Integrand size = 22, antiderivative size = 346 \[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=\frac {c (9 b c-13 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a^2 x^4}-\frac {\left (63 b^2 c^2-148 a b c d+93 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^3 x^3}+\frac {\left (315 b^3 c^3-749 a b^2 c^2 d+481 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^4 c x^2}-\frac {\left (945 b^4 c^4-2310 a b^3 c^3 d+1564 a^2 b^2 c^2 d^2-90 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^5 c^2 x}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}+\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}} \]
1/128*(-a*d+b*c)^3*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*arctanh(c^(1/2)*(b*x+ a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(11/2)/c^(5/2)-1/5*c*(d*x+c)^(3/2)*(b*x+ a)^(1/2)/a/x^5+1/40*c*(-13*a*d+9*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/x^4- 1/240*(93*a^2*d^2-148*a*b*c*d+63*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/ x^3+1/960*(-15*a^3*d^3+481*a^2*b*c*d^2-749*a*b^2*c^2*d+315*b^3*c^3)*(b*x+a )^(1/2)*(d*x+c)^(1/2)/a^4/c/x^2-1/1920*(-45*a^4*d^4-90*a^3*b*c*d^3+1564*a^ 2*b^2*c^2*d^2-2310*a*b^3*c^3*d+945*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^ 5/c^2/x
Time = 0.71 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.75 \[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (945 b^4 c^4 x^4-210 a b^3 c^3 x^3 (3 c+11 d x)+2 a^2 b^2 c^2 x^2 \left (252 c^2+749 c d x+782 d^2 x^2\right )-2 a^3 b c x \left (216 c^3+592 c^2 d x+481 c d^2 x^2+45 d^3 x^3\right )+3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )\right )}{1920 a^5 c^2 x^5}+\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{11/2} c^{5/2}} \]
-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*b^4*c^4*x^4 - 210*a*b^3*c^3*x^3* (3*c + 11*d*x) + 2*a^2*b^2*c^2*x^2*(252*c^2 + 749*c*d*x + 782*d^2*x^2) - 2 *a^3*b*c*x*(216*c^3 + 592*c^2*d*x + 481*c*d^2*x^2 + 45*d^3*x^3) + 3*a^4*(1 28*c^4 + 336*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(a^5 *c^2*x^5) + ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[( Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(11/2)*c^(5/2))
Time = 0.51 (sec) , antiderivative size = 377, 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 = {109, 27, 166, 27, 168, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (9 b c-13 a d)+2 d (3 b c-5 a d) x)}{2 x^5 \sqrt {a+b x}}dx}{5 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (9 b c-13 a d)+2 d (3 b c-5 a d) x)}{x^5 \sqrt {a+b x}}dx}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {\frac {\int -\frac {c \left (63 b^2 c^2-148 a b d c+93 a^2 d^2\right )+2 d \left (27 b^2 c^2-63 a b d c+40 a^2 d^2\right ) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {c \left (63 b^2 c^2-148 a b d c+93 a^2 d^2\right )+2 d \left (27 b^2 c^2-63 a b d c+40 a^2 d^2\right ) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {c \left (315 b^3 c^3-749 a b^2 d c^2+481 a^2 b d^2 c-15 a^3 d^3+4 b d \left (63 b^2 c^2-148 a b d c+93 a^2 d^2\right ) x\right )}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {315 b^3 c^3-749 a b^2 d c^2+481 a^2 b d^2 c-15 a^3 d^3+4 b d \left (63 b^2 c^2-148 a b d c+93 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {945 b^4 c^4-2310 a b^3 d c^3+1564 a^2 b^2 d^2 c^2-90 a^3 b d^3 c-45 a^4 d^4+2 b d \left (315 b^3 c^3-749 a b^2 d c^2+481 a^2 b d^2 c-15 a^3 d^3\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {945 b^4 c^4-2310 a b^3 d c^3+1564 a^2 b^2 d^2 c^2-90 a^3 b d^3 c-45 a^4 d^4+2 b d \left (315 b^3 c^3-749 a b^2 d c^2+481 a^2 b d^2 c-15 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {\int \frac {15 (b c-a d)^3 \left (63 b^2 c^2+14 a b d c+3 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {15 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {15 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{3 a x^3}-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{2 a c x^2}-\frac {\frac {15 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{a c x}}{4 a c}}{6 a}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x} (9 b c-13 a d)}{4 a x^4}}{10 a}-\frac {c \sqrt {a+b x} (c+d x)^{3/2}}{5 a x^5}\) |
-1/5*(c*Sqrt[a + b*x]*(c + d*x)^(3/2))/(a*x^5) - (-1/4*(c*(9*b*c - 13*a*d) *Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x^4) - (-1/3*((63*b^2*c^2 - 148*a*b*c*d + 93*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x^3) - (-1/2*((315*b^3*c^3 - 749*a*b^2*c^2*d + 481*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x ])/(a*c*x^2) - (-(((945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b*c*d^3 - 45*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (15 *(b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt [a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(3/2)))/(4*a*c))/(6*a))/(8 *a))/(10*a)
3.8.22.3.1 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_)^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. \(812\) vs. \(2(302)=604\).
Time = 1.65 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.35
| method | result | size |
| default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (45 \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^{5} d^{5} x^{5}+75 \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} b c \,d^{4} x^{5}+450 \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^{2} c^{2} d^{3} x^{5}-2250 \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^{3} c^{3} d^{2} x^{5}+2625 \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^{4} c^{4} d \,x^{5}-945 \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^{5} c^{5} x^{5}-90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} d^{4} x^{4}-180 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}+3128 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}-4620 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}+1890 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}-1924 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+2996 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}-1260 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}-2368 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+1008 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x -864 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{5} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{5} \sqrt {a c}}\) | \(813\) |
-1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^5/c^2*(45*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*d^5*x^5+75*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*c*d^4*x^5+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^3*b^2*c^2*d^3*x^5-2 250*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^ 3*c^3*d^2*x^5+2625*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^4*c^4*d*x^5-945*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^5*c^5*x^5-90*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4* d^4*x^4-180*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c*d^3*x^4+3128*((b*x +a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^2*c^2*d^2*x^4-4620*((b*x+a)*(d*x+c))^ (1/2)*(a*c)^(1/2)*a*b^3*c^3*d*x^4+1890*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2) *b^4*c^4*x^4+60*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3-1924*((b *x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3+2996*((b*x+a)*(d*x+c))^ (1/2)*(a*c)^(1/2)*a^2*b^2*c^3*d*x^3-1260*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/ 2)*a*b^3*c^4*x^3+1488*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2- 2368*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2+1008*((b*x+a)*(d* x+c))^(1/2)*(a*c)^(1/2)*a^2*b^2*c^4*x^2+2016*((b*x+a)*(d*x+c))^(1/2)*(a*c) ^(1/2)*a^4*c^3*d*x-864*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^4*x+768 *((b*x+a)*(d*x+c))^(1/2)*a^4*c^4*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^5/ (a*c)^(1/2)
Time = 2.73 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (384 \, a^{5} c^{5} + {\left (945 \, a b^{4} c^{5} - 2310 \, a^{2} b^{3} c^{4} d + 1564 \, a^{3} b^{2} c^{3} d^{2} - 90 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (315 \, a^{2} b^{3} c^{5} - 749 \, a^{3} b^{2} c^{4} d + 481 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (63 \, a^{3} b^{2} c^{5} - 148 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} - 144 \, {\left (3 \, a^{4} b c^{5} - 7 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{6} c^{3} x^{5}}, -\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (384 \, a^{5} c^{5} + {\left (945 \, a b^{4} c^{5} - 2310 \, a^{2} b^{3} c^{4} d + 1564 \, a^{3} b^{2} c^{3} d^{2} - 90 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (315 \, a^{2} b^{3} c^{5} - 749 \, a^{3} b^{2} c^{4} d + 481 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (63 \, a^{3} b^{2} c^{5} - 148 \, a^{4} b c^{4} d + 93 \, a^{5} c^{3} d^{2}\right )} x^{2} - 144 \, {\left (3 \, a^{4} b c^{5} - 7 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{6} c^{3} x^{5}}\right ] \]
[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3* b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*sqrt(a*c)*x^5*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)*sq rt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(384*a^5*c^5 + (945*a*b^4*c^5 - 2310*a^2*b^3*c^4*d + 1564*a^3*b^2*c^3*d^2 - 90*a^4*b*c ^2*d^3 - 45*a^5*c*d^4)*x^4 - 2*(315*a^2*b^3*c^5 - 749*a^3*b^2*c^4*d + 481* a^4*b*c^3*d^2 - 15*a^5*c^2*d^3)*x^3 + 8*(63*a^3*b^2*c^5 - 148*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 - 144*(3*a^4*b*c^5 - 7*a^5*c^4*d)*x)*sqrt(b*x + a)*s qrt(d*x + c))/(a^6*c^3*x^5), -1/3840*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 1 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*sqrt( -a*c)*x^5*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*(384*a^5*c^ 5 + (945*a*b^4*c^5 - 2310*a^2*b^3*c^4*d + 1564*a^3*b^2*c^3*d^2 - 90*a^4*b* c^2*d^3 - 45*a^5*c*d^4)*x^4 - 2*(315*a^2*b^3*c^5 - 749*a^3*b^2*c^4*d + 481 *a^4*b*c^3*d^2 - 15*a^5*c^2*d^3)*x^3 + 8*(63*a^3*b^2*c^5 - 148*a^4*b*c^4*d + 93*a^5*c^3*d^2)*x^2 - 144*(3*a^4*b*c^5 - 7*a^5*c^4*d)*x)*sqrt(b*x + a)* sqrt(d*x + c))/(a^6*c^3*x^5)]
\[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{6} \sqrt {a + b x}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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. 5928 vs. \(2 (302) = 604\).
Time = 15.19 (sec) , antiderivative size = 5928, normalized size of antiderivative = 17.13 \[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=\text {Too large to display} \]
1/1920*(15*(63*sqrt(b*d)*b^6*c^5*abs(b) - 175*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 150*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) - 30*sqrt(b*d)*a^3*b^3*c^2*d^3*abs (b) - 5*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) - 3*sqrt(b*d)*a^5*b*d^5*abs(b))*arc tan(-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^5*b*c^2) - 2*(945 *sqrt(b*d)*b^24*c^14*abs(b) - 11760*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 67189 *sqrt(b*d)*a^2*b^22*c^12*d^2*abs(b) - 233080*sqrt(b*d)*a^3*b^21*c^11*d^3*a bs(b) + 546885*sqrt(b*d)*a^4*b^20*c^10*d^4*abs(b) - 914520*sqrt(b*d)*a^5*b ^19*c^9*d^5*abs(b) + 1117785*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) - 1006128*s qrt(b*d)*a^7*b^17*c^7*d^7*abs(b) + 661395*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b ) - 308640*sqrt(b*d)*a^9*b^15*c^5*d^9*abs(b) + 95775*sqrt(b*d)*a^10*b^14*c ^4*d^10*abs(b) - 16600*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) + 439*sqrt(b*d) *a^12*b^12*c^2*d^12*abs(b) + 360*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) - 45*sq rt(b*d)*a^14*b^10*d^14*abs(b) - 8505*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^22*c^13*abs(b) + 79065*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^21*c ^12*d*abs(b) - 316630*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^20*c^11*d^2*abs(b) + 692110*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^19*c^10* d^3*abs(b) - 818075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (...
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^6\,\sqrt {a+b\,x}} \,d x \]