Integrand size = 22, antiderivative size = 266 \[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(9 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 c^2 x^3}-\frac {\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^2 c^4 x}-\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{9/2}} \]
-1/64*(-a*d+b*c)^2*(35*a^2*d^2+10*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2)*(b*x+ a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c^(9/2)-1/4*a*(b*x+a)^(1/2)*(d*x+c )^(1/2)/c/x^4-1/24*(-7*a*d+9*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2/x^3-1/96 *(35*a^2*d^2-46*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^3/x^2+1 /192*(105*a^3*d^3-145*a^2*b*c*d^2+15*a*b^2*c^2*d+9*b^3*c^3)*(b*x+a)^(1/2)* (d*x+c)^(1/2)/a^2/c^4/x
Time = 0.48 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 b^3 c^3 x^3+3 a b^2 c^2 x^2 (-2 c+5 d x)+a^2 b c x \left (-72 c^2+92 c d x-145 d^2 x^2\right )+a^3 \left (-48 c^3+56 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )\right )}{192 a^2 c^4 x^4}-\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{5/2} c^{9/2}} \]
(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 + 5*d* x) + a^2*b*c*x*(-72*c^2 + 92*c*d*x - 145*d^2*x^2) + a^3*(-48*c^3 + 56*c^2* d*x - 70*c*d^2*x^2 + 105*d^3*x^3)))/(192*a^2*c^4*x^4) - ((b*c - a*d)^2*(3* b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c ]*Sqrt[a + b*x])])/(64*a^(5/2)*c^(9/2))
Time = 0.39 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.06, 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, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int -\frac {a (9 b c-7 a d)+2 b (4 b c-3 a d) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (9 b c-7 a d)+2 b (4 b c-3 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {\int -\frac {a \left (3 b^2 c^2-46 a b d c+35 a^2 d^2-4 b d (9 b c-7 a d) x\right )}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 b^2 c^2-46 a b d c+35 a^2 d^2-4 b d (9 b c-7 a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {9 b^3 c^3+15 a b^2 d c^2-145 a^2 b d^2 c+105 a^3 d^3+2 b d \left (3 b^2 c^2-46 a b d c+35 a^2 d^2\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}+\frac {35 a d^2}{c}-46 b d\right )}{2 x^2}}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {9 b^3 c^3+15 a b^2 d c^2-145 a^2 b d^2 c+105 a^3 d^3+2 b d \left (3 b^2 c^2-46 a b d c+35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}+\frac {35 a d^2}{c}-46 b d\right )}{2 x^2}}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {3 (b c-a d)^2 \left (3 b^2 c^2+10 a b d c+35 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}+\frac {35 a d^2}{c}-46 b d\right )}{2 x^2}}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}+\frac {35 a d^2}{c}-46 b d\right )}{2 x^2}}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}+\frac {35 a d^2}{c}-46 b d\right )}{2 x^2}}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3 (b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}+\frac {35 a d^2}{c}-46 b d\right )}{2 x^2}}{6 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (9 b c-7 a d)}{3 c x^3}}{8 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
-1/4*(a*Sqrt[a + b*x]*Sqrt[c + d*x])/(c*x^4) + (-1/3*((9*b*c - 7*a*d)*Sqrt [a + b*x]*Sqrt[c + d*x])/(c*x^3) + (-1/2*(((3*b^2*c)/a - 46*b*d + (35*a*d^ 2)/c)*Sqrt[a + b*x]*Sqrt[c + d*x])/x^2 - (-(((9*b^3*c^3 + 15*a*b^2*c^2*d - 145*a^2*b*c*d^2 + 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (3 *(b*c - a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35*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*c))/(8 *c)
3.7.33.3.1 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_)^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. \(592\) vs. \(2(228)=456\).
Time = 0.55 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.23
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-180 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}+54 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}+12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}+290 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}-18 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-184 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{2} c^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^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)*a^4*d^4*x^4-180*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+54*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+1 2*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+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^4-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3*x^3+290*(a*c)^ (1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3-30*(a*c)^(1/2)*((b*x+a)*(d*x +c))^(1/2)*a*b^2*c^2*d*x^3-18*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^3* x^3+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-184*((b*x+a)*(d* x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2+12*((b*x+a)*(d*x+c))^(1/2)*(a*c)^( 1/2)*a*b^2*c^3*x^2-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x+144 *((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c^3*x+96*((b*x+a)*(d*x+c))^(1/2 )*a^3*c^3*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)
Time = 1.59 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (48 \, a^{4} c^{4} - {\left (9 \, a b^{3} c^{4} + 15 \, a^{2} b^{2} c^{3} d - 145 \, a^{3} b c^{2} d^{2} + 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (9 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{3} c^{5} x^{4}}, \frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (48 \, a^{4} c^{4} - {\left (9 \, a b^{3} c^{4} + 15 \, a^{2} b^{2} c^{3} d - 145 \, a^{3} b c^{2} d^{2} + 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (9 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{3} c^{5} x^{4}}\right ] \]
[1/768*(3*(3*b^4*c^4 + 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 35*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 + 15*a^2*b^ 2*c^3*d - 145*a^3*b*c^2*d^2 + 105*a^4*c*d^3)*x^3 + 2*(3*a^2*b^2*c^4 - 46*a ^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c^4 - 7*a^4*c^3*d)*x)*sqrt(b *x + a)*sqrt(d*x + c))/(a^3*c^5*x^4), 1/384*(3*(3*b^4*c^4 + 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 35*a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 - (9*a*b^3*c^4 + 15*a^2*b^2*c^3*d - 145*a^3*b*c^2*d^2 + 105*a^4*c*d^3)*x^3 + 2*(3*a^2*b^2* c^4 - 46*a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c^4 - 7*a^4*c^3*d) *x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^5*x^4)]
\[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x^{5} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 3690 vs. \(2 (228) = 456\).
Time = 1.43 (sec) , antiderivative size = 3690, normalized size of antiderivative = 13.87 \[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
-1/192*b*(3*(3*sqrt(b*d)*b^5*c^4 + 4*sqrt(b*d)*a*b^4*c^3*d + 18*sqrt(b*d)* a^2*b^3*c^2*d^2 - 60*sqrt(b*d)*a^3*b^2*c*d^3 + 35*sqrt(b*d)*a^4*b*d^4)*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^2*b*c^4) - 2*(9*s qrt(b*d)*b^19*c^11 - 57*sqrt(b*d)*a*b^18*c^10*d - 13*sqrt(b*d)*a^2*b^17*c^ 9*d^2 + 1181*sqrt(b*d)*a^3*b^16*c^8*d^3 - 5110*sqrt(b*d)*a^4*b^15*c^7*d^4 + 11606*sqrt(b*d)*a^5*b^14*c^6*d^5 - 16618*sqrt(b*d)*a^6*b^13*c^5*d^6 + 15 818*sqrt(b*d)*a^7*b^12*c^4*d^7 - 10051*sqrt(b*d)*a^8*b^11*c^3*d^8 + 4115*s qrt(b*d)*a^9*b^10*c^2*d^9 - 985*sqrt(b*d)*a^10*b^9*c*d^10 + 105*sqrt(b*d)* a^11*b^8*d^11 - 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 + 198*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^16*c^9*d + 1229*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 - 7864*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 + 17490*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 - 17692*sqrt(b*d) *(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^1 2*c^5*d^5 + 3602*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6 + 10056*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7 - 10771*...
Timed out. \[ \int \frac {(a+b x)^{3/2}}{x^5 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x^5\,\sqrt {c+d\,x}} \,d x \]