3*c^2*(c + d*x)^3)/(a + b*x)^3 - (85*a^4*c*(c + d*x)^4)/(a + b*x)^ 4 + (15*a^5*(c + d*x)^5)/(a + b*x)^5))/(b*c*x - a*d*x)^6) + 15*ArcTanh[(Sq rt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])]))/(1536*a^(7/2)*c^(7/2))
Time = 0.33 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {105, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6}dx}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5}dx}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}}dx}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\) |
-1/6*((a + b*x)^(5/2)*(c + d*x)^(7/2))/(c*x^6) + (5*(b*c - 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)*Sqrt[c ])))/(4*a)))/(6*a)))/(8*c)))/(10*c)))/(12*c)
3.7.73.3.1 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_)^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. \(2(230)=460\).
Time = 1.00 (sec) , antiderivative size = 1068, normalized size of antiderivative = 3.81
1/3072*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(20*((b*x+a)*(d*x+c))^(1/2)*(a* c)^(1/2)*a^5*c*d^4*x^4-1280*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^5* x-864*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^3*d^2*x^2-864*((b*x+a)*(d* x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c^5*x^2+20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^( 1/2)*a*b^4*c^5*x^4-16*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^2*d^3*x^3- 1280*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^4*d*x-112*((b*x+a)*(d*x+c)) ^(1/2)*(a*c)^(1/2)*a^2*b^3*c^4*d*x^4-2544*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1 /2)*a^4*b*c^3*d^2*x^3-2544*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c^4 *d*x^3-3392*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^4*d*x^2-112*((b*x+ a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^2*d^3*x^4-2376*((b*x+a)*(d*x+c))^(1/ 2)*(a*c)^(1/2)*a^3*b^2*c^3*d^2*x^4-90*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+22 5*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^6-90*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^6-16*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^5*x^ 3-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*d^5*x^5-30*((b*x+a)*(d*x+c))^ (1/2)*(a*c)^(1/2)*b^5*c^5*x^5-512*((b*x+a)*(d*x+c))^(1/2)*a^5*c^5*(a*c)^(1 /2)+170*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c*d^4*x^5-396*((b*x+a...
Time = 6.67 (sec) , antiderivative size = 908, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=\left [\frac {15 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, 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} - 6 \, a^{5} b c d^{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 (256 \, a^{6} c^{6} + {\left (15 \, a b^{5} c^{6} - 85 \, a^{2} b^{4} c^{5} d + 198 \, a^{3} b^{3} c^{4} d^{2} + 198 \, a^{4} b^{2} c^{3} d^{3} - 85 \, a^{5} b c^{2} d^{4} + 15 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (5 \, a^{2} b^{4} c^{6} - 28 \, a^{3} b^{3} c^{5} d - 594 \, a^{4} b^{2} c^{4} d^{2} - 28 \, a^{5} b c^{3} d^{3} + 5 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (a^{3} b^{3} c^{6} + 159 \, a^{4} b^{2} c^{5} d + 159 \, a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (27 \, a^{4} b^{2} c^{6} + 106 \, a^{5} b c^{5} d + 27 \, a^{6} c^{4} d^{2}\right )} x^{2} + 640 \, {\left (a^{5} b c^{6} + a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6144 \, a^{4} c^{4} x^{6}}, -\frac {15 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, 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} - 6 \, a^{5} b c d^{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 (256 \, a^{6} c^{6} + {\left (15 \, a b^{5} c^{6} - 85 \, a^{2} b^{4} c^{5} d + 198 \, a^{3} b^{3} c^{4} d^{2} + 198 \, a^{4} b^{2} c^{3} d^{3} - 85 \, a^{5} b c^{2} d^{4} + 15 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (5 \, a^{2} b^{4} c^{6} - 28 \, a^{3} b^{3} c^{5} d - 594 \, a^{4} b^{2} c^{4} d^{2} - 28 \, a^{5} b c^{3} d^{3} + 5 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (a^{3} b^{3} c^{6} + 159 \, a^{4} b^{2} c^{5} d + 159 \, a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (27 \, a^{4} b^{2} c^{6} + 106 \, a^{5} b c^{5} d + 27 \, a^{6} c^{4} d^{2}\right )} x^{2} + 640 \, {\left (a^{5} b c^{6} + a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3072 \, a^{4} c^{4} x^{6}}\right ] \]
[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^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* (256*a^6*c^6 + (15*a*b^5*c^6 - 85*a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 19 8*a^4*b^2*c^3*d^3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^4*c^ 6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*b*c^3*d^3 + 5*a^6*c^2* d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^ 3*d^3)*x^3 + 16*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^6), -1/3072*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-a*c)*x^6*arcta n(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*(256*a^6*c^6 + (15*a*b^5*c ^6 - 85*a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^3 - 85*a^5 *b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^4*c^6 - 28*a^3*b^3*c^5*d - 594 *a^4*b^2*c^4*d^2 - 28*a^5*b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 16*(27*a^4*b^ 2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*x^2 + 640*(a^5*b*c^6 + a^6*c^5*d )*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^6)]
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{7}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=\text {Exception raised: ValueError} \]
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. 8500 vs. \(2 (230) = 460\).
Time = 3.79 (sec) , antiderivative size = 8500, normalized size of antiderivative = 30.36 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=\text {Too large to display} \]
1/1536*(15*(sqrt(b*d)*b^7*c^6*abs(b) - 6*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 15 *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) - 6*sqrt(b*d)*a^5*b^2*c*d^5*abs(b) + s qrt(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^3*b*c^3) - 2*(15*sqrt(b*d)*b^29*c^17*abs(b) - 265*sqrt(b*d)*a *b^28*c^16*d*abs(b) + 2208*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 11088*sqrt (b*d)*a^3*b^26*c^14*d^3*abs(b) + 36732*sqrt(b*d)*a^4*b^25*c^13*d^4*abs(b) - 83412*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 129840*sqrt(b*d)*a^6*b^23*c^1 1*d^6*abs(b) - 129536*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 55506*sqrt(b*d) *a^8*b^21*c^9*d^8*abs(b) + 55506*sqrt(b*d)*a^9*b^20*c^8*d^9*abs(b) - 12953 6*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) + 129840*sqrt(b*d)*a^11*b^18*c^6*d^1 1*abs(b) - 83412*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) + 36732*sqrt(b*d)*a^1 3*b^16*c^4*d^13*abs(b) - 11088*sqrt(b*d)*a^14*b^15*c^3*d^14*abs(b) + 2208* sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) - 265*sqrt(b*d)*a^16*b^13*c*d^16*abs(b ) + 15*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 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^27*c^16*abs(b) + 2400*sqr t(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) - 16056*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^25*c^14*d^2*abs(b) + 62880*sqrt(b*d...
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \]