\(\int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx\) [217]

Optimal result

Integrand size = 17, antiderivative size = 182 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}-\frac {d \sqrt {c+d x}}{24 b (b c-a d) (a+b x)^3}+\frac {5 d^2 \sqrt {c+d x}}{96 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 \sqrt {c+d x}}{64 b (b c-a d)^3 (a+b x)}+\frac {5 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{3/2} (b c-a d)^{7/2}} \] Output:

-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)+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)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {\sqrt {c+d x} \left (-15 a^3 d^3+a^2 b d^2 (118 c+73 d x)+a b^2 d \left (-136 c^2-36 c d x+55 d^2 x^2\right )+b^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )\right )}{192 b (-b c+a d)^3 (a+b x)^4}+\frac {5 d^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{64 b^{3/2} (-b c+a d)^{7/2}} \] Input:

Integrate[Sqrt[c + d*x]/(a + b*x)^5,x]
 

Output:

(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))
 

Rubi [A] (verified)

Time = 0.25 (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

\(\displaystyle \frac {d \left (-\frac {5 d \left (-\frac {3 d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 b}-\frac {\sqrt {c+d x}}{4 b (a+b x)^4}\)

Input:

Int[Sqrt[c + d*x]/(a + b*x)^5,x]
 

Output:

-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)
 

Defintions of rubi rules used

rule 51

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]
 

rule 52

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]
 

rule 73

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]
 

rule 221

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]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {-\frac {5 \left (\left (-b^{3} x^{3}-\frac {11}{3} a \,b^{2} x^{2}-\frac {73}{15} a^{2} b x +a^{3}\right ) d^{3}-\frac {118 c \left (-\frac {5}{59} b^{2} x^{2}-\frac {18}{59} a b x +a^{2}\right ) b \,d^{2}}{15}+\frac {136 c^{2} \left (-\frac {b x}{17}+a \right ) b^{2} d}{15}-\frac {16 b^{3} c^{3}}{5}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {x d +c}}{64}+\frac {5 d^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{4} b \left (a d -b c \right )^{3}}\)	\(170\)
derivativedivides	\(2 d^{4} \left (\frac {\frac {5 b^{2} \left (x d +c \right )^{\frac {7}{2}}}{128 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {55 b \left (x d +c \right )^{\frac {5}{2}}}{384 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {73 \left (x d +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {5 \sqrt {x d +c}}{128 b}}{\left (\left (x d +c \right ) b +a d -b c \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\)	\(217\)
default	\(2 d^{4} \left (\frac {\frac {5 b^{2} \left (x d +c \right )^{\frac {7}{2}}}{128 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {55 b \left (x d +c \right )^{\frac {5}{2}}}{384 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {73 \left (x d +c \right )^{\frac {3}{2}}}{384 \left (a d -b c \right )}-\frac {5 \sqrt {x d +c}}{128 b}}{\left (\left (x d +c \right ) b +a d -b c \right )^{4}}+\frac {5 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\)	\(217\)

Input:

int((d*x+c)^(1/2)/(b*x+a)^5,x,method=_RETURNVERBOSE)
 

Output:

5/64/((a*d-b*c)*b)^(1/2)*(-((-b^3*x^3-11/3*a*b^2*x^2-73/15*a^2*b*x+a^3)*d^ 
3-118/15*c*(-5/59*b^2*x^2-18/59*a*b*x+a^2)*b*d^2+136/15*c^2*(-1/17*b*x+a)* 
b^2*d-16/5*b^3*c^3)*((a*d-b*c)*b)^(1/2)*(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
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (154) = 308\).

Time = 0.13 (sec) , antiderivative size = 1176, normalized size of antiderivative = 6.46 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/(b*x+a)^5,x, algorithm="fricas")
 

Output:

[-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^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)/(b*x+a)**5,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)/(b*x+a)^5,x, algorithm="maxima")
 

Output:

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (154) = 308\).

Time = 0.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=-\frac {5 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 55 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} + 73 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} + 15 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 55 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} - 146 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} - 45 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} + 73 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} + 45 \, \sqrt {d x + c} a^{2} b c d^{6} - 15 \, \sqrt {d x + c} a^{3} d^{7}}{192 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \] Input:

integrate((d*x+c)^(1/2)/(b*x+a)^5,x, algorithm="giac")
 

Output:

-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)
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {\frac {73\,d^4\,{\left (c+d\,x\right )}^{3/2}}{192\,\left (a\,d-b\,c\right )}-\frac {5\,d^4\,\sqrt {c+d\,x}}{64\,b}+\frac {5\,b^2\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^3}+\frac {55\,b\,d^4\,{\left (c+d\,x\right )}^{5/2}}{192\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3}+\frac {5\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{7/2}} \] Input:

int((c + d*x)^(1/2)/(a + b*x)^5,x)
 

Output:

((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))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.51 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx=\frac {15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{4} d^{4}+60 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{3} b \,d^{4} x +90 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b^{2} d^{4} x^{2}+60 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{3} d^{4} x^{3}+15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{4} d^{4} x^{4}-15 \sqrt {d x +c}\, a^{4} b \,d^{4}+133 \sqrt {d x +c}\, a^{3} b^{2} c \,d^{3}+73 \sqrt {d x +c}\, a^{3} b^{2} d^{4} x -254 \sqrt {d x +c}\, a^{2} b^{3} c^{2} d^{2}-109 \sqrt {d x +c}\, a^{2} b^{3} c \,d^{3} x +55 \sqrt {d x +c}\, a^{2} b^{3} d^{4} x^{2}+184 \sqrt {d x +c}\, a \,b^{4} c^{3} d +44 \sqrt {d x +c}\, a \,b^{4} c^{2} d^{2} x -65 \sqrt {d x +c}\, a \,b^{4} c \,d^{3} x^{2}+15 \sqrt {d x +c}\, a \,b^{4} d^{4} x^{3}-48 \sqrt {d x +c}\, b^{5} c^{4}-8 \sqrt {d x +c}\, b^{5} c^{3} d x +10 \sqrt {d x +c}\, b^{5} c^{2} d^{2} x^{2}-15 \sqrt {d x +c}\, b^{5} c \,d^{3} x^{3}}{192 b^{2} \left (a^{4} b^{4} d^{4} x^{4}-4 a^{3} b^{5} c \,d^{3} x^{4}+6 a^{2} b^{6} c^{2} d^{2} x^{4}-4 a \,b^{7} c^{3} d \,x^{4}+b^{8} c^{4} x^{4}+4 a^{5} b^{3} d^{4} x^{3}-16 a^{4} b^{4} c \,d^{3} x^{3}+24 a^{3} b^{5} c^{2} d^{2} x^{3}-16 a^{2} b^{6} c^{3} d \,x^{3}+4 a \,b^{7} c^{4} x^{3}+6 a^{6} b^{2} d^{4} x^{2}-24 a^{5} b^{3} c \,d^{3} x^{2}+36 a^{4} b^{4} c^{2} d^{2} x^{2}-24 a^{3} b^{5} c^{3} d \,x^{2}+6 a^{2} b^{6} c^{4} x^{2}+4 a^{7} b \,d^{4} x -16 a^{6} b^{2} c \,d^{3} x +24 a^{5} b^{3} c^{2} d^{2} x -16 a^{4} b^{4} c^{3} d x +4 a^{3} b^{5} c^{4} x +a^{8} d^{4}-4 a^{7} b c \,d^{3}+6 a^{6} b^{2} c^{2} d^{2}-4 a^{5} b^{3} c^{3} d +a^{4} b^{4} c^{4}\right )} \] Input:

int((d*x+c)^(1/2)/(b*x+a)^5,x)
 

Output:

(15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c 
)))*a**4*d**4 + 60*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b) 
*sqrt(a*d - b*c)))*a**3*b*d**4*x + 90*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c 
 + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**2*d**4*x**2 + 60*sqrt(b)*sqr 
t(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**3*d**4 
*x**3 + 15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a* 
d - b*c)))*b**4*d**4*x**4 - 15*sqrt(c + d*x)*a**4*b*d**4 + 133*sqrt(c + d* 
x)*a**3*b**2*c*d**3 + 73*sqrt(c + d*x)*a**3*b**2*d**4*x - 254*sqrt(c + d*x 
)*a**2*b**3*c**2*d**2 - 109*sqrt(c + d*x)*a**2*b**3*c*d**3*x + 55*sqrt(c + 
 d*x)*a**2*b**3*d**4*x**2 + 184*sqrt(c + d*x)*a*b**4*c**3*d + 44*sqrt(c + 
d*x)*a*b**4*c**2*d**2*x - 65*sqrt(c + d*x)*a*b**4*c*d**3*x**2 + 15*sqrt(c 
+ d*x)*a*b**4*d**4*x**3 - 48*sqrt(c + d*x)*b**5*c**4 - 8*sqrt(c + d*x)*b** 
5*c**3*d*x + 10*sqrt(c + d*x)*b**5*c**2*d**2*x**2 - 15*sqrt(c + d*x)*b**5* 
c*d**3*x**3)/(192*b**2*(a**8*d**4 - 4*a**7*b*c*d**3 + 4*a**7*b*d**4*x + 6* 
a**6*b**2*c**2*d**2 - 16*a**6*b**2*c*d**3*x + 6*a**6*b**2*d**4*x**2 - 4*a* 
*5*b**3*c**3*d + 24*a**5*b**3*c**2*d**2*x - 24*a**5*b**3*c*d**3*x**2 + 4*a 
**5*b**3*d**4*x**3 + a**4*b**4*c**4 - 16*a**4*b**4*c**3*d*x + 36*a**4*b**4 
*c**2*d**2*x**2 - 16*a**4*b**4*c*d**3*x**3 + a**4*b**4*d**4*x**4 + 4*a**3* 
b**5*c**4*x - 24*a**3*b**5*c**3*d*x**2 + 24*a**3*b**5*c**2*d**2*x**3 - 4*a 
**3*b**5*c*d**3*x**4 + 6*a**2*b**6*c**4*x**2 - 16*a**2*b**6*c**3*d*x**3...