Integrand size = 22, antiderivative size = 241 \[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}+\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}} \]
1/8*(-a*d+b*c)*(-a^2*d^2-10*a*b*c*d+35*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1 /2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(3/2)-1/24*b*(3*a^2*d^2-100*a*b*c*d+1 05*b^2*c^2)*(d*x+c)^(1/2)/a^4/c/(b*x+a)^(1/2)-1/3*c*(d*x+c)^(1/2)/a/x^3/(b *x+a)^(1/2)+7/12*(-a*d+b*c)*(d*x+c)^(1/2)/a^2/x^2/(b*x+a)^(1/2)-1/24*(-3*a *d+35*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^3/c/x/(b*x+a)^(1/2)
Time = 10.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=-\frac {\sqrt {c+d x} \left (105 b^3 c^2 x^3+5 a b^2 c x^2 (7 c-20 d x)+a^2 b x \left (-14 c^2-38 c d x+3 d^2 x^2\right )+a^3 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )}{24 a^4 c x^3 \sqrt {a+b x}}+\frac {\left (35 b^3 c^3-45 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}} \]
-1/24*(Sqrt[c + d*x]*(105*b^3*c^2*x^3 + 5*a*b^2*c*x^2*(7*c - 20*d*x) + a^2 *b*x*(-14*c^2 - 38*c*d*x + 3*d^2*x^2) + a^3*(8*c^2 + 14*c*d*x + 3*d^2*x^2) ))/(a^4*c*x^3*Sqrt[a + b*x]) + ((35*b^3*c^3 - 45*a*b^2*c^2*d + 9*a^2*b*c*d ^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8 *a^(9/2)*c^(3/2))
Time = 0.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {109, 27, 168, 27, 168, 27, 169, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int \frac {(b c-a d) (7 c+6 d x)}{2 x^3 (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(b c-a d) \int \frac {7 c+6 d x}{x^3 (a+b x)^{3/2} \sqrt {c+d x}}dx}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {\int \frac {c (35 b c-3 a d+28 b d x)}{2 x^2 (a+b x)^{3/2} \sqrt {c+d x}}dx}{2 a c}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {\int \frac {35 b c-3 a d+28 b d x}{x^2 (a+b x)^{3/2} \sqrt {c+d x}}dx}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\int \frac {3 \left (35 b^2 c^2-10 a b d c-a^2 d^2\right )+2 b d (35 b c-3 a d) x}{2 x (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {c+d x} (35 b c-3 a d)}{a c x \sqrt {a+b x}}}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\int \frac {3 \left (35 b^2 c^2-10 a b d c-a^2 d^2\right )+2 b d (35 b c-3 a d) x}{x (a+b x)^{3/2} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {c+d x} (35 b c-3 a d)}{a c x \sqrt {a+b x}}}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {2 \int \frac {3 (b c-a d) \left (35 b^2 c^2-10 a b d c-a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x} (35 b c-3 a d)}{a c x \sqrt {a+b x}}}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {3 \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}+\frac {2 b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x} (35 b c-3 a d)}{a c x \sqrt {a+b x}}}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {6 \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}+\frac {2 b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x} (35 b c-3 a d)}{a c x \sqrt {a+b x}}}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {2 b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {6 \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}}{2 a c}-\frac {\sqrt {c+d x} (35 b c-3 a d)}{a c x \sqrt {a+b x}}}{4 a}-\frac {7 \sqrt {c+d x}}{2 a x^2 \sqrt {a+b x}}\right )}{6 a}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}\) |
-1/3*(c*Sqrt[c + d*x])/(a*x^3*Sqrt[a + b*x]) - ((b*c - a*d)*((-7*Sqrt[c + d*x])/(2*a*x^2*Sqrt[a + b*x]) - (-(((35*b*c - 3*a*d)*Sqrt[c + d*x])/(a*c*x *Sqrt[a + b*x])) - ((2*b*(105*b^2*c^2 - 100*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x])/(a*(b*c - a*d)*Sqrt[a + b*x]) - (6*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^ 2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sqrt [c]))/(2*a*c))/(4*a)))/(6*a)
3.8.66.3.1 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 + 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_))^(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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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. \(706\) vs. \(2(203)=406\).
Time = 0.58 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.93
| method | result | size |
| default | \(\frac {\sqrt {d x +c}\, \left (3 \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 \,d^{3} x^{4}+27 \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 \,d^{2} x^{4}-135 \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^{2} d \,x^{4}+105 \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^{3} x^{4}+3 \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^{3} x^{3}+27 \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^{2} x^{3}-135 \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 \,x^{3}+105 \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} x^{3}-6 a^{2} b \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+200 a \,b^{2} c d \,x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-210 b^{3} c^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-6 a^{3} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+76 a^{2} b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-70 a \,b^{2} c^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-28 a^{3} c d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+28 a^{2} b \,c^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-16 a^{3} c^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{48 a^{4} c \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{3} \sqrt {a c}\, \sqrt {b x +a}}\) | \(707\) |
1/48*(d*x+c)^(1/2)*(3*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*d^3*x^4+27*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*d^2*x^4-135*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^2*d*x^4+105*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^3*x^4+3*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^3*x^3+27*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^2*x^3-135*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* x^3+105*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*x^3-6*a^2*b*d^2*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+200*a*b^2* c*d*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*b^3*c^2*x^3*(a*c)^(1/2)*(( b*x+a)*(d*x+c))^(1/2)-6*a^3*d^2*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+76 *a^2*b*c*d*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-70*a*b^2*c^2*x^2*(a*c)^ (1/2)*((b*x+a)*(d*x+c))^(1/2)-28*a^3*c*d*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 1/2)+28*a^2*b*c^2*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-16*a^3*c^2*(a*c)^( 1/2)*((b*x+a)*(d*x+c))^(1/2))/a^4/c/((b*x+a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2 )/(b*x+a)^(1/2)
Time = 1.02 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} + {\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}, -\frac {3 \, {\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} + {\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}\right ] \]
[1/96*(3*((35*b^4*c^3 - 45*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 + a^3*b*d^3)*x^4 + (35*a*b^3*c^3 - 45*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 + a^4*d^3)*x^3)*sqrt(a* c)*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*(8*a^4*c^3 + (105*a*b^3*c^3 - 100*a^2*b^2*c^2*d + 3*a^3*b*c*d^2) *x^3 + (35*a^2*b^2*c^3 - 38*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 14*(a^3*b*c^3 - a^4*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*b*c^2*x^4 + a^6*c^2*x^3 ), -1/48*(3*((35*b^4*c^3 - 45*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 + a^3*b*d^3)*x ^4 + (35*a*b^3*c^3 - 45*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 + a^4*d^3)*x^3)*sqrt (-a*c)*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*(8*a^4*c^3 + ( 105*a*b^3*c^3 - 100*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)*x^3 + (35*a^2*b^2*c^3 - 38*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 14*(a^3*b*c^3 - a^4*c^2*d)*x)*sqrt(b* x + a)*sqrt(d*x + c))/(a^5*b*c^2*x^4 + a^6*c^2*x^3)]
\[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{4} \left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 2310 vs. \(2 (203) = 406\).
Time = 4.91 (sec) , antiderivative size = 2310, normalized size of antiderivative = 9.59 \[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=\text {Too large to display} \]
-4*(sqrt(b*d)*b^3*c^2*abs(b) - 2*sqrt(b*d)*a*b^2*c*d*abs(b) + sqrt(b*d)*a^ 2*b*d^2*abs(b))/((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)*a^4) + 1/8*(35*sqrt(b*d)*b^3*c^3*abs(b) - 45*sq rt(b*d)*a*b^2*c^2*d*abs(b) + 9*sqrt(b*d)*a^2*b*c*d^2*abs(b) + sqrt(b*d)*a^ 3*d^3*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) - 1/12*(57*sqrt(b*d)*b^13*c^8*abs(b) - 394*sqrt(b*d)*a*b^12*c^7*d*a bs(b) + 1170*sqrt(b*d)*a^2*b^11*c^6*d^2*abs(b) - 1938*sqrt(b*d)*a^3*b^10*c ^5*d^3*abs(b) + 1940*sqrt(b*d)*a^4*b^9*c^4*d^4*abs(b) - 1182*sqrt(b*d)*a^5 *b^8*c^3*d^5*abs(b) + 414*sqrt(b*d)*a^6*b^7*c^2*d^6*abs(b) - 70*sqrt(b*d)* a^7*b^6*c*d^7*abs(b) + 3*sqrt(b*d)*a^8*b^5*d^8*abs(b) - 285*sqrt(b*d)*(sqr t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^11*c^7*abs (b) + 1071*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^10*c^6*d*abs(b) - 1161*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^9*c^5*d^2*abs(b) - 381*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 ^3*b^8*c^4*d^3*abs(b) + 1689*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^7*c^3*d^4*abs(b) - 1251*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^6*c^2 *d^5*abs(b) + 333*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*...
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^4\,{\left (a+b\,x\right )}^{3/2}} \,d x \]