\(\int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx\) [320]

Optimal result

Integrand size = 19, antiderivative size = 171 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac {256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}} \] Output:

-2/11*(d*x+c)^(3/2)/(-a*d+b*c)/(b*x+a)^(11/2)+16/99*d*(d*x+c)^(3/2)/(-a*d+ 
b*c)^2/(b*x+a)^(9/2)-32/231*d^2*(d*x+c)^(3/2)/(-a*d+b*c)^3/(b*x+a)^(7/2)+1 
28/1155*d^3*(d*x+c)^(3/2)/(-a*d+b*c)^4/(b*x+a)^(5/2)-256/3465*d^4*(d*x+c)^ 
(3/2)/(-a*d+b*c)^5/(b*x+a)^(3/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{3/2} \left (1155 a^4 d^4+924 a^3 b d^3 (-3 c+2 d x)+198 a^2 b^2 d^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )+44 a b^3 d \left (-35 c^3+30 c^2 d x-24 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (315 c^4-280 c^3 d x+240 c^2 d^2 x^2-192 c d^3 x^3+128 d^4 x^4\right )\right )}{3465 (b c-a d)^5 (a+b x)^{11/2}} \] Input:

Integrate[Sqrt[c + d*x]/(a + b*x)^(13/2),x]
 

Output:

(-2*(c + d*x)^(3/2)*(1155*a^4*d^4 + 924*a^3*b*d^3*(-3*c + 2*d*x) + 198*a^2 
*b^2*d^2*(15*c^2 - 12*c*d*x + 8*d^2*x^2) + 44*a*b^3*d*(-35*c^3 + 30*c^2*d* 
x - 24*c*d^2*x^2 + 16*d^3*x^3) + b^4*(315*c^4 - 280*c^3*d*x + 240*c^2*d^2* 
x^2 - 192*c*d^3*x^3 + 128*d^4*x^4)))/(3465*(b*c - a*d)^5*(a + b*x)^(11/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {55, 55, 55, 55, 48}

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.

\(\displaystyle \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {8 d \int \frac {\sqrt {c+d x}}{(a+b x)^{11/2}}dx}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {8 d \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}}dx}{3 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}}dx}{7 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}}dx}{5 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}-\frac {8 d \left (-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}-\frac {2 d \left (-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}-\frac {4 d \left (\frac {4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}\right )}{3 (b c-a d)}\right )}{11 (b c-a d)}\)

Input:

Int[Sqrt[c + d*x]/(a + b*x)^(13/2),x]
 

Output:

(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) - (8*d*((-2*(c + d* 
x)^(3/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) - (2*d*((-2*(c + d*x)^(3/2))/(7* 
(b*c - a*d)*(a + b*x)^(7/2)) - (4*d*((-2*(c + d*x)^(3/2))/(5*(b*c - a*d)*( 
a + b*x)^(5/2)) + (4*d*(c + d*x)^(3/2))/(15*(b*c - a*d)^2*(a + b*x)^(3/2)) 
))/(7*(b*c - a*d))))/(3*(b*c - a*d))))/(11*(b*c - a*d))
 

Defintions of rubi rules used

rule 48

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] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

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*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {\sqrt {x d +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {x d +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {x d +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {x d +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {x d +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {x d +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {x d +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\)	\(248\)
gosper	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (128 d^{4} x^{4} b^{4}+704 a \,b^{3} d^{4} x^{3}-192 b^{4} c \,d^{3} x^{3}+1584 a^{2} b^{2} d^{4} x^{2}-1056 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+1848 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1320 a \,b^{3} c^{2} d^{2} x -280 b^{4} c^{3} d x +1155 d^{4} a^{4}-2772 a^{3} b c \,d^{3}+2970 a^{2} b^{2} c^{2} d^{2}-1540 a \,b^{3} c^{3} d +315 c^{4} b^{4}\right )}{3465 \left (b x +a \right )^{\frac {11}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}\)	\(256\)
orering	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (128 d^{4} x^{4} b^{4}+704 a \,b^{3} d^{4} x^{3}-192 b^{4} c \,d^{3} x^{3}+1584 a^{2} b^{2} d^{4} x^{2}-1056 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+1848 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1320 a \,b^{3} c^{2} d^{2} x -280 b^{4} c^{3} d x +1155 d^{4} a^{4}-2772 a^{3} b c \,d^{3}+2970 a^{2} b^{2} c^{2} d^{2}-1540 a \,b^{3} c^{3} d +315 c^{4} b^{4}\right )}{3465 \left (b x +a \right )^{\frac {11}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}\)	\(256\)

Input:

int((d*x+c)^(1/2)/(b*x+a)^(13/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5/b*(d*x+c)^(1/2)/(b*x+a)^(11/2)+1/10*(a*d-b*c)/b*(-2/11/(-a*d+b*c)/(b* 
x+a)^(11/2)*(d*x+c)^(1/2)-10/11*d/(-a*d+b*c)*(-2/9*(d*x+c)^(1/2)/(-a*d+b*c 
)/(b*x+a)^(9/2)-8/9*d/(-a*d+b*c)*(-2/7*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(7 
/2)-6/7*d/(-a*d+b*c)*(-2/5*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(5/2)-4/5*d/(- 
a*d+b*c)*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(3/2)+4/3*d*(d*x+c)^(1/2)/ 
(-a*d+b*c)^2/(b*x+a)^(1/2))))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (141) = 282\).

Time = 5.14 (sec) , antiderivative size = 781, normalized size of antiderivative = 4.57 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=-\frac {2 \, {\left (128 \, b^{4} d^{5} x^{5} + 315 \, b^{4} c^{5} - 1540 \, a b^{3} c^{4} d + 2970 \, a^{2} b^{2} c^{3} d^{2} - 2772 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} - 64 \, {\left (b^{4} c d^{4} - 11 \, a b^{3} d^{5}\right )} x^{4} + 16 \, {\left (3 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} + 99 \, a^{2} b^{2} d^{5}\right )} x^{3} - 8 \, {\left (5 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} - 231 \, a^{3} b d^{5}\right )} x^{2} + {\left (35 \, b^{4} c^{4} d - 220 \, a b^{3} c^{3} d^{2} + 594 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} + 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3465 \, {\left (a^{6} b^{5} c^{5} - 5 \, a^{7} b^{4} c^{4} d + 10 \, a^{8} b^{3} c^{3} d^{2} - 10 \, a^{9} b^{2} c^{2} d^{3} + 5 \, a^{10} b c d^{4} - a^{11} d^{5} + {\left (b^{11} c^{5} - 5 \, a b^{10} c^{4} d + 10 \, a^{2} b^{9} c^{3} d^{2} - 10 \, a^{3} b^{8} c^{2} d^{3} + 5 \, a^{4} b^{7} c d^{4} - a^{5} b^{6} d^{5}\right )} x^{6} + 6 \, {\left (a b^{10} c^{5} - 5 \, a^{2} b^{9} c^{4} d + 10 \, a^{3} b^{8} c^{3} d^{2} - 10 \, a^{4} b^{7} c^{2} d^{3} + 5 \, a^{5} b^{6} c d^{4} - a^{6} b^{5} d^{5}\right )} x^{5} + 15 \, {\left (a^{2} b^{9} c^{5} - 5 \, a^{3} b^{8} c^{4} d + 10 \, a^{4} b^{7} c^{3} d^{2} - 10 \, a^{5} b^{6} c^{2} d^{3} + 5 \, a^{6} b^{5} c d^{4} - a^{7} b^{4} d^{5}\right )} x^{4} + 20 \, {\left (a^{3} b^{8} c^{5} - 5 \, a^{4} b^{7} c^{4} d + 10 \, a^{5} b^{6} c^{3} d^{2} - 10 \, a^{6} b^{5} c^{2} d^{3} + 5 \, a^{7} b^{4} c d^{4} - a^{8} b^{3} d^{5}\right )} x^{3} + 15 \, {\left (a^{4} b^{7} c^{5} - 5 \, a^{5} b^{6} c^{4} d + 10 \, a^{6} b^{5} c^{3} d^{2} - 10 \, a^{7} b^{4} c^{2} d^{3} + 5 \, a^{8} b^{3} c d^{4} - a^{9} b^{2} d^{5}\right )} x^{2} + 6 \, {\left (a^{5} b^{6} c^{5} - 5 \, a^{6} b^{5} c^{4} d + 10 \, a^{7} b^{4} c^{3} d^{2} - 10 \, a^{8} b^{3} c^{2} d^{3} + 5 \, a^{9} b^{2} c d^{4} - a^{10} b d^{5}\right )} x\right )}} \] Input:

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),x, algorithm="fricas")
 

Output:

-2/3465*(128*b^4*d^5*x^5 + 315*b^4*c^5 - 1540*a*b^3*c^4*d + 2970*a^2*b^2*c 
^3*d^2 - 2772*a^3*b*c^2*d^3 + 1155*a^4*c*d^4 - 64*(b^4*c*d^4 - 11*a*b^3*d^ 
5)*x^4 + 16*(3*b^4*c^2*d^3 - 22*a*b^3*c*d^4 + 99*a^2*b^2*d^5)*x^3 - 8*(5*b 
^4*c^3*d^2 - 33*a*b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 - 231*a^3*b*d^5)*x^2 + (3 
5*b^4*c^4*d - 220*a*b^3*c^3*d^2 + 594*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 + 
1155*a^4*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^5*c^5 - 5*a^7*b^4*c^4* 
d + 10*a^8*b^3*c^3*d^2 - 10*a^9*b^2*c^2*d^3 + 5*a^10*b*c*d^4 - a^11*d^5 + 
(b^11*c^5 - 5*a*b^10*c^4*d + 10*a^2*b^9*c^3*d^2 - 10*a^3*b^8*c^2*d^3 + 5*a 
^4*b^7*c*d^4 - a^5*b^6*d^5)*x^6 + 6*(a*b^10*c^5 - 5*a^2*b^9*c^4*d + 10*a^3 
*b^8*c^3*d^2 - 10*a^4*b^7*c^2*d^3 + 5*a^5*b^6*c*d^4 - a^6*b^5*d^5)*x^5 + 1 
5*(a^2*b^9*c^5 - 5*a^3*b^8*c^4*d + 10*a^4*b^7*c^3*d^2 - 10*a^5*b^6*c^2*d^3 
 + 5*a^6*b^5*c*d^4 - a^7*b^4*d^5)*x^4 + 20*(a^3*b^8*c^5 - 5*a^4*b^7*c^4*d 
+ 10*a^5*b^6*c^3*d^2 - 10*a^6*b^5*c^2*d^3 + 5*a^7*b^4*c*d^4 - a^8*b^3*d^5) 
*x^3 + 15*(a^4*b^7*c^5 - 5*a^5*b^6*c^4*d + 10*a^6*b^5*c^3*d^2 - 10*a^7*b^4 
*c^2*d^3 + 5*a^8*b^3*c*d^4 - a^9*b^2*d^5)*x^2 + 6*(a^5*b^6*c^5 - 5*a^6*b^5 
*c^4*d + 10*a^7*b^4*c^3*d^2 - 10*a^8*b^3*c^2*d^3 + 5*a^9*b^2*c*d^4 - a^10* 
b*d^5)*x)
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {13}{2}}}\, dx \] Input:

integrate((d*x+c)**(1/2)/(b*x+a)**(13/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(c + d*x)/(a + b*x)**(13/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),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. 1345 vs. \(2 (141) = 282\).

Time = 0.56 (sec) , antiderivative size = 1345, normalized size of antiderivative = 7.87 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/(b*x+a)^(13/2),x, algorithm="giac")
 

Output:

-512/3465*(sqrt(b*d)*b^16*c^6*d^5 - 6*sqrt(b*d)*a*b^15*c^5*d^6 + 15*sqrt(b 
*d)*a^2*b^14*c^4*d^7 - 20*sqrt(b*d)*a^3*b^13*c^3*d^8 + 15*sqrt(b*d)*a^4*b^ 
12*c^2*d^9 - 6*sqrt(b*d)*a^5*b^11*c*d^10 + sqrt(b*d)*a^6*b^10*d^11 - 11*sq 
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2* 
b^14*c^5*d^5 + 55*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^13*c^4*d^6 - 110*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^12*c^3*d^7 + 110*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^ 
11*c^2*d^8 - 55*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^10*c*d^9 + 11*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^9*d^10 + 55*sqrt(b*d)*(sqrt( 
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^12*c^4*d^5 - 
 220*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b 
*d))^4*a*b^11*c^3*d^6 + 330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2* 
c + (b*x + a)*b*d - a*b*d))^4*a^2*b^10*c^2*d^7 - 220*sqrt(b*d)*(sqrt(b*d)* 
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^9*c*d^8 + 55* 
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^ 
4*a^4*b^8*d^9 - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x 
 + a)*b*d - a*b*d))^6*b^10*c^3*d^5 + 495*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a 
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^9*c^2*d^6 - 495*sqrt(b*d...
 

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {2310\,a^4\,c\,d^4-5544\,a^3\,b\,c^2\,d^3+5940\,a^2\,b^2\,c^3\,d^2-3080\,a\,b^3\,c^4\,d+630\,b^4\,c^5}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (2310\,a^4\,d^5-1848\,a^3\,b\,c\,d^4+1188\,a^2\,b^2\,c^2\,d^3-440\,a\,b^3\,c^3\,d^2+70\,b^4\,c^4\,d\right )}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {256\,d^5\,x^5}{3465\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {16\,d^2\,x^2\,\left (231\,a^3\,d^3-99\,a^2\,b\,c\,d^2+33\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{3465\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{3465\,b^2\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3465\,b^3\,{\left (a\,d-b\,c\right )}^5}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \] Input:

int((c + d*x)^(1/2)/(a + b*x)^(13/2),x)
 

Output:

((c + d*x)^(1/2)*((630*b^4*c^5 + 2310*a^4*c*d^4 - 5544*a^3*b*c^2*d^3 + 594 
0*a^2*b^2*c^3*d^2 - 3080*a*b^3*c^4*d)/(3465*b^5*(a*d - b*c)^5) + (x*(2310* 
a^4*d^5 + 70*b^4*c^4*d - 440*a*b^3*c^3*d^2 + 1188*a^2*b^2*c^2*d^3 - 1848*a 
^3*b*c*d^4))/(3465*b^5*(a*d - b*c)^5) + (256*d^5*x^5)/(3465*b*(a*d - b*c)^ 
5) + (16*d^2*x^2*(231*a^3*d^3 - 5*b^3*c^3 + 33*a*b^2*c^2*d - 99*a^2*b*c*d^ 
2))/(3465*b^4*(a*d - b*c)^5) + (128*d^4*x^4*(11*a*d - b*c))/(3465*b^2*(a*d 
 - b*c)^5) + (32*d^3*x^3*(99*a^2*d^2 + 3*b^2*c^2 - 22*a*b*c*d))/(3465*b^3* 
(a*d - b*c)^5)))/(x^5*(a + b*x)^(1/2) + (a^5*(a + b*x)^(1/2))/b^5 + (10*a^ 
2*x^3*(a + b*x)^(1/2))/b^2 + (10*a^3*x^2*(a + b*x)^(1/2))/b^3 + (5*a*x^4*( 
a + b*x)^(1/2))/b + (5*a^4*x*(a + b*x)^(1/2))/b^4)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1027, normalized size of antiderivative = 6.01 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)/(b*x+a)^(13/2),x)
 

Output:

(2*( - 128*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*d**5 - 640*sqrt(d)*sqrt(b)*s 
qrt(a + b*x)*a**4*b*d**5*x - 1280*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**2* 
d**5*x**2 - 1280*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**3*d**5*x**3 - 640*s 
qrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**4*d**5*x**4 - 128*sqrt(d)*sqrt(b)*sqrt(a 
 + b*x)*b**5*d**5*x**5 + 1155*sqrt(c + d*x)*a**4*b**2*c*d**4 + 1155*sqrt(c 
 + d*x)*a**4*b**2*d**5*x - 2772*sqrt(c + d*x)*a**3*b**3*c**2*d**3 - 924*sq 
rt(c + d*x)*a**3*b**3*c*d**4*x + 1848*sqrt(c + d*x)*a**3*b**3*d**5*x**2 + 
2970*sqrt(c + d*x)*a**2*b**4*c**3*d**2 + 594*sqrt(c + d*x)*a**2*b**4*c**2* 
d**3*x - 792*sqrt(c + d*x)*a**2*b**4*c*d**4*x**2 + 1584*sqrt(c + d*x)*a**2 
*b**4*d**5*x**3 - 1540*sqrt(c + d*x)*a*b**5*c**4*d - 220*sqrt(c + d*x)*a*b 
**5*c**3*d**2*x + 264*sqrt(c + d*x)*a*b**5*c**2*d**3*x**2 - 352*sqrt(c + d 
*x)*a*b**5*c*d**4*x**3 + 704*sqrt(c + d*x)*a*b**5*d**5*x**4 + 315*sqrt(c + 
 d*x)*b**6*c**5 + 35*sqrt(c + d*x)*b**6*c**4*d*x - 40*sqrt(c + d*x)*b**6*c 
**3*d**2*x**2 + 48*sqrt(c + d*x)*b**6*c**2*d**3*x**3 - 64*sqrt(c + d*x)*b* 
*6*c*d**4*x**4 + 128*sqrt(c + d*x)*b**6*d**5*x**5))/(3465*sqrt(a + b*x)*b* 
*2*(a**10*d**5 - 5*a**9*b*c*d**4 + 5*a**9*b*d**5*x + 10*a**8*b**2*c**2*d** 
3 - 25*a**8*b**2*c*d**4*x + 10*a**8*b**2*d**5*x**2 - 10*a**7*b**3*c**3*d** 
2 + 50*a**7*b**3*c**2*d**3*x - 50*a**7*b**3*c*d**4*x**2 + 10*a**7*b**3*d** 
5*x**3 + 5*a**6*b**4*c**4*d - 50*a**6*b**4*c**3*d**2*x + 100*a**6*b**4*c** 
2*d**3*x**2 - 50*a**6*b**4*c*d**4*x**3 + 5*a**6*b**4*d**5*x**4 - a**5*b...