\(\int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx\) [341]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=-\frac {2 (c+d x)^{7/2}}{13 (b c-a d) (a+b x)^{13/2}}+\frac {12 d (c+d x)^{7/2}}{143 (b c-a d)^2 (a+b x)^{11/2}}-\frac {16 d^2 (c+d x)^{7/2}}{429 (b c-a d)^3 (a+b x)^{9/2}}+\frac {32 d^3 (c+d x)^{7/2}}{3003 (b c-a d)^4 (a+b x)^{7/2}} \] Output:

-2/13*(d*x+c)^(7/2)/(-a*d+b*c)/(b*x+a)^(13/2)+12/143*d*(d*x+c)^(7/2)/(-a*d 
+b*c)^2/(b*x+a)^(11/2)-16/429*d^2*(d*x+c)^(7/2)/(-a*d+b*c)^3/(b*x+a)^(9/2) 
+32/3003*d^3*(d*x+c)^(7/2)/(-a*d+b*c)^4/(b*x+a)^(7/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=\frac {2 (c+d x)^{7/2} \left (429 a^3 d^3+143 a^2 b d^2 (-7 c+2 d x)+13 a b^2 d \left (63 c^2-28 c d x+8 d^2 x^2\right )+b^3 \left (-231 c^3+126 c^2 d x-56 c d^2 x^2+16 d^3 x^3\right )\right )}{3003 (b c-a d)^4 (a+b x)^{13/2}} \] Input:

Integrate[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]
 

Output:

(2*(c + d*x)^(7/2)*(429*a^3*d^3 + 143*a^2*b*d^2*(-7*c + 2*d*x) + 13*a*b^2* 
d*(63*c^2 - 28*c*d*x + 8*d^2*x^2) + b^3*(-231*c^3 + 126*c^2*d*x - 56*c*d^2 
*x^2 + 16*d^3*x^3)))/(3003*(b*c - a*d)^4*(a + b*x)^(13/2))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {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.

\(\displaystyle \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {6 d \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}}dx}{13 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {6 d \left (-\frac {4 d \int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}}dx}{11 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)}\right )}{13 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {6 d \left (-\frac {4 d \left (-\frac {2 d \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)}\right )}{13 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {2 (c+d x)^{7/2}}{13 (a+b x)^{13/2} (b c-a d)}-\frac {6 d \left (-\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)}-\frac {4 d \left (\frac {4 d (c+d x)^{7/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}\right )}{13 (b c-a d)}\)

Input:

Int[(c + d*x)^(5/2)/(a + b*x)^(15/2),x]
 

Output:

(-2*(c + d*x)^(7/2))/(13*(b*c - a*d)*(a + b*x)^(13/2)) - (6*d*((-2*(c + d* 
x)^(7/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) - (4*d*((-2*(c + d*x)^(7/2))/( 
9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(7/2))/(63*(b*c - a*d)^2*( 
a + b*x)^(7/2))))/(11*(b*c - a*d))))/(13*(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 = 171, normalized size of antiderivative = 1.26

method	result	size
gosper	\(\frac {2 \left (x d +c \right )^{\frac {7}{2}} \left (16 d^{3} x^{3} b^{3}+104 x^{2} a \,b^{2} d^{3}-56 x^{2} b^{3} c \,d^{2}+286 x \,a^{2} b \,d^{3}-364 x a \,b^{2} c \,d^{2}+126 x \,b^{3} c^{2} d +429 a^{3} d^{3}-1001 a^{2} b c \,d^{2}+819 a \,b^{2} c^{2} d -231 b^{3} c^{3}\right )}{3003 \left (b x +a \right )^{\frac {13}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(171\)
orering	\(\frac {2 \left (x d +c \right )^{\frac {7}{2}} \left (16 d^{3} x^{3} b^{3}+104 x^{2} a \,b^{2} d^{3}-56 x^{2} b^{3} c \,d^{2}+286 x \,a^{2} b \,d^{3}-364 x a \,b^{2} c \,d^{2}+126 x \,b^{3} c^{2} d +429 a^{3} d^{3}-1001 a^{2} b c \,d^{2}+819 a \,b^{2} c^{2} d -231 b^{3} c^{3}\right )}{3003 \left (b x +a \right )^{\frac {13}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(171\)
default	\(-\frac {\left (x d +c \right )^{\frac {5}{2}}}{4 b \left (b x +a \right )^{\frac {13}{2}}}+\frac {5 \left (a d -b c \right ) \left (-\frac {\left (x d +c \right )^{\frac {3}{2}}}{5 b \left (b x +a \right )^{\frac {13}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {x d +c}}{6 b \left (b x +a \right )^{\frac {13}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {x d +c}}{13 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {13}{2}}}-\frac {12 d \left (-\frac {2 \sqrt {x d +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {x d +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {x d +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {x d +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {x d +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {x d +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{13 \left (-a d +b c \right )}\right )}{12 b}\right )}{10 b}\right )}{8 b}\)	\(354\)

Input:

int((d*x+c)^(5/2)/(b*x+a)^(15/2),x,method=_RETURNVERBOSE)
 

Output:

2/3003*(d*x+c)^(7/2)*(16*b^3*d^3*x^3+104*a*b^2*d^3*x^2-56*b^3*c*d^2*x^2+28 
6*a^2*b*d^3*x-364*a*b^2*c*d^2*x+126*b^3*c^2*d*x+429*a^3*d^3-1001*a^2*b*c*d 
^2+819*a*b^2*c^2*d-231*b^3*c^3)/(b*x+a)^(13/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^ 
2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (112) = 224\).

Time = 11.54 (sec) , antiderivative size = 765, normalized size of antiderivative = 5.62 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=\frac {2 \, {\left (16 \, b^{3} d^{6} x^{6} - 231 \, b^{3} c^{6} + 819 \, a b^{2} c^{5} d - 1001 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} - 8 \, {\left (b^{3} c d^{5} - 13 \, a b^{2} d^{6}\right )} x^{5} + 2 \, {\left (3 \, b^{3} c^{2} d^{4} - 26 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} - {\left (5 \, b^{3} c^{3} d^{3} - 39 \, a b^{2} c^{2} d^{4} + 143 \, a^{2} b c d^{5} - 429 \, a^{3} d^{6}\right )} x^{3} - {\left (371 \, b^{3} c^{4} d^{2} - 1469 \, a b^{2} c^{3} d^{3} + 2145 \, a^{2} b c^{2} d^{4} - 1287 \, a^{3} c d^{5}\right )} x^{2} - {\left (567 \, b^{3} c^{5} d - 2093 \, a b^{2} c^{4} d^{2} + 2717 \, a^{2} b c^{3} d^{3} - 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3003 \, {\left (a^{7} b^{4} c^{4} - 4 \, a^{8} b^{3} c^{3} d + 6 \, a^{9} b^{2} c^{2} d^{2} - 4 \, a^{10} b c d^{3} + a^{11} d^{4} + {\left (b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}\right )} x^{7} + 7 \, {\left (a b^{10} c^{4} - 4 \, a^{2} b^{9} c^{3} d + 6 \, a^{3} b^{8} c^{2} d^{2} - 4 \, a^{4} b^{7} c d^{3} + a^{5} b^{6} d^{4}\right )} x^{6} + 21 \, {\left (a^{2} b^{9} c^{4} - 4 \, a^{3} b^{8} c^{3} d + 6 \, a^{4} b^{7} c^{2} d^{2} - 4 \, a^{5} b^{6} c d^{3} + a^{6} b^{5} d^{4}\right )} x^{5} + 35 \, {\left (a^{3} b^{8} c^{4} - 4 \, a^{4} b^{7} c^{3} d + 6 \, a^{5} b^{6} c^{2} d^{2} - 4 \, a^{6} b^{5} c d^{3} + a^{7} b^{4} d^{4}\right )} x^{4} + 35 \, {\left (a^{4} b^{7} c^{4} - 4 \, a^{5} b^{6} c^{3} d + 6 \, a^{6} b^{5} c^{2} d^{2} - 4 \, a^{7} b^{4} c d^{3} + a^{8} b^{3} d^{4}\right )} x^{3} + 21 \, {\left (a^{5} b^{6} c^{4} - 4 \, a^{6} b^{5} c^{3} d + 6 \, a^{7} b^{4} c^{2} d^{2} - 4 \, a^{8} b^{3} c d^{3} + a^{9} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (a^{6} b^{5} c^{4} - 4 \, a^{7} b^{4} c^{3} d + 6 \, a^{8} b^{3} c^{2} d^{2} - 4 \, a^{9} b^{2} c d^{3} + a^{10} b d^{4}\right )} x\right )}} \] Input:

integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),x, algorithm="fricas")
 

Output:

2/3003*(16*b^3*d^6*x^6 - 231*b^3*c^6 + 819*a*b^2*c^5*d - 1001*a^2*b*c^4*d^ 
2 + 429*a^3*c^3*d^3 - 8*(b^3*c*d^5 - 13*a*b^2*d^6)*x^5 + 2*(3*b^3*c^2*d^4 
- 26*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 - (5*b^3*c^3*d^3 - 39*a*b^2*c^2*d^4 
+ 143*a^2*b*c*d^5 - 429*a^3*d^6)*x^3 - (371*b^3*c^4*d^2 - 1469*a*b^2*c^3*d 
^3 + 2145*a^2*b*c^2*d^4 - 1287*a^3*c*d^5)*x^2 - (567*b^3*c^5*d - 2093*a*b^ 
2*c^4*d^2 + 2717*a^2*b*c^3*d^3 - 1287*a^3*c^2*d^4)*x)*sqrt(b*x + a)*sqrt(d 
*x + c)/(a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^ 
3 + a^11*d^4 + (b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8* 
c*d^3 + a^4*b^7*d^4)*x^7 + 7*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2 
*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*x^6 + 21*(a^2*b^9*c^4 - 4*a^3*b^8*c^ 
3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*x^5 + 35*(a^3*b^8 
*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4 
)*x^4 + 35*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4* 
c*d^3 + a^8*b^3*d^4)*x^3 + 21*(a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c 
^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*x^2 + 7*(a^6*b^5*c^4 - 4*a^7*b^4*c 
^3*d + 6*a^8*b^3*c^2*d^2 - 4*a^9*b^2*c*d^3 + a^10*b*d^4)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(5/2)/(b*x+a)**(15/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),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. 2868 vs. \(2 (112) = 224\).

Time = 0.69 (sec) , antiderivative size = 2868, normalized size of antiderivative = 21.09 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)/(b*x+a)^(15/2),x, algorithm="giac")
 

Output:

64/3003*(sqrt(b*d)*b^18*c^9*d^6*abs(b) - 9*sqrt(b*d)*a*b^17*c^8*d^7*abs(b) 
 + 36*sqrt(b*d)*a^2*b^16*c^7*d^8*abs(b) - 84*sqrt(b*d)*a^3*b^15*c^6*d^9*ab 
s(b) + 126*sqrt(b*d)*a^4*b^14*c^5*d^10*abs(b) - 126*sqrt(b*d)*a^5*b^13*c^4 
*d^11*abs(b) + 84*sqrt(b*d)*a^6*b^12*c^3*d^12*abs(b) - 36*sqrt(b*d)*a^7*b^ 
11*c^2*d^13*abs(b) + 9*sqrt(b*d)*a^8*b^10*c*d^14*abs(b) - sqrt(b*d)*a^9*b^ 
9*d^15*abs(b) - 13*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x 
+ a)*b*d - a*b*d))^2*b^16*c^8*d^6*abs(b) + 104*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^15*c^7*d^7*abs(b) - 3 
64*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^14*c^6*d^8*abs(b) + 728*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq 
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^13*c^5*d^9*abs(b) - 910*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 
^12*c^4*d^10*abs(b) + 728*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^11*c^3*d^11*abs(b) - 364*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^10*c^2* 
d^12*abs(b) + 104*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^9*c*d^13*abs(b) - 13*sqrt(b*d)*(sqrt(b*d)*sqrt(b 
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^8*d^14*abs(b) + 78* 
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^ 
4*b^14*c^7*d^6*abs(b) - 546*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b...
 

Mupad [B] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.38 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (2574\,a^3\,c\,d^5-4290\,a^2\,b\,c^2\,d^4+2938\,a\,b^2\,c^3\,d^3-742\,b^3\,c^4\,d^2\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}-\frac {-858\,a^3\,c^3\,d^3+2002\,a^2\,b\,c^4\,d^2-1638\,a\,b^2\,c^5\,d+462\,b^3\,c^6}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x^3\,\left (858\,a^3\,d^6-286\,a^2\,b\,c\,d^5+78\,a\,b^2\,c^2\,d^4-10\,b^3\,c^3\,d^3\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,d^6\,x^6}{3003\,b^3\,{\left (a\,d-b\,c\right )}^4}-\frac {x\,\left (-2574\,a^3\,c^2\,d^4+5434\,a^2\,b\,c^3\,d^3-4186\,a\,b^2\,c^4\,d^2+1134\,b^3\,c^5\,d\right )}{3003\,b^6\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^5\,x^5\,\left (13\,a\,d-b\,c\right )}{3003\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d^4\,x^4\,\left (143\,a^2\,d^2-26\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3003\,b^5\,{\left (a\,d-b\,c\right )}^4}\right )}{x^6\,\sqrt {a+b\,x}+\frac {a^6\,\sqrt {a+b\,x}}{b^6}+\frac {15\,a^2\,x^4\,\sqrt {a+b\,x}}{b^2}+\frac {20\,a^3\,x^3\,\sqrt {a+b\,x}}{b^3}+\frac {15\,a^4\,x^2\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a\,x^5\,\sqrt {a+b\,x}}{b}+\frac {6\,a^5\,x\,\sqrt {a+b\,x}}{b^5}} \] Input:

int((c + d*x)^(5/2)/(a + b*x)^(15/2),x)
 

Output:

((c + d*x)^(1/2)*((x^2*(2574*a^3*c*d^5 - 742*b^3*c^4*d^2 + 2938*a*b^2*c^3* 
d^3 - 4290*a^2*b*c^2*d^4))/(3003*b^6*(a*d - b*c)^4) - (462*b^3*c^6 - 858*a 
^3*c^3*d^3 + 2002*a^2*b*c^4*d^2 - 1638*a*b^2*c^5*d)/(3003*b^6*(a*d - b*c)^ 
4) + (x^3*(858*a^3*d^6 - 10*b^3*c^3*d^3 + 78*a*b^2*c^2*d^4 - 286*a^2*b*c*d 
^5))/(3003*b^6*(a*d - b*c)^4) + (32*d^6*x^6)/(3003*b^3*(a*d - b*c)^4) - (x 
*(1134*b^3*c^5*d - 2574*a^3*c^2*d^4 - 4186*a*b^2*c^4*d^2 + 5434*a^2*b*c^3* 
d^3))/(3003*b^6*(a*d - b*c)^4) + (16*d^5*x^5*(13*a*d - b*c))/(3003*b^4*(a* 
d - b*c)^4) + (4*d^4*x^4*(143*a^2*d^2 + 3*b^2*c^2 - 26*a*b*c*d))/(3003*b^5 
*(a*d - b*c)^4)))/(x^6*(a + b*x)^(1/2) + (a^6*(a + b*x)^(1/2))/b^6 + (15*a 
^2*x^4*(a + b*x)^(1/2))/b^2 + (20*a^3*x^3*(a + b*x)^(1/2))/b^3 + (15*a^4*x 
^2*(a + b*x)^(1/2))/b^4 + (6*a*x^5*(a + b*x)^(1/2))/b + (6*a^5*x*(a + b*x) 
^(1/2))/b^5)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1078, normalized size of antiderivative = 7.93 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{15/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(5/2)/(b*x+a)^(15/2),x)
 

Output:

(2*( - 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**6*d**6 - 96*sqrt(d)*sqrt(b)*sqr 
t(a + b*x)*a**5*b*d**6*x - 240*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**2*d** 
6*x**2 - 320*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**3*d**6*x**3 - 240*sqrt( 
d)*sqrt(b)*sqrt(a + b*x)*a**2*b**4*d**6*x**4 - 96*sqrt(d)*sqrt(b)*sqrt(a + 
 b*x)*a*b**5*d**6*x**5 - 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**6*d**6*x**6 + 
 429*sqrt(c + d*x)*a**3*b**4*c**3*d**3 + 1287*sqrt(c + d*x)*a**3*b**4*c**2 
*d**4*x + 1287*sqrt(c + d*x)*a**3*b**4*c*d**5*x**2 + 429*sqrt(c + d*x)*a** 
3*b**4*d**6*x**3 - 1001*sqrt(c + d*x)*a**2*b**5*c**4*d**2 - 2717*sqrt(c + 
d*x)*a**2*b**5*c**3*d**3*x - 2145*sqrt(c + d*x)*a**2*b**5*c**2*d**4*x**2 - 
 143*sqrt(c + d*x)*a**2*b**5*c*d**5*x**3 + 286*sqrt(c + d*x)*a**2*b**5*d** 
6*x**4 + 819*sqrt(c + d*x)*a*b**6*c**5*d + 2093*sqrt(c + d*x)*a*b**6*c**4* 
d**2*x + 1469*sqrt(c + d*x)*a*b**6*c**3*d**3*x**2 + 39*sqrt(c + d*x)*a*b** 
6*c**2*d**4*x**3 - 52*sqrt(c + d*x)*a*b**6*c*d**5*x**4 + 104*sqrt(c + d*x) 
*a*b**6*d**6*x**5 - 231*sqrt(c + d*x)*b**7*c**6 - 567*sqrt(c + d*x)*b**7*c 
**5*d*x - 371*sqrt(c + d*x)*b**7*c**4*d**2*x**2 - 5*sqrt(c + d*x)*b**7*c** 
3*d**3*x**3 + 6*sqrt(c + d*x)*b**7*c**2*d**4*x**4 - 8*sqrt(c + d*x)*b**7*c 
*d**5*x**5 + 16*sqrt(c + d*x)*b**7*d**6*x**6))/(3003*sqrt(a + b*x)*b**4*(a 
**10*d**4 - 4*a**9*b*c*d**3 + 6*a**9*b*d**4*x + 6*a**8*b**2*c**2*d**2 - 24 
*a**8*b**2*c*d**3*x + 15*a**8*b**2*d**4*x**2 - 4*a**7*b**3*c**3*d + 36*a** 
7*b**3*c**2*d**2*x - 60*a**7*b**3*c*d**3*x**2 + 20*a**7*b**3*d**4*x**3 ...