Integrand size = 17, antiderivative size = 182 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 \sqrt {c+d x}}{64 b (b c-a d)^3 (a+b x)}+\frac {5 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}} \]
5/64*d^4*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c )^(7/2)-1/4*(d*x+c)^(1/2)/b/(b*x+a)^4-1/24*d*(d*x+c)^(1/2)/b/(-a*d+b*c)/(b *x+a)^3+5/96*d^2*(d*x+c)^(1/2)/b/(-a*d+b*c)^2/(b*x+a)^2-5/64*d^3*(d*x+c)^( 1/2)/b/(-a*d+b*c)^3/(b*x+a)
Time = 0.92 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {\sqrt {c+d x} \left (-15 a^3 d^3+a^2 b d^2 (118 c+73 d x)+a b^2 d \left (-136 c^2-36 c d x+55 d^2 x^2\right )+b^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )\right )}{192 b (-b c+a d)^3 (a+b x)^4}+\frac {5 d^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{64 b^{3/2} (-b c+a d)^{7/2}} \]
(Sqrt[c + d*x]*(-15*a^3*d^3 + a^2*b*d^2*(118*c + 73*d*x) + a*b^2*d*(-136*c ^2 - 36*c*d*x + 55*d^2*x^2) + b^3*(48*c^3 + 8*c^2*d*x - 10*c*d^2*x^2 + 15* d^3*x^3)))/(192*b*(-(b*c) + a*d)^3*(a + b*x)^4) + (5*d^4*ArcTan[(Sqrt[b]*S qrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(64*b^(3/2)*(-(b*c) + a*d)^(7/2))
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {51, 52, 52, 52, 73, 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.
\(\displaystyle \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {d \int \frac {1}{(a+b x)^4 \sqrt {c+d x}}dx}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {d \left (-\frac {5 d \int \frac {1}{(a+b x)^3 \sqrt {c+d x}}dx}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {d \left (-\frac {5 d \left (-\frac {3 d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}}dx}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {d \left (-\frac {5 d \left (-\frac {3 d \left (-\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{2 (b c-a d)}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {d \left (-\frac {5 d \left (-\frac {3 d \left (-\frac {\int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b c-a d}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {d \left (-\frac {5 d \left (-\frac {3 d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\) |
-1/4*Sqrt[c + d*x]/(b*(a + b*x)^4) + (d*(-1/3*Sqrt[c + d*x]/((b*c - a*d)*( a + b*x)^3) - (5*d*(-1/2*Sqrt[c + d*x]/((b*c - a*d)*(a + b*x)^2) - (3*d*(- (Sqrt[c + d*x]/((b*c - a*d)*(a + b*x))) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x ])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b*c - a*d)^(3/2))))/(4*(b*c - a*d))))/(6*(b *c - a*d))))/(8*b)
3.14.85.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Time = 0.33 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-\frac {5 \sqrt {\left (a d -b c \right ) b}\, \left (\left (-b^{3} x^{3}-\frac {11}{3} a \,b^{2} x^{2}-\frac {73}{15} a^{2} b x +a^{3}\right ) d^{3}-\frac {118 b \left (-\frac {5}{59} b^{2} x^{2}-\frac {18}{59} a b x +a^{2}\right ) c \,d^{2}}{15}+\frac {136 b^{2} \left (-\frac {b x}{17}+a \right ) c^{2} d}{15}-\frac {16 b^{3} c^{3}}{5}\right ) \sqrt {d x +c}}{64}+\frac {5 d^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{4} b \left (a d -b c \right )^{3}}\) | \(170\) |
derivativedivides | \(2 d^{4} \left (\frac {\frac {5 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {55 b \left (d x +c \right )^{\frac {5}{2}}}{384 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {73 \left (d x +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {5 \sqrt {d x +c}}{128 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(217\) |
default | \(2 d^{4} \left (\frac {\frac {5 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {55 b \left (d x +c \right )^{\frac {5}{2}}}{384 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {73 \left (d x +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {5 \sqrt {d x +c}}{128 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(217\) |
5/64/((a*d-b*c)*b)^(1/2)*(-((a*d-b*c)*b)^(1/2)*((-b^3*x^3-11/3*a*b^2*x^2-7 3/15*a^2*b*x+a^3)*d^3-118/15*b*(-5/59*b^2*x^2-18/59*a*b*x+a^2)*c*d^2+136/1 5*b^2*(-1/17*b*x+a)*c^2*d-16/5*b^3*c^3)*(d*x+c)^(1/2)+d^4*(b*x+a)^4*arctan (b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))/(b*x+a)^4/b/(a*d-b*c)^3
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (154) = 308\).
Time = 0.25 (sec) , antiderivative size = 1176, normalized size of antiderivative = 6.46 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\left [-\frac {15 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (48 \, b^{5} c^{4} - 184 \, a b^{4} c^{3} d + 254 \, a^{2} b^{3} c^{2} d^{2} - 133 \, a^{3} b^{2} c d^{3} + 15 \, a^{4} b d^{4} + 15 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} - 5 \, {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (8 \, b^{5} c^{3} d - 44 \, a b^{4} c^{2} d^{2} + 109 \, a^{2} b^{3} c d^{3} - 73 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{384 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{4} + 4 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x\right )}}, -\frac {15 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (48 \, b^{5} c^{4} - 184 \, a b^{4} c^{3} d + 254 \, a^{2} b^{3} c^{2} d^{2} - 133 \, a^{3} b^{2} c d^{3} + 15 \, a^{4} b d^{4} + 15 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} - 5 \, {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (8 \, b^{5} c^{3} d - 44 \, a b^{4} c^{2} d^{2} + 109 \, a^{2} b^{3} c d^{3} - 73 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{192 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{4} + 4 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x\right )}}\right ] \]
[-1/384*(15*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d ^4*x + a^4*d^4)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2* c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*(48*b^5*c^4 - 184*a*b^4*c^3*d + 2 54*a^2*b^3*c^2*d^2 - 133*a^3*b^2*c*d^3 + 15*a^4*b*d^4 + 15*(b^5*c*d^3 - a* b^4*d^4)*x^3 - 5*(2*b^5*c^2*d^2 - 13*a*b^4*c*d^3 + 11*a^2*b^3*d^4)*x^2 + ( 8*b^5*c^3*d - 44*a*b^4*c^2*d^2 + 109*a^2*b^3*c*d^3 - 73*a^3*b^2*d^4)*x)*sq rt(d*x + c))/(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^ 3*c*d^3 + a^8*b^2*d^4 + (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4* a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^4 + 4*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3* b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^3 + 6*(a^2*b^8*c^4 - 4*a^3* b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^2 + 4*(a^ 3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^ 3*d^4)*x), -1/192*(15*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d) *sqrt(d*x + c)/(b*d*x + b*c)) + (48*b^5*c^4 - 184*a*b^4*c^3*d + 254*a^2*b^ 3*c^2*d^2 - 133*a^3*b^2*c*d^3 + 15*a^4*b*d^4 + 15*(b^5*c*d^3 - a*b^4*d^4)* x^3 - 5*(2*b^5*c^2*d^2 - 13*a*b^4*c*d^3 + 11*a^2*b^3*d^4)*x^2 + (8*b^5*c^3 *d - 44*a*b^4*c^2*d^2 + 109*a^2*b^3*c*d^3 - 73*a^3*b^2*d^4)*x)*sqrt(d*x + c))/(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4 + (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^...
Timed out. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\text {Exception raised: ValueError} \]
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. 311 vs. \(2 (154) = 308\).
Time = 0.31 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=-\frac {5 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 55 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} + 73 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} + 15 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 55 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} - 146 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} - 45 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} + 73 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} + 45 \, \sqrt {d x + c} a^{2} b c d^{6} - 15 \, \sqrt {d x + c} a^{3} d^{7}}{192 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]
-5/64*d^4*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^3 - 3*a*b^3 *c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*sqrt(-b^2*c + a*b*d)) - 1/192*(15*(d *x + c)^(7/2)*b^3*d^4 - 55*(d*x + c)^(5/2)*b^3*c*d^4 + 73*(d*x + c)^(3/2)* b^3*c^2*d^4 + 15*sqrt(d*x + c)*b^3*c^3*d^4 + 55*(d*x + c)^(5/2)*a*b^2*d^5 - 146*(d*x + c)^(3/2)*a*b^2*c*d^5 - 45*sqrt(d*x + c)*a*b^2*c^2*d^5 + 73*(d *x + c)^(3/2)*a^2*b*d^6 + 45*sqrt(d*x + c)*a^2*b*c*d^6 - 15*sqrt(d*x + c)* a^3*d^7)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*((d*x + c)*b - b*c + a*d)^4)
Time = 0.23 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {\frac {73\,d^4\,{\left (c+d\,x\right )}^{3/2}}{192\,\left (a\,d-b\,c\right )}-\frac {5\,d^4\,\sqrt {c+d\,x}}{64\,b}+\frac {5\,b^2\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^3}+\frac {55\,b\,d^4\,{\left (c+d\,x\right )}^{5/2}}{192\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3}+\frac {5\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{7/2}} \]
((73*d^4*(c + d*x)^(3/2))/(192*(a*d - b*c)) - (5*d^4*(c + d*x)^(1/2))/(64* b) + (5*b^2*d^4*(c + d*x)^(7/2))/(64*(a*d - b*c)^3) + (55*b*d^4*(c + d*x)^ (5/2))/(192*(a*d - b*c)^2))/(b^4*(c + d*x)^4 - (4*b^4*c - 4*a*b^3*d)*(c + d*x)^3 - (c + d*x)*(4*b^4*c^3 - 4*a^3*b*d^3 + 12*a^2*b^2*c*d^2 - 12*a*b^3* c^2*d) + a^4*d^4 + b^4*c^4 + (c + d*x)^2*(6*b^4*c^2 + 6*a^2*b^2*d^2 - 12*a *b^3*c*d) + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3) + (5*d^4*at an((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(64*b^(3/2)*(a*d - b*c)^( 7/2))