\(\int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx\) [253]

Optimal result

Integrand size = 17, antiderivative size = 180 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{4 (b c-a d) (a+b x)^4}+\frac {7 d \sqrt {c+d x}}{24 (b c-a d)^2 (a+b x)^3}-\frac {35 d^2 \sqrt {c+d x}}{96 (b c-a d)^3 (a+b x)^2}+\frac {35 d^3 \sqrt {c+d x}}{64 (b c-a d)^4 (a+b x)}-\frac {35 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 \sqrt {b} (b c-a d)^{9/2}} \] Output:

-1/4*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^4+7/24*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/ 
(b*x+a)^3-35/96*d^2*(d*x+c)^(1/2)/(-a*d+b*c)^3/(b*x+a)^2+35/64*d^3*(d*x+c) 
^(1/2)/(-a*d+b*c)^4/(b*x+a)-35/64*d^4*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+ 
b*c)^(1/2))/b^(1/2)/(-a*d+b*c)^(9/2)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\frac {1}{192} \left (\frac {\sqrt {c+d x} \left (279 a^3 d^3+a^2 b d^2 (-326 c+511 d x)+a b^2 d \left (200 c^2-252 c d x+385 d^2 x^2\right )+b^3 \left (-48 c^3+56 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )\right )}{(b c-a d)^4 (a+b x)^4}+\frac {105 d^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{9/2}}\right ) \] Input:

Integrate[1/((a + b*x)^5*Sqrt[c + d*x]),x]
 

Output:

((Sqrt[c + d*x]*(279*a^3*d^3 + a^2*b*d^2*(-326*c + 511*d*x) + a*b^2*d*(200 
*c^2 - 252*c*d*x + 385*d^2*x^2) + b^3*(-48*c^3 + 56*c^2*d*x - 70*c*d^2*x^2 
 + 105*d^3*x^3)))/((b*c - a*d)^4*(a + b*x)^4) + (105*d^4*ArcTan[(Sqrt[b]*S 
qrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(9/2)))/192
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {52, 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.

\(\displaystyle \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {7 d \int \frac {1}{(a+b x)^4 \sqrt {c+d x}}dx}{8 (b c-a d)}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {7 d \left (-\frac {5 d \int \frac {1}{(a+b x)^3 \sqrt {c+d x}}dx}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 (b c-a d)}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {7 d \left (-\frac {5 d \left (-\frac {3 d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}}dx}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 (b c-a d)}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {7 d \left (-\frac {5 d \left (-\frac {3 d \left (-\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{2 (b c-a d)}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 (b c-a d)}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {7 d \left (-\frac {5 d \left (-\frac {3 d \left (-\frac {\int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b c-a d}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 (b c-a d)}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {7 d \left (-\frac {5 d \left (-\frac {3 d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 (b c-a d)}-\frac {\sqrt {c+d x}}{3 (a+b x)^3 (b c-a d)}\right )}{8 (b c-a d)}-\frac {\sqrt {c+d x}}{4 (a+b x)^4 (b c-a d)}\)

Input:

Int[1/((a + b*x)^5*Sqrt[c + d*x]),x]
 

Output:

-1/4*Sqrt[c + d*x]/((b*c - a*d)*(a + b*x)^4) - (7*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*c - a*d))
 

Defintions of rubi rules used

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 = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {\frac {35 d^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}+\frac {93 \left (\frac {\left (35 x^{3} d^{3}-\frac {70}{3} c \,d^{2} x^{2}+\frac {56}{3} c^{2} d x -16 c^{3}\right ) b^{3}}{93}+\frac {200 a d \left (\frac {77}{40} d^{2} x^{2}-\frac {63}{50} c d x +c^{2}\right ) b^{2}}{279}-\frac {326 a^{2} d^{2} \left (-\frac {511 x d}{326}+c \right ) b}{279}+a^{3} d^{3}\right ) \sqrt {x d +c}\, \sqrt {\left (a d -b c \right ) b}}{64}}{\left (a d -b c \right )^{4} \left (b x +a \right )^{4} \sqrt {\left (a d -b c \right ) b}}\)	\(169\)
derivativedivides	\(2 d^{4} \left (\frac {\sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{4}}+\frac {\frac {7 \sqrt {x d +c}}{48 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {x d +c}}{24 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}\right )}{8 \left (a d -b c \right )}}{a d -b c}\right )\)	\(236\)
default	\(2 d^{4} \left (\frac {\sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{4}}+\frac {\frac {7 \sqrt {x d +c}}{48 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {x d +c}}{24 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{6 \left (a d -b c \right )}\right )}{8 \left (a d -b c \right )}}{a d -b c}\right )\)	\(236\)

Input:

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

Output:

93/64/((a*d-b*c)*b)^(1/2)*(35/93*d^4*(b*x+a)^4*arctan(b*(d*x+c)^(1/2)/((a* 
d-b*c)*b)^(1/2))+(1/93*(35*x^3*d^3-70/3*c*d^2*x^2+56/3*c^2*d*x-16*c^3)*b^3 
+200/279*a*d*(77/40*d^2*x^2-63/50*c*d*x+c^2)*b^2-326/279*a^2*d^2*(-511/326 
*x*d+c)*b+a^3*d^3)*(d*x+c)^(1/2)*((a*d-b*c)*b)^(1/2))/(a*d-b*c)^4/(b*x+a)^ 
4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (152) = 304\).

Time = 0.10 (sec) , antiderivative size = 1325, normalized size of antiderivative = 7.36 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/384*(105*(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 - 248*a*b^4*c^3*d + 5 
26*a^2*b^3*c^2*d^2 - 605*a^3*b^2*c*d^3 + 279*a^4*b*d^4 - 105*(b^5*c*d^3 - 
a*b^4*d^4)*x^3 + 35*(2*b^5*c^2*d^2 - 13*a*b^4*c*d^3 + 11*a^2*b^3*d^4)*x^2 
- 7*(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^5 - 5*a^5*b^5*c^4*d + 10*a^6*b^4*c^3*d^2 - 10 
*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5 + (b^10*c^5 - 5*a*b^9*c^4*d 
 + 10*a^2*b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5 
)*x^4 + 4*(a*b^9*c^5 - 5*a^2*b^8*c^4*d + 10*a^3*b^7*c^3*d^2 - 10*a^4*b^6*c 
^2*d^3 + 5*a^5*b^5*c*d^4 - a^6*b^4*d^5)*x^3 + 6*(a^2*b^8*c^5 - 5*a^3*b^7*c 
^4*d + 10*a^4*b^6*c^3*d^2 - 10*a^5*b^5*c^2*d^3 + 5*a^6*b^4*c*d^4 - a^7*b^3 
*d^5)*x^2 + 4*(a^3*b^7*c^5 - 5*a^4*b^6*c^4*d + 10*a^5*b^5*c^3*d^2 - 10*a^6 
*b^4*c^2*d^3 + 5*a^7*b^3*c*d^4 - a^8*b^2*d^5)*x), 1/192*(105*(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 - 248*a*b^4*c^3*d + 526*a^2*b^3*c^2*d^2 - 605*a^3*b^2*c*d^3 + 279 
*a^4*b*d^4 - 105*(b^5*c*d^3 - a*b^4*d^4)*x^3 + 35*(2*b^5*c^2*d^2 - 13*a*b^ 
4*c*d^3 + 11*a^2*b^3*d^4)*x^2 - 7*(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^5 - 5*a^5*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(b*x+a)^5/(d*x+c)^(1/2),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. 331 vs. \(2 (152) = 304\).

Time = 0.13 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\frac {35 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d}} + \frac {105 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 385 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} + 511 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} - 279 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 385 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} - 1022 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} + 837 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} + 511 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} - 837 \, \sqrt {d x + c} a^{2} b c d^{6} + 279 \, \sqrt {d x + c} a^{3} d^{7}}{192 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \] Input:

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

Output:

35/64*d^4*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^4 - 4*a*b^3 
*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b^2*c + a*b*d) 
) + 1/192*(105*(d*x + c)^(7/2)*b^3*d^4 - 385*(d*x + c)^(5/2)*b^3*c*d^4 + 5 
11*(d*x + c)^(3/2)*b^3*c^2*d^4 - 279*sqrt(d*x + c)*b^3*c^3*d^4 + 385*(d*x 
+ c)^(5/2)*a*b^2*d^5 - 1022*(d*x + c)^(3/2)*a*b^2*c*d^5 + 837*sqrt(d*x + c 
)*a*b^2*c^2*d^5 + 511*(d*x + c)^(3/2)*a^2*b*d^6 - 837*sqrt(d*x + c)*a^2*b* 
c*d^6 + 279*sqrt(d*x + c)*a^3*d^7)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c 
^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*((d*x + c)*b - b*c + a*d)^4)
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx=\frac {\frac {93\,d^4\,\sqrt {c+d\,x}}{64\,\left (a\,d-b\,c\right )}+\frac {385\,b^2\,d^4\,{\left (c+d\,x\right )}^{5/2}}{192\,{\left (a\,d-b\,c\right )}^3}+\frac {35\,b^3\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^4}+\frac {511\,b\,d^4\,{\left (c+d\,x\right )}^{3/2}}{192\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3}+\frac {35\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{9/2}} \] Input:

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

Output:

((93*d^4*(c + d*x)^(1/2))/(64*(a*d - b*c)) + (385*b^2*d^4*(c + d*x)^(5/2)) 
/(192*(a*d - b*c)^3) + (35*b^3*d^4*(c + d*x)^(7/2))/(64*(a*d - b*c)^4) + ( 
511*b*d^4*(c + d*x)^(3/2))/(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) + (35*d^4*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(64* 
b^(1/2)*(a*d - b*c)^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 903, normalized size of antiderivative = 5.02 \[ \int \frac {1}{(a+b x)^5 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(105*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b* 
c)))*a**4*d**4 + 420*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 + 630*sqrt(b)*sqrt(a*d - b*c)*atan((sqr 
t(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**2*d**4*x**2 + 420*sqrt(b) 
*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**3* 
d**4*x**3 + 105*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sq 
rt(a*d - b*c)))*b**4*d**4*x**4 + 279*sqrt(c + d*x)*a**4*b*d**4 - 605*sqrt( 
c + d*x)*a**3*b**2*c*d**3 + 511*sqrt(c + d*x)*a**3*b**2*d**4*x + 526*sqrt( 
c + d*x)*a**2*b**3*c**2*d**2 - 763*sqrt(c + d*x)*a**2*b**3*c*d**3*x + 385* 
sqrt(c + d*x)*a**2*b**3*d**4*x**2 - 248*sqrt(c + d*x)*a*b**4*c**3*d + 308* 
sqrt(c + d*x)*a*b**4*c**2*d**2*x - 455*sqrt(c + d*x)*a*b**4*c*d**3*x**2 + 
105*sqrt(c + d*x)*a*b**4*d**4*x**3 + 48*sqrt(c + d*x)*b**5*c**4 - 56*sqrt( 
c + d*x)*b**5*c**3*d*x + 70*sqrt(c + d*x)*b**5*c**2*d**2*x**2 - 105*sqrt(c 
 + d*x)*b**5*c*d**3*x**3)/(192*b*(a**9*d**5 - 5*a**8*b*c*d**4 + 4*a**8*b*d 
**5*x + 10*a**7*b**2*c**2*d**3 - 20*a**7*b**2*c*d**4*x + 6*a**7*b**2*d**5* 
x**2 - 10*a**6*b**3*c**3*d**2 + 40*a**6*b**3*c**2*d**3*x - 30*a**6*b**3*c* 
d**4*x**2 + 4*a**6*b**3*d**5*x**3 + 5*a**5*b**4*c**4*d - 40*a**5*b**4*c**3 
*d**2*x + 60*a**5*b**4*c**2*d**3*x**2 - 20*a**5*b**4*c*d**4*x**3 + a**5*b* 
*4*d**5*x**4 - a**4*b**5*c**5 + 20*a**4*b**5*c**4*d*x - 60*a**4*b**5*c**3* 
d**2*x**2 + 40*a**4*b**5*c**2*d**3*x**3 - 5*a**4*b**5*c*d**4*x**4 - 4*a...