\(\int \frac {(c+d x)^3 (a x+b x^2)^{3/2}}{x^{10}} \, dx\) [91]

Optimal result

Integrand size = 24, antiderivative size = 286 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {2 c^3 \left (a x+b x^2\right )^{5/2}}{15 a x^{10}}+\frac {2 c^2 (2 b c-9 a d) \left (a x+b x^2\right )^{5/2}}{39 a^2 x^9}-\frac {2 c \left (117 a^2 d^2+8 b c (2 b c-9 a d)\right ) \left (a x+b x^2\right )^{5/2}}{429 a^3 x^8}-\frac {2 \left (143 d^3-\frac {2 b c \left (117 a^2 d^2+8 b c (2 b c-9 a d)\right )}{a^3}\right ) \left (a x+b x^2\right )^{5/2}}{1287 a x^7}+\frac {8 b \left (143 a^3 d^3-2 b c \left (117 a^2 d^2+8 b c (2 b c-9 a d)\right )\right ) \left (a x+b x^2\right )^{5/2}}{9009 a^5 x^6}-\frac {16 b^2 \left (143 a^3 d^3-2 b c \left (117 a^2 d^2+8 b c (2 b c-9 a d)\right )\right ) \left (a x+b x^2\right )^{5/2}}{45045 a^6 x^5} \] Output:

-2/15*c^3*(b*x^2+a*x)^(5/2)/a/x^10+2/39*c^2*(-9*a*d+2*b*c)*(b*x^2+a*x)^(5/ 
2)/a^2/x^9-2/429*c*(117*a^2*d^2+8*b*c*(-9*a*d+2*b*c))*(b*x^2+a*x)^(5/2)/a^ 
3/x^8-2/1287*(143*d^3-2*b*c*(117*a^2*d^2+8*b*c*(-9*a*d+2*b*c))/a^3)*(b*x^2 
+a*x)^(5/2)/a/x^7+8/9009*b*(143*a^3*d^3-2*b*c*(117*a^2*d^2+8*b*c*(-9*a*d+2 
*b*c)))*(b*x^2+a*x)^(5/2)/a^5/x^6-16/45045*b^2*(143*a^3*d^3-2*b*c*(117*a^2 
*d^2+8*b*c*(-9*a*d+2*b*c)))*(b*x^2+a*x)^(5/2)/a^6/x^5
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.69 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {2 (x (a+b x))^{5/2} \left (-256 b^5 c^3 x^5+128 a b^4 c^2 x^4 (5 c+9 d x)-16 a^2 b^3 c x^3 \left (70 c^2+180 c d x+117 d^2 x^2\right )+8 a^3 b^2 x^2 \left (210 c^3+630 c^2 d x+585 c d^2 x^2+143 d^3 x^3\right )-10 a^4 b x \left (231 c^3+756 c^2 d x+819 c d^2 x^2+286 d^3 x^3\right )+7 a^5 \left (429 c^3+1485 c^2 d x+1755 c d^2 x^2+715 d^3 x^3\right )\right )}{45045 a^6 x^{10}} \] Input:

Integrate[((c + d*x)^3*(a*x + b*x^2)^(3/2))/x^10,x]
 

Output:

(-2*(x*(a + b*x))^(5/2)*(-256*b^5*c^3*x^5 + 128*a*b^4*c^2*x^4*(5*c + 9*d*x 
) - 16*a^2*b^3*c*x^3*(70*c^2 + 180*c*d*x + 117*d^2*x^2) + 8*a^3*b^2*x^2*(2 
10*c^3 + 630*c^2*d*x + 585*c*d^2*x^2 + 143*d^3*x^3) - 10*a^4*b*x*(231*c^3 
+ 756*c^2*d*x + 819*c*d^2*x^2 + 286*d^3*x^3) + 7*a^5*(429*c^3 + 1485*c^2*d 
*x + 1755*c*d^2*x^2 + 715*d^3*x^3)))/(45045*a^6*x^10)
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1262, 27, 2169, 27, 1220, 1129, 1129, 1129, 1129, 1123}

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.

\(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2} (c+d x)^3}{x^{10}} \, dx\)

\(\Big \downarrow \) 1262

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a x\right )^{3/2} \left (6 b c^3+18 b d x c^2+d^2 (18 b c-11 a d) x^2\right )}{2 x^{10}}dx}{3 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (b x^2+a x\right )^{3/2} \left (6 b c^3+18 b d x c^2+d^2 (18 b c-11 a d) x^2\right )}{x^{10}}dx}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {-\frac {\int -\frac {\left (48 b^2 c^3+d \left (144 b^2 c^2-234 a b d c+143 a^2 d^2\right ) x\right ) \left (b x^2+a x\right )^{3/2}}{2 x^{10}}dx}{4 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (48 b^2 c^3+d \left (144 b^2 c^2-234 a b d c+143 a^2 d^2\right ) x\right ) \left (b x^2+a x\right )^{3/2}}{x^{10}}dx}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {\frac {-\frac {\left (-143 a^3 d^3+234 a^2 b c d^2-144 a b^2 c^2 d+32 b^3 c^3\right ) \int \frac {\left (b x^2+a x\right )^{3/2}}{x^9}dx}{a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{5 a x^{10}}}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-143 a^3 d^3+234 a^2 b c d^2-144 a b^2 c^2 d+32 b^3 c^3\right ) \left (-\frac {8 b \int \frac {\left (b x^2+a x\right )^{3/2}}{x^8}dx}{13 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{13 a x^9}\right )}{a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{5 a x^{10}}}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-143 a^3 d^3+234 a^2 b c d^2-144 a b^2 c^2 d+32 b^3 c^3\right ) \left (-\frac {8 b \left (-\frac {6 b \int \frac {\left (b x^2+a x\right )^{3/2}}{x^7}dx}{11 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{11 a x^8}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{13 a x^9}\right )}{a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{5 a x^{10}}}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-143 a^3 d^3+234 a^2 b c d^2-144 a b^2 c^2 d+32 b^3 c^3\right ) \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a x\right )^{3/2}}{x^6}dx}{9 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{11 a x^8}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{13 a x^9}\right )}{a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{5 a x^{10}}}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-143 a^3 d^3+234 a^2 b c d^2-144 a b^2 c^2 d+32 b^3 c^3\right ) \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a x\right )^{3/2}}{x^5}dx}{7 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{7 a x^6}\right )}{9 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{11 a x^8}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{13 a x^9}\right )}{a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{5 a x^{10}}}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\frac {-\frac {\left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b \left (a x+b x^2\right )^{5/2}}{35 a^2 x^5}-\frac {2 \left (a x+b x^2\right )^{5/2}}{7 a x^6}\right )}{9 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{11 a x^8}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{13 a x^9}\right ) \left (-143 a^3 d^3+234 a^2 b c d^2-144 a b^2 c^2 d+32 b^3 c^3\right )}{a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{5 a x^{10}}}{8 b}-\frac {d^2 \left (a x+b x^2\right )^{5/2} (18 b c-11 a d)}{4 b x^9}}{6 b}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{3 b x^8}\)

Input:

Int[((c + d*x)^3*(a*x + b*x^2)^(3/2))/x^10,x]
 

Output:

-1/3*(d^3*(a*x + b*x^2)^(5/2))/(b*x^8) + (-1/4*(d^2*(18*b*c - 11*a*d)*(a*x 
 + b*x^2)^(5/2))/(b*x^9) + ((-32*b^2*c^3*(a*x + b*x^2)^(5/2))/(5*a*x^10) - 
 ((32*b^3*c^3 - 144*a*b^2*c^2*d + 234*a^2*b*c*d^2 - 143*a^3*d^3)*((-2*(a*x 
 + b*x^2)^(5/2))/(13*a*x^9) - (8*b*((-2*(a*x + b*x^2)^(5/2))/(11*a*x^8) - 
(6*b*((-2*(a*x + b*x^2)^(5/2))/(9*a*x^7) - (4*b*((-2*(a*x + b*x^2)^(5/2))/ 
(7*a*x^6) + (4*b*(a*x + b*x^2)^(5/2))/(35*a^2*x^5)))/(9*a)))/(11*a)))/(13* 
a)))/a)/(8*b))/(6*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1123

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b 
*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 
0] && EqQ[m + 2*p + 2, 0]
 

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

rule 1220

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x 
^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e 
*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))   Int[(d + e*x 
)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0 
]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 
]
 

rule 1262

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n + e*g^n*( 
m + p + n)*(d + e*x)^(n - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; 
 FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
 

rule 2169

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - 
b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right )^{2} \left (\left (\frac {5}{3} d^{3} x^{3}+\frac {45}{11} c \,d^{2} x^{2}+\frac {45}{13} c^{2} d x +c^{3}\right ) a^{5}-\frac {10 \left (\frac {26}{21} d^{3} x^{3}+\frac {39}{11} c \,d^{2} x^{2}+\frac {36}{11} c^{2} d x +c^{3}\right ) x b \,a^{4}}{13}+\frac {80 \left (\frac {143}{210} d^{3} x^{3}+\frac {39}{14} c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right ) x^{2} b^{2} a^{3}}{143}-\frac {160 \left (\frac {117}{70} d^{2} x^{2}+\frac {18}{7} c d x +c^{2}\right ) x^{3} b^{3} c \,a^{2}}{429}+\frac {640 \left (\frac {9 d x}{5}+c \right ) x^{4} b^{4} c^{2} a}{3003}-\frac {256 b^{5} c^{3} x^{5}}{3003}\right )}{15 x^{8} a^{6}}\)	\(190\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (1144 a^{3} b^{2} d^{3} x^{5}-1872 a^{2} b^{3} c \,d^{2} x^{5}+1152 a \,b^{4} c^{2} d \,x^{5}-256 b^{5} c^{3} x^{5}-2860 a^{4} b \,d^{3} x^{4}+4680 a^{3} b^{2} c \,d^{2} x^{4}-2880 a^{2} b^{3} c^{2} d \,x^{4}+640 a \,b^{4} c^{3} x^{4}+5005 a^{5} d^{3} x^{3}-8190 a^{4} b c \,d^{2} x^{3}+5040 a^{3} b^{2} c^{2} d \,x^{3}-1120 a^{2} b^{3} c^{3} x^{3}+12285 a^{5} c \,d^{2} x^{2}-7560 a^{4} b \,c^{2} d \,x^{2}+1680 a^{3} b^{2} c^{3} x^{2}+10395 a^{5} c^{2} d x -2310 a^{4} b \,c^{3} x +3003 a^{5} c^{3}\right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{45045 x^{9} a^{6}}\)	\(253\)
orering	\(-\frac {2 \left (b x +a \right ) \left (1144 a^{3} b^{2} d^{3} x^{5}-1872 a^{2} b^{3} c \,d^{2} x^{5}+1152 a \,b^{4} c^{2} d \,x^{5}-256 b^{5} c^{3} x^{5}-2860 a^{4} b \,d^{3} x^{4}+4680 a^{3} b^{2} c \,d^{2} x^{4}-2880 a^{2} b^{3} c^{2} d \,x^{4}+640 a \,b^{4} c^{3} x^{4}+5005 a^{5} d^{3} x^{3}-8190 a^{4} b c \,d^{2} x^{3}+5040 a^{3} b^{2} c^{2} d \,x^{3}-1120 a^{2} b^{3} c^{3} x^{3}+12285 a^{5} c \,d^{2} x^{2}-7560 a^{4} b \,c^{2} d \,x^{2}+1680 a^{3} b^{2} c^{3} x^{2}+10395 a^{5} c^{2} d x -2310 a^{4} b \,c^{3} x +3003 a^{5} c^{3}\right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{45045 x^{9} a^{6}}\)	\(253\)
trager	\(-\frac {2 \left (1144 a^{3} b^{4} d^{3} x^{7}-1872 a^{2} b^{5} c \,d^{2} x^{7}+1152 a \,b^{6} c^{2} d \,x^{7}-256 b^{7} c^{3} x^{7}-572 a^{4} b^{3} d^{3} x^{6}+936 a^{3} b^{4} c \,d^{2} x^{6}-576 a^{2} b^{5} c^{2} d \,x^{6}+128 a \,b^{6} c^{3} x^{6}+429 a^{5} b^{2} d^{3} x^{5}-702 a^{4} b^{3} c \,d^{2} x^{5}+432 a^{3} b^{4} c^{2} d \,x^{5}-96 a^{2} b^{5} c^{3} x^{5}+7150 a^{6} b \,d^{3} x^{4}+585 a^{5} b^{2} c \,d^{2} x^{4}-360 a^{4} b^{3} c^{2} d \,x^{4}+80 a^{3} b^{4} c^{3} x^{4}+5005 a^{7} d^{3} x^{3}+16380 a^{6} b c \,d^{2} x^{3}+315 a^{5} b^{2} c^{2} d \,x^{3}-70 a^{4} b^{3} c^{3} x^{3}+12285 a^{7} c \,d^{2} x^{2}+13230 a^{6} b \,c^{2} d \,x^{2}+63 a^{5} b^{2} c^{3} x^{2}+10395 a^{7} c^{2} d x +3696 a^{6} b \,c^{3} x +3003 a^{7} c^{3}\right ) \sqrt {b \,x^{2}+a x}}{45045 a^{6} x^{8}}\)	\(364\)
risch	\(-\frac {2 \left (b x +a \right ) \left (1144 a^{3} b^{4} d^{3} x^{7}-1872 a^{2} b^{5} c \,d^{2} x^{7}+1152 a \,b^{6} c^{2} d \,x^{7}-256 b^{7} c^{3} x^{7}-572 a^{4} b^{3} d^{3} x^{6}+936 a^{3} b^{4} c \,d^{2} x^{6}-576 a^{2} b^{5} c^{2} d \,x^{6}+128 a \,b^{6} c^{3} x^{6}+429 a^{5} b^{2} d^{3} x^{5}-702 a^{4} b^{3} c \,d^{2} x^{5}+432 a^{3} b^{4} c^{2} d \,x^{5}-96 a^{2} b^{5} c^{3} x^{5}+7150 a^{6} b \,d^{3} x^{4}+585 a^{5} b^{2} c \,d^{2} x^{4}-360 a^{4} b^{3} c^{2} d \,x^{4}+80 a^{3} b^{4} c^{3} x^{4}+5005 a^{7} d^{3} x^{3}+16380 a^{6} b c \,d^{2} x^{3}+315 a^{5} b^{2} c^{2} d \,x^{3}-70 a^{4} b^{3} c^{3} x^{3}+12285 a^{7} c \,d^{2} x^{2}+13230 a^{6} b \,c^{2} d \,x^{2}+63 a^{5} b^{2} c^{3} x^{2}+10395 a^{7} c^{2} d x +3696 a^{6} b \,c^{3} x +3003 a^{7} c^{3}\right )}{45045 x^{7} \sqrt {x \left (b x +a \right )}\, a^{6}}\)	\(367\)
default	\(c^{3} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{15 a \,x^{10}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{13 a \,x^{9}}-\frac {8 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{11 a \,x^{8}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{9 a \,x^{7}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\right )+d^{3} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{9 a \,x^{7}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )+3 c \,d^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{11 a \,x^{8}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{9 a \,x^{7}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )}{11 a}\right )+3 c^{2} d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{13 a \,x^{9}}-\frac {8 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{11 a \,x^{8}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{9 a \,x^{7}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )\)	\(442\)

Input:

int((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^10,x,method=_RETURNVERBOSE)
 

Output:

-2/15*(x*(b*x+a))^(1/2)*(b*x+a)^2*((5/3*d^3*x^3+45/11*c*d^2*x^2+45/13*c^2* 
d*x+c^3)*a^5-10/13*(26/21*d^3*x^3+39/11*c*d^2*x^2+36/11*c^2*d*x+c^3)*x*b*a 
^4+80/143*(143/210*d^3*x^3+39/14*c*d^2*x^2+3*c^2*d*x+c^3)*x^2*b^2*a^3-160/ 
429*(117/70*d^2*x^2+18/7*c*d*x+c^2)*x^3*b^3*c*a^2+640/3003*(9/5*d*x+c)*x^4 
*b^4*c^2*a-256/3003*b^5*c^3*x^5)/x^8/a^6
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=-\frac {2 \, {\left (3003 \, a^{7} c^{3} - 8 \, {\left (32 \, b^{7} c^{3} - 144 \, a b^{6} c^{2} d + 234 \, a^{2} b^{5} c d^{2} - 143 \, a^{3} b^{4} d^{3}\right )} x^{7} + 4 \, {\left (32 \, a b^{6} c^{3} - 144 \, a^{2} b^{5} c^{2} d + 234 \, a^{3} b^{4} c d^{2} - 143 \, a^{4} b^{3} d^{3}\right )} x^{6} - 3 \, {\left (32 \, a^{2} b^{5} c^{3} - 144 \, a^{3} b^{4} c^{2} d + 234 \, a^{4} b^{3} c d^{2} - 143 \, a^{5} b^{2} d^{3}\right )} x^{5} + 5 \, {\left (16 \, a^{3} b^{4} c^{3} - 72 \, a^{4} b^{3} c^{2} d + 117 \, a^{5} b^{2} c d^{2} + 1430 \, a^{6} b d^{3}\right )} x^{4} - 35 \, {\left (2 \, a^{4} b^{3} c^{3} - 9 \, a^{5} b^{2} c^{2} d - 468 \, a^{6} b c d^{2} - 143 \, a^{7} d^{3}\right )} x^{3} + 63 \, {\left (a^{5} b^{2} c^{3} + 210 \, a^{6} b c^{2} d + 195 \, a^{7} c d^{2}\right )} x^{2} + 231 \, {\left (16 \, a^{6} b c^{3} + 45 \, a^{7} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{45045 \, a^{6} x^{8}} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^10,x, algorithm="fricas")
 

Output:

-2/45045*(3003*a^7*c^3 - 8*(32*b^7*c^3 - 144*a*b^6*c^2*d + 234*a^2*b^5*c*d 
^2 - 143*a^3*b^4*d^3)*x^7 + 4*(32*a*b^6*c^3 - 144*a^2*b^5*c^2*d + 234*a^3* 
b^4*c*d^2 - 143*a^4*b^3*d^3)*x^6 - 3*(32*a^2*b^5*c^3 - 144*a^3*b^4*c^2*d + 
 234*a^4*b^3*c*d^2 - 143*a^5*b^2*d^3)*x^5 + 5*(16*a^3*b^4*c^3 - 72*a^4*b^3 
*c^2*d + 117*a^5*b^2*c*d^2 + 1430*a^6*b*d^3)*x^4 - 35*(2*a^4*b^3*c^3 - 9*a 
^5*b^2*c^2*d - 468*a^6*b*c*d^2 - 143*a^7*d^3)*x^3 + 63*(a^5*b^2*c^3 + 210* 
a^6*b*c^2*d + 195*a^7*c*d^2)*x^2 + 231*(16*a^6*b*c^3 + 45*a^7*c^2*d)*x)*sq 
rt(b*x^2 + a*x)/(a^6*x^8)
 

Sympy [F]

\[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )^{3}}{x^{10}}\, dx \] Input:

integrate((d*x+c)**3*(b*x**2+a*x)**(3/2)/x**10,x)
 

Output:

Integral((x*(a + b*x))**(3/2)*(c + d*x)**3/x**10, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (262) = 524\).

Time = 0.06 (sec) , antiderivative size = 702, normalized size of antiderivative = 2.45 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=\frac {512 \, \sqrt {b x^{2} + a x} b^{7} c^{3}}{45045 \, a^{6} x} - \frac {256 \, \sqrt {b x^{2} + a x} b^{6} c^{2} d}{5005 \, a^{5} x} + \frac {32 \, \sqrt {b x^{2} + a x} b^{5} c d^{2}}{385 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{4} d^{3}}{315 \, a^{3} x} - \frac {256 \, \sqrt {b x^{2} + a x} b^{6} c^{3}}{45045 \, a^{5} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} b^{5} c^{2} d}{5005 \, a^{4} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} b^{4} c d^{2}}{385 \, a^{3} x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} b^{3} d^{3}}{315 \, a^{2} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} b^{5} c^{3}}{15015 \, a^{4} x^{3}} - \frac {96 \, \sqrt {b x^{2} + a x} b^{4} c^{2} d}{5005 \, a^{3} x^{3}} + \frac {12 \, \sqrt {b x^{2} + a x} b^{3} c d^{2}}{385 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{2} d^{3}}{105 \, a x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} b^{4} c^{3}}{9009 \, a^{3} x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} b^{3} c^{2} d}{1001 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{2} c d^{2}}{77 \, a x^{4}} + \frac {\sqrt {b x^{2} + a x} b d^{3}}{63 \, x^{4}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{3} c^{3}}{1287 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{2} c^{2} d}{143 \, a x^{5}} + \frac {\sqrt {b x^{2} + a x} b c d^{2}}{44 \, x^{5}} + \frac {\sqrt {b x^{2} + a x} a d^{3}}{9 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{2} c^{3}}{715 \, a x^{6}} + \frac {9 \, \sqrt {b x^{2} + a x} b c^{2} d}{715 \, x^{6}} + \frac {9 \, \sqrt {b x^{2} + a x} a c d^{2}}{44 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} d^{3}}{3 \, x^{6}} + \frac {\sqrt {b x^{2} + a x} b c^{3}}{390 \, x^{7}} + \frac {9 \, \sqrt {b x^{2} + a x} a c^{2} d}{65 \, x^{7}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c d^{2}}{4 \, x^{7}} + \frac {\sqrt {b x^{2} + a x} a c^{3}}{30 \, x^{8}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{2} d}{5 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{3}}{6 \, x^{9}} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^10,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

512/45045*sqrt(b*x^2 + a*x)*b^7*c^3/(a^6*x) - 256/5005*sqrt(b*x^2 + a*x)*b 
^6*c^2*d/(a^5*x) + 32/385*sqrt(b*x^2 + a*x)*b^5*c*d^2/(a^4*x) - 16/315*sqr 
t(b*x^2 + a*x)*b^4*d^3/(a^3*x) - 256/45045*sqrt(b*x^2 + a*x)*b^6*c^3/(a^5* 
x^2) + 128/5005*sqrt(b*x^2 + a*x)*b^5*c^2*d/(a^4*x^2) - 16/385*sqrt(b*x^2 
+ a*x)*b^4*c*d^2/(a^3*x^2) + 8/315*sqrt(b*x^2 + a*x)*b^3*d^3/(a^2*x^2) + 6 
4/15015*sqrt(b*x^2 + a*x)*b^5*c^3/(a^4*x^3) - 96/5005*sqrt(b*x^2 + a*x)*b^ 
4*c^2*d/(a^3*x^3) + 12/385*sqrt(b*x^2 + a*x)*b^3*c*d^2/(a^2*x^3) - 2/105*s 
qrt(b*x^2 + a*x)*b^2*d^3/(a*x^3) - 32/9009*sqrt(b*x^2 + a*x)*b^4*c^3/(a^3* 
x^4) + 16/1001*sqrt(b*x^2 + a*x)*b^3*c^2*d/(a^2*x^4) - 2/77*sqrt(b*x^2 + a 
*x)*b^2*c*d^2/(a*x^4) + 1/63*sqrt(b*x^2 + a*x)*b*d^3/x^4 + 4/1287*sqrt(b*x 
^2 + a*x)*b^3*c^3/(a^2*x^5) - 2/143*sqrt(b*x^2 + a*x)*b^2*c^2*d/(a*x^5) + 
1/44*sqrt(b*x^2 + a*x)*b*c*d^2/x^5 + 1/9*sqrt(b*x^2 + a*x)*a*d^3/x^5 - 2/7 
15*sqrt(b*x^2 + a*x)*b^2*c^3/(a*x^6) + 9/715*sqrt(b*x^2 + a*x)*b*c^2*d/x^6 
 + 9/44*sqrt(b*x^2 + a*x)*a*c*d^2/x^6 - 1/3*(b*x^2 + a*x)^(3/2)*d^3/x^6 + 
1/390*sqrt(b*x^2 + a*x)*b*c^3/x^7 + 9/65*sqrt(b*x^2 + a*x)*a*c^2*d/x^7 - 3 
/4*(b*x^2 + a*x)^(3/2)*c*d^2/x^7 + 1/30*sqrt(b*x^2 + a*x)*a*c^3/x^8 - 3/5* 
(b*x^2 + a*x)^(3/2)*c^2*d/x^8 - 1/6*(b*x^2 + a*x)^(3/2)*c^3/x^9
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1066 vs. \(2 (262) = 524\).

Time = 0.14 (sec) , antiderivative size = 1066, normalized size of antiderivative = 3.73 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^10,x, algorithm="giac")
 

Output:

2/45045*(60060*(sqrt(b)*x - sqrt(b*x^2 + a*x))^12*b^3*d^3 + 270270*(sqrt(b 
)*x - sqrt(b*x^2 + a*x))^11*b^(7/2)*c*d^2 + 225225*(sqrt(b)*x - sqrt(b*x^2 
 + a*x))^11*a*b^(5/2)*d^3 + 432432*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*b^4* 
c^2*d + 1189188*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*a*b^3*c*d^2 + 369369*(s 
qrt(b)*x - sqrt(b*x^2 + a*x))^10*a^2*b^2*d^3 + 240240*(sqrt(b)*x - sqrt(b* 
x^2 + a*x))^9*b^(9/2)*c^3 + 2162160*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a*b^ 
(7/2)*c^2*d + 2297295*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a^2*b^(5/2)*c*d^2 
+ 330330*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a^3*b^(3/2)*d^3 + 1338480*(sqrt 
(b)*x - sqrt(b*x^2 + a*x))^8*a*b^4*c^3 + 4787640*(sqrt(b)*x - sqrt(b*x^2 + 
 a*x))^8*a^2*b^3*c^2*d + 2490345*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^3*b^2 
*c*d^2 + 167310*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^4*b*d^3 + 3333330*(sqr 
t(b)*x - sqrt(b*x^2 + a*x))^7*a^2*b^(7/2)*c^3 + 6081075*(sqrt(b)*x - sqrt( 
b*x^2 + a*x))^7*a^3*b^(5/2)*c^2*d + 1621620*(sqrt(b)*x - sqrt(b*x^2 + a*x) 
)^7*a^4*b^(3/2)*c*d^2 + 45045*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^5*sqrt(b 
)*d^3 + 4844840*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*b^3*c^3 + 4819815*(s 
qrt(b)*x - sqrt(b*x^2 + a*x))^6*a^4*b^2*c^2*d + 630630*(sqrt(b)*x - sqrt(b 
*x^2 + a*x))^6*a^5*b*c*d^2 + 5005*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^6*d^ 
3 + 4513509*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^4*b^(5/2)*c^3 + 2432430*(s 
qrt(b)*x - sqrt(b*x^2 + a*x))^5*a^5*b^(3/2)*c^2*d + 135135*(sqrt(b)*x - sq 
rt(b*x^2 + a*x))^5*a^6*sqrt(b)*c*d^2 + 2788695*(sqrt(b)*x - sqrt(b*x^2 ...
 

Mupad [B] (verification not implemented)

Time = 16.44 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.18 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=\frac {4\,b^3\,c^3\,\sqrt {b\,x^2+a\,x}}{1287\,a^2\,x^5}-\frac {2\,a\,d^3\,\sqrt {b\,x^2+a\,x}}{9\,x^5}-\frac {32\,b\,c^3\,\sqrt {b\,x^2+a\,x}}{195\,x^7}-\frac {20\,b\,d^3\,\sqrt {b\,x^2+a\,x}}{63\,x^4}-\frac {2\,b^2\,c^3\,\sqrt {b\,x^2+a\,x}}{715\,a\,x^6}-\frac {2\,a\,c^3\,\sqrt {b\,x^2+a\,x}}{15\,x^8}-\frac {32\,b^4\,c^3\,\sqrt {b\,x^2+a\,x}}{9009\,a^3\,x^4}+\frac {64\,b^5\,c^3\,\sqrt {b\,x^2+a\,x}}{15015\,a^4\,x^3}-\frac {256\,b^6\,c^3\,\sqrt {b\,x^2+a\,x}}{45045\,a^5\,x^2}+\frac {512\,b^7\,c^3\,\sqrt {b\,x^2+a\,x}}{45045\,a^6\,x}-\frac {2\,b^2\,d^3\,\sqrt {b\,x^2+a\,x}}{105\,a\,x^3}+\frac {8\,b^3\,d^3\,\sqrt {b\,x^2+a\,x}}{315\,a^2\,x^2}-\frac {16\,b^4\,d^3\,\sqrt {b\,x^2+a\,x}}{315\,a^3\,x}-\frac {6\,a\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{11\,x^6}-\frac {6\,a\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{13\,x^7}-\frac {8\,b\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{11\,x^5}-\frac {84\,b\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{143\,x^6}-\frac {2\,b^2\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{77\,a\,x^4}-\frac {2\,b^2\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{143\,a\,x^5}+\frac {12\,b^3\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{385\,a^2\,x^3}+\frac {16\,b^3\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{1001\,a^2\,x^4}-\frac {16\,b^4\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{385\,a^3\,x^2}-\frac {96\,b^4\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{5005\,a^3\,x^3}+\frac {32\,b^5\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{385\,a^4\,x}+\frac {128\,b^5\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{5005\,a^4\,x^2}-\frac {256\,b^6\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{5005\,a^5\,x} \] Input:

int(((a*x + b*x^2)^(3/2)*(c + d*x)^3)/x^10,x)
 

Output:

(4*b^3*c^3*(a*x + b*x^2)^(1/2))/(1287*a^2*x^5) - (2*a*d^3*(a*x + b*x^2)^(1 
/2))/(9*x^5) - (32*b*c^3*(a*x + b*x^2)^(1/2))/(195*x^7) - (20*b*d^3*(a*x + 
 b*x^2)^(1/2))/(63*x^4) - (2*b^2*c^3*(a*x + b*x^2)^(1/2))/(715*a*x^6) - (2 
*a*c^3*(a*x + b*x^2)^(1/2))/(15*x^8) - (32*b^4*c^3*(a*x + b*x^2)^(1/2))/(9 
009*a^3*x^4) + (64*b^5*c^3*(a*x + b*x^2)^(1/2))/(15015*a^4*x^3) - (256*b^6 
*c^3*(a*x + b*x^2)^(1/2))/(45045*a^5*x^2) + (512*b^7*c^3*(a*x + b*x^2)^(1/ 
2))/(45045*a^6*x) - (2*b^2*d^3*(a*x + b*x^2)^(1/2))/(105*a*x^3) + (8*b^3*d 
^3*(a*x + b*x^2)^(1/2))/(315*a^2*x^2) - (16*b^4*d^3*(a*x + b*x^2)^(1/2))/( 
315*a^3*x) - (6*a*c*d^2*(a*x + b*x^2)^(1/2))/(11*x^6) - (6*a*c^2*d*(a*x + 
b*x^2)^(1/2))/(13*x^7) - (8*b*c*d^2*(a*x + b*x^2)^(1/2))/(11*x^5) - (84*b* 
c^2*d*(a*x + b*x^2)^(1/2))/(143*x^6) - (2*b^2*c*d^2*(a*x + b*x^2)^(1/2))/( 
77*a*x^4) - (2*b^2*c^2*d*(a*x + b*x^2)^(1/2))/(143*a*x^5) + (12*b^3*c*d^2* 
(a*x + b*x^2)^(1/2))/(385*a^2*x^3) + (16*b^3*c^2*d*(a*x + b*x^2)^(1/2))/(1 
001*a^2*x^4) - (16*b^4*c*d^2*(a*x + b*x^2)^(1/2))/(385*a^3*x^2) - (96*b^4* 
c^2*d*(a*x + b*x^2)^(1/2))/(5005*a^3*x^3) + (32*b^5*c*d^2*(a*x + b*x^2)^(1 
/2))/(385*a^4*x) + (128*b^5*c^2*d*(a*x + b*x^2)^(1/2))/(5005*a^4*x^2) - (2 
56*b^6*c^2*d*(a*x + b*x^2)^(1/2))/(5005*a^5*x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.17 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^{10}} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{7} d^{3} x^{3}}{9}+\frac {512 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c^{3} x^{7}}{45045}+\frac {16 \sqrt {b}\, a^{3} b^{4} d^{3} x^{8}}{315}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{7} c^{3}}{15}-\frac {512 \sqrt {b}\, b^{7} c^{3} x^{8}}{45045}-\frac {6 \sqrt {x}\, \sqrt {b x +a}\, a^{7} c^{2} d x}{13}-\frac {6 \sqrt {x}\, \sqrt {b x +a}\, a^{7} c \,d^{2} x^{2}}{11}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,c^{3} x}{195}-\frac {20 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,d^{3} x^{4}}{63}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c^{3} x^{2}}{715}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} d^{3} x^{5}}{105}+\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c^{3} x^{3}}{1287}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} d^{3} x^{6}}{315}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{3} x^{4}}{9009}-\frac {84 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,c^{2} d \,x^{2}}{143}-\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b c \,d^{2} x^{3}}{11}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c^{2} d \,x^{3}}{143}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c \,d^{2} x^{4}}{77}+\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c^{2} d \,x^{4}}{1001}+\frac {12 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c \,d^{2} x^{5}}{385}-\frac {96 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{2} d \,x^{5}}{5005}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c \,d^{2} x^{6}}{385}+\frac {128 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{2} d \,x^{6}}{5005}+\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c \,d^{2} x^{7}}{385}-\frac {256 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{2} d \,x^{7}}{5005}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} d^{3} x^{7}}{315}+\frac {64 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{3} x^{5}}{15015}-\frac {256 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{3} x^{6}}{45045}-\frac {32 \sqrt {b}\, a^{2} b^{5} c \,d^{2} x^{8}}{385}+\frac {256 \sqrt {b}\, a \,b^{6} c^{2} d \,x^{8}}{5005}}{a^{6} x^{8}} \] Input:

int((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^10,x)
 

Output:

(2*( - 3003*sqrt(x)*sqrt(a + b*x)*a**7*c**3 - 10395*sqrt(x)*sqrt(a + b*x)* 
a**7*c**2*d*x - 12285*sqrt(x)*sqrt(a + b*x)*a**7*c*d**2*x**2 - 5005*sqrt(x 
)*sqrt(a + b*x)*a**7*d**3*x**3 - 3696*sqrt(x)*sqrt(a + b*x)*a**6*b*c**3*x 
- 13230*sqrt(x)*sqrt(a + b*x)*a**6*b*c**2*d*x**2 - 16380*sqrt(x)*sqrt(a + 
b*x)*a**6*b*c*d**2*x**3 - 7150*sqrt(x)*sqrt(a + b*x)*a**6*b*d**3*x**4 - 63 
*sqrt(x)*sqrt(a + b*x)*a**5*b**2*c**3*x**2 - 315*sqrt(x)*sqrt(a + b*x)*a** 
5*b**2*c**2*d*x**3 - 585*sqrt(x)*sqrt(a + b*x)*a**5*b**2*c*d**2*x**4 - 429 
*sqrt(x)*sqrt(a + b*x)*a**5*b**2*d**3*x**5 + 70*sqrt(x)*sqrt(a + b*x)*a**4 
*b**3*c**3*x**3 + 360*sqrt(x)*sqrt(a + b*x)*a**4*b**3*c**2*d*x**4 + 702*sq 
rt(x)*sqrt(a + b*x)*a**4*b**3*c*d**2*x**5 + 572*sqrt(x)*sqrt(a + b*x)*a**4 
*b**3*d**3*x**6 - 80*sqrt(x)*sqrt(a + b*x)*a**3*b**4*c**3*x**4 - 432*sqrt( 
x)*sqrt(a + b*x)*a**3*b**4*c**2*d*x**5 - 936*sqrt(x)*sqrt(a + b*x)*a**3*b* 
*4*c*d**2*x**6 - 1144*sqrt(x)*sqrt(a + b*x)*a**3*b**4*d**3*x**7 + 96*sqrt( 
x)*sqrt(a + b*x)*a**2*b**5*c**3*x**5 + 576*sqrt(x)*sqrt(a + b*x)*a**2*b**5 
*c**2*d*x**6 + 1872*sqrt(x)*sqrt(a + b*x)*a**2*b**5*c*d**2*x**7 - 128*sqrt 
(x)*sqrt(a + b*x)*a*b**6*c**3*x**6 - 1152*sqrt(x)*sqrt(a + b*x)*a*b**6*c** 
2*d*x**7 + 256*sqrt(x)*sqrt(a + b*x)*b**7*c**3*x**7 + 1144*sqrt(b)*a**3*b* 
*4*d**3*x**8 - 1872*sqrt(b)*a**2*b**5*c*d**2*x**8 + 1152*sqrt(b)*a*b**6*c* 
*2*d*x**8 - 256*sqrt(b)*b**7*c**3*x**8))/(45045*a**6*x**8)