\(\int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx\) [991]

Optimal result

Integrand size = 34, antiderivative size = 124 \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=-\frac {2 (b c-a d)^2 (c+d x)^{5/2}}{11 (a+b x)^{11/2}}+\frac {26 d (b c-a d) (c+d x)^{5/2}}{33 (a+b x)^{9/2}}-\frac {302 d^2 (c+d x)^{5/2}}{231 (a+b x)^{7/2}}+\frac {1066 d^3 (c+d x)^{5/2}}{1155 (b c-a d) (a+b x)^{5/2}} \] Output:

-2/11*(-a*d+b*c)^2*(d*x+c)^(5/2)/(b*x+a)^(11/2)+26/33*d*(-a*d+b*c)*(d*x+c) 
^(5/2)/(b*x+a)^(9/2)-302/231*d^2*(d*x+c)^(5/2)/(b*x+a)^(7/2)+1066/1155*d^3 
*(d*x+c)^(5/2)/(-a*d+b*c)/(b*x+a)^(5/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{11/2} \left (105 b^3-\frac {1848 d^3 (a+b x)^3}{(c+d x)^3}+\frac {1980 b d^2 (a+b x)^2}{(c+d x)^2}-\frac {770 b^2 d (a+b x)}{c+d x}\right )}{1155 (b c-a d) (a+b x)^{11/2}} \] Input:

Integrate[((c + d*x)^(3/2)*(b*c - 2*a*d - b*d*x)^3)/(a + b*x)^(13/2),x]
 

Output:

(-2*(c + d*x)^(11/2)*(105*b^3 - (1848*d^3*(a + b*x)^3)/(c + d*x)^3 + (1980 
*b*d^2*(a + b*x)^2)/(c + d*x)^2 - (770*b^2*d*(a + b*x))/(c + d*x)))/(1155* 
(b*c - a*d)*(a + b*x)^(11/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(263\) vs. \(2(124)=248\).

Time = 0.38 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {105, 105, 105, 100, 27, 87, 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.

\(\displaystyle \int \frac {(c+d x)^{3/2} (-2 a d+b c-b d x)^3}{(a+b x)^{13/2}} \, dx\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {6 d \int \frac {\sqrt {c+d x} (b c-2 a d-b d x)^3}{(a+b x)^{11/2}}dx}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {6 d \left (\frac {2 d \int \frac {(b c-2 a d-b d x)^3}{(a+b x)^{9/2} \sqrt {c+d x}}dx}{9 b}-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^4}{9 b (a+b x)^{9/2} (b c-a d)}\right )}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {6 d \left (\frac {2 d \left (-\frac {12}{7} d \int \frac {(b c-2 a d-b d x)^2}{(a+b x)^{7/2} \sqrt {c+d x}}dx-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^3}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 b}-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^4}{9 b (a+b x)^{9/2} (b c-a d)}\right )}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {6 d \left (\frac {2 d \left (-\frac {12}{7} d \left (\frac {2 \int -\frac {b^2 d (b c-a d) (14 b c-19 a d-5 b d x)}{2 (a+b x)^{5/2} \sqrt {c+d x}}dx}{5 b^2 (b c-a d)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 (a+b x)^{5/2}}\right )-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^3}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 b}-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^4}{9 b (a+b x)^{9/2} (b c-a d)}\right )}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 d \left (\frac {2 d \left (-\frac {12}{7} d \left (-\frac {1}{5} d \int \frac {14 b c-19 a d-5 b d x}{(a+b x)^{5/2} \sqrt {c+d x}}dx-\frac {2 \sqrt {c+d x} (b c-a d)}{5 (a+b x)^{5/2}}\right )-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^3}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 b}-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^4}{9 b (a+b x)^{9/2} (b c-a d)}\right )}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {6 d \left (\frac {2 d \left (-\frac {12}{7} d \left (-\frac {1}{5} d \left (-\frac {43}{3} d \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}}dx-\frac {28 \sqrt {c+d x}}{3 (a+b x)^{3/2}}\right )-\frac {2 \sqrt {c+d x} (b c-a d)}{5 (a+b x)^{5/2}}\right )-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^3}{7 (a+b x)^{7/2} (b c-a d)}\right )}{9 b}-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^4}{9 b (a+b x)^{9/2} (b c-a d)}\right )}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {6 d \left (\frac {2 d \left (-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^3}{7 (a+b x)^{7/2} (b c-a d)}-\frac {12}{7} d \left (-\frac {2 \sqrt {c+d x} (b c-a d)}{5 (a+b x)^{5/2}}-\frac {1}{5} d \left (\frac {86 d \sqrt {c+d x}}{3 \sqrt {a+b x} (b c-a d)}-\frac {28 \sqrt {c+d x}}{3 (a+b x)^{3/2}}\right )\right )\right )}{9 b}-\frac {2 \sqrt {c+d x} (-2 a d+b c-b d x)^4}{9 b (a+b x)^{9/2} (b c-a d)}\right )}{11 b}-\frac {2 (c+d x)^{3/2} (-2 a d+b c-b d x)^4}{11 b (a+b x)^{11/2} (b c-a d)}\)

Input:

Int[((c + d*x)^(3/2)*(b*c - 2*a*d - b*d*x)^3)/(a + b*x)^(13/2),x]
 

Output:

(-2*(c + d*x)^(3/2)*(b*c - 2*a*d - b*d*x)^4)/(11*b*(b*c - a*d)*(a + b*x)^( 
11/2)) + (6*d*((-2*Sqrt[c + d*x]*(b*c - 2*a*d - b*d*x)^4)/(9*b*(b*c - a*d) 
*(a + b*x)^(9/2)) + (2*d*((-2*Sqrt[c + d*x]*(b*c - 2*a*d - b*d*x)^3)/(7*(b 
*c - a*d)*(a + b*x)^(7/2)) - (12*d*((-2*(b*c - a*d)*Sqrt[c + d*x])/(5*(a + 
 b*x)^(5/2)) - (d*((-28*Sqrt[c + d*x])/(3*(a + b*x)^(3/2)) + (86*d*Sqrt[c 
+ d*x])/(3*(b*c - a*d)*Sqrt[a + b*x])))/5))/7))/(9*b)))/(11*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 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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 105

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05

method	result	size
gosper	\(-\frac {2 \left (533 d^{3} x^{3} b^{3}+2354 x^{2} a \,b^{2} d^{3}-755 x^{2} b^{3} c \,d^{2}+3564 x \,a^{2} b \,d^{3}-2420 x a \,b^{2} c \,d^{2}+455 x \,b^{3} c^{2} d +1848 a^{3} d^{3}-1980 a^{2} b c \,d^{2}+770 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right ) \left (x d +c \right )^{\frac {5}{2}}}{1155 \left (a d -b c \right ) \left (b x +a \right )^{\frac {11}{2}}}\)	\(130\)
orering	\(\frac {2 \left (533 d^{3} x^{3} b^{3}+2354 x^{2} a \,b^{2} d^{3}-755 x^{2} b^{3} c \,d^{2}+3564 x \,a^{2} b \,d^{3}-2420 x a \,b^{2} c \,d^{2}+455 x \,b^{3} c^{2} d +1848 a^{3} d^{3}-1980 a^{2} b c \,d^{2}+770 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right ) \left (x d +c \right )^{\frac {5}{2}} \left (-b d x -2 a d +b c \right )^{3}}{1155 \left (b x +a \right )^{\frac {11}{2}} \left (a d -b c \right ) \left (b d x +2 a d -b c \right )^{3}}\)	\(160\)
default	\(-\frac {2 \left (533 b^{3} d^{4} x^{4}+2354 a \,b^{2} d^{4} x^{3}-222 b^{3} c \,d^{3} x^{3}+3564 a^{2} b \,d^{4} x^{2}-66 a \,b^{2} c \,d^{3} x^{2}-300 b^{3} c^{2} d^{2} x^{2}+1848 a^{3} d^{4} x +1584 a^{2} b c \,d^{3} x -1650 a \,b^{2} c^{2} d^{2} x +350 b^{3} c^{3} d x +1848 a^{3} c \,d^{3}-1980 a^{2} b \,c^{2} d^{2}+770 a \,b^{2} c^{3} d -105 c^{4} b^{3}\right ) \left (x d +c \right )^{\frac {3}{2}}}{1155 \left (b x +a \right )^{\frac {11}{2}} \left (a d -b c \right )}\)	\(184\)

Input:

int((d*x+c)^(3/2)*(-b*d*x-2*a*d+b*c)^3/(b*x+a)^(13/2),x,method=_RETURNVERB 
OSE)
 

Output:

-2/1155*(533*b^3*d^3*x^3+2354*a*b^2*d^3*x^2-755*b^3*c*d^2*x^2+3564*a^2*b*d 
^3*x-2420*a*b^2*c*d^2*x+455*b^3*c^2*d*x+1848*a^3*d^3-1980*a^2*b*c*d^2+770* 
a*b^2*c^2*d-105*b^3*c^3)*(d*x+c)^(5/2)/(a*d-b*c)/(b*x+a)^(11/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (100) = 200\).

Time = 6.39 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.91 \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=\frac {2 \, {\left (533 \, b^{3} d^{5} x^{5} - 105 \, b^{3} c^{5} + 770 \, a b^{2} c^{4} d - 1980 \, a^{2} b c^{3} d^{2} + 1848 \, a^{3} c^{2} d^{3} + {\left (311 \, b^{3} c d^{4} + 2354 \, a b^{2} d^{5}\right )} x^{4} - 2 \, {\left (261 \, b^{3} c^{2} d^{3} - 1144 \, a b^{2} c d^{4} - 1782 \, a^{2} b d^{5}\right )} x^{3} + 2 \, {\left (25 \, b^{3} c^{3} d^{2} - 858 \, a b^{2} c^{2} d^{3} + 2574 \, a^{2} b c d^{4} + 924 \, a^{3} d^{5}\right )} x^{2} + {\left (245 \, b^{3} c^{4} d - 880 \, a b^{2} c^{3} d^{2} - 396 \, a^{2} b c^{2} d^{3} + 3696 \, a^{3} c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1155 \, {\left (a^{6} b c - a^{7} d + {\left (b^{7} c - a b^{6} d\right )} x^{6} + 6 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} x^{5} + 15 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} x^{4} + 20 \, {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} x^{3} + 15 \, {\left (a^{4} b^{3} c - a^{5} b^{2} d\right )} x^{2} + 6 \, {\left (a^{5} b^{2} c - a^{6} b d\right )} x\right )}} \] Input:

integrate((d*x+c)^(3/2)*(-b*d*x-2*a*d+b*c)^3/(b*x+a)^(13/2),x, algorithm=" 
fricas")
 

Output:

2/1155*(533*b^3*d^5*x^5 - 105*b^3*c^5 + 770*a*b^2*c^4*d - 1980*a^2*b*c^3*d 
^2 + 1848*a^3*c^2*d^3 + (311*b^3*c*d^4 + 2354*a*b^2*d^5)*x^4 - 2*(261*b^3* 
c^2*d^3 - 1144*a*b^2*c*d^4 - 1782*a^2*b*d^5)*x^3 + 2*(25*b^3*c^3*d^2 - 858 
*a*b^2*c^2*d^3 + 2574*a^2*b*c*d^4 + 924*a^3*d^5)*x^2 + (245*b^3*c^4*d - 88 
0*a*b^2*c^3*d^2 - 396*a^2*b*c^2*d^3 + 3696*a^3*c*d^4)*x)*sqrt(b*x + a)*sqr 
t(d*x + c)/(a^6*b*c - a^7*d + (b^7*c - a*b^6*d)*x^6 + 6*(a*b^6*c - a^2*b^5 
*d)*x^5 + 15*(a^2*b^5*c - a^3*b^4*d)*x^4 + 20*(a^3*b^4*c - a^4*b^3*d)*x^3 
+ 15*(a^4*b^3*c - a^5*b^2*d)*x^2 + 6*(a^5*b^2*c - a^6*b*d)*x)
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)**(3/2)*(-b*d*x-2*a*d+b*c)**3/(b*x+a)**(13/2),x)
 

Output:

-Integral(-b**3*c**4*sqrt(c + d*x)/(a**6*sqrt(a + b*x) + 6*a**5*b*x*sqrt(a 
 + b*x) + 15*a**4*b**2*x**2*sqrt(a + b*x) + 20*a**3*b**3*x**3*sqrt(a + b*x 
) + 15*a**2*b**4*x**4*sqrt(a + b*x) + 6*a*b**5*x**5*sqrt(a + b*x) + b**6*x 
**6*sqrt(a + b*x)), x) - Integral(8*a**3*c*d**3*sqrt(c + d*x)/(a**6*sqrt(a 
 + b*x) + 6*a**5*b*x*sqrt(a + b*x) + 15*a**4*b**2*x**2*sqrt(a + b*x) + 20* 
a**3*b**3*x**3*sqrt(a + b*x) + 15*a**2*b**4*x**4*sqrt(a + b*x) + 6*a*b**5* 
x**5*sqrt(a + b*x) + b**6*x**6*sqrt(a + b*x)), x) - Integral(8*a**3*d**4*x 
*sqrt(c + d*x)/(a**6*sqrt(a + b*x) + 6*a**5*b*x*sqrt(a + b*x) + 15*a**4*b* 
*2*x**2*sqrt(a + b*x) + 20*a**3*b**3*x**3*sqrt(a + b*x) + 15*a**2*b**4*x** 
4*sqrt(a + b*x) + 6*a*b**5*x**5*sqrt(a + b*x) + b**6*x**6*sqrt(a + b*x)), 
x) - Integral(b**3*d**4*x**4*sqrt(c + d*x)/(a**6*sqrt(a + b*x) + 6*a**5*b* 
x*sqrt(a + b*x) + 15*a**4*b**2*x**2*sqrt(a + b*x) + 20*a**3*b**3*x**3*sqrt 
(a + b*x) + 15*a**2*b**4*x**4*sqrt(a + b*x) + 6*a*b**5*x**5*sqrt(a + b*x) 
+ b**6*x**6*sqrt(a + b*x)), x) - Integral(6*a*b**2*c**3*d*sqrt(c + d*x)/(a 
**6*sqrt(a + b*x) + 6*a**5*b*x*sqrt(a + b*x) + 15*a**4*b**2*x**2*sqrt(a + 
b*x) + 20*a**3*b**3*x**3*sqrt(a + b*x) + 15*a**2*b**4*x**4*sqrt(a + b*x) + 
 6*a*b**5*x**5*sqrt(a + b*x) + b**6*x**6*sqrt(a + b*x)), x) - Integral(6*a 
*b**2*d**4*x**3*sqrt(c + d*x)/(a**6*sqrt(a + b*x) + 6*a**5*b*x*sqrt(a + b* 
x) + 15*a**4*b**2*x**2*sqrt(a + b*x) + 20*a**3*b**3*x**3*sqrt(a + b*x) + 1 
5*a**2*b**4*x**4*sqrt(a + b*x) + 6*a*b**5*x**5*sqrt(a + b*x) + b**6*x**...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)*(-b*d*x-2*a*d+b*c)^3/(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. 3480 vs. \(2 (100) = 200\).

Time = 1.19 (sec) , antiderivative size = 3480, normalized size of antiderivative = 28.06 \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)*(-b*d*x-2*a*d+b*c)^3/(b*x+a)^(13/2),x, algorithm=" 
giac")
 

Output:

4/1155*(533*sqrt(b*d)*b^20*c^10*d^5*abs(b) - 5330*sqrt(b*d)*a*b^19*c^9*d^6 
*abs(b) + 23985*sqrt(b*d)*a^2*b^18*c^8*d^7*abs(b) - 63960*sqrt(b*d)*a^3*b^ 
17*c^7*d^8*abs(b) + 111930*sqrt(b*d)*a^4*b^16*c^6*d^9*abs(b) - 134316*sqrt 
(b*d)*a^5*b^15*c^5*d^10*abs(b) + 111930*sqrt(b*d)*a^6*b^14*c^4*d^11*abs(b) 
 - 63960*sqrt(b*d)*a^7*b^13*c^3*d^12*abs(b) + 23985*sqrt(b*d)*a^8*b^12*c^2 
*d^13*abs(b) - 5330*sqrt(b*d)*a^9*b^11*c*d^14*abs(b) + 533*sqrt(b*d)*a^10* 
b^10*d^15*abs(b) - 4708*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d))^2*b^18*c^9*d^5*abs(b) + 42372*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^17*c^8*d^6*abs 
(b) - 169488*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^16*c^7*d^7*abs(b) + 395472*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^15*c^6*d^8*abs(b) 
- 593208*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^14*c^5*d^9*abs(b) + 593208*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^13*c^4*d^10*abs(b) - 3 
95472*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^12*c^3*d^11*abs(b) + 169488*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^11*c^2*d^12*abs(b) - 423 
72*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^10*c*d^13*abs(b) + 4708*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -...
 

Mupad [B] (verification not implemented)

Time = 2.46 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.19 \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (3696\,a^3\,d^5+10296\,a^2\,b\,c\,d^4-3432\,a\,b^2\,c^2\,d^3+100\,b^3\,c^3\,d^2\right )}{1155\,b^6\,c-1155\,a\,b^5\,d}-\frac {-3696\,a^3\,c^2\,d^3+3960\,a^2\,b\,c^3\,d^2-1540\,a\,b^2\,c^4\,d+210\,b^3\,c^5}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {x\,\left (7392\,a^3\,c\,d^4-792\,a^2\,b\,c^2\,d^3-1760\,a\,b^2\,c^3\,d^2+490\,b^3\,c^4\,d\right )}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {1066\,b^3\,d^5\,x^5}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {2\,b^2\,d^4\,x^4\,\left (2354\,a\,d+311\,b\,c\right )}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {4\,b\,d^3\,x^3\,\left (1782\,a^2\,d^2+1144\,a\,b\,c\,d-261\,b^2\,c^2\right )}{1155\,b^6\,c-1155\,a\,b^5\,d}\right )}{\frac {\left (1155\,a^6\,d-1155\,a^5\,b\,c\right )\,\sqrt {a+b\,x}}{1155\,b^6\,c-1155\,a\,b^5\,d}-x^5\,\sqrt {a+b\,x}+\frac {5775\,a^4\,b\,x\,\left (a\,d-b\,c\right )\,\sqrt {a+b\,x}}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {5775\,a\,b^4\,x^4\,\left (a\,d-b\,c\right )\,\sqrt {a+b\,x}}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {11550\,a^3\,b^2\,x^2\,\left (a\,d-b\,c\right )\,\sqrt {a+b\,x}}{1155\,b^6\,c-1155\,a\,b^5\,d}+\frac {11550\,a^2\,b^3\,x^3\,\left (a\,d-b\,c\right )\,\sqrt {a+b\,x}}{1155\,b^6\,c-1155\,a\,b^5\,d}} \] Input:

int(-((c + d*x)^(3/2)*(2*a*d - b*c + b*d*x)^3)/(a + b*x)^(13/2),x)
 

Output:

-((c + d*x)^(1/2)*((x^2*(3696*a^3*d^5 + 100*b^3*c^3*d^2 - 3432*a*b^2*c^2*d 
^3 + 10296*a^2*b*c*d^4))/(1155*b^6*c - 1155*a*b^5*d) - (210*b^3*c^5 - 3696 
*a^3*c^2*d^3 + 3960*a^2*b*c^3*d^2 - 1540*a*b^2*c^4*d)/(1155*b^6*c - 1155*a 
*b^5*d) + (x*(7392*a^3*c*d^4 + 490*b^3*c^4*d - 1760*a*b^2*c^3*d^2 - 792*a^ 
2*b*c^2*d^3))/(1155*b^6*c - 1155*a*b^5*d) + (1066*b^3*d^5*x^5)/(1155*b^6*c 
 - 1155*a*b^5*d) + (2*b^2*d^4*x^4*(2354*a*d + 311*b*c))/(1155*b^6*c - 1155 
*a*b^5*d) + (4*b*d^3*x^3*(1782*a^2*d^2 - 261*b^2*c^2 + 1144*a*b*c*d))/(115 
5*b^6*c - 1155*a*b^5*d)))/(((1155*a^6*d - 1155*a^5*b*c)*(a + b*x)^(1/2))/( 
1155*b^6*c - 1155*a*b^5*d) - x^5*(a + b*x)^(1/2) + (5775*a^4*b*x*(a*d - b* 
c)*(a + b*x)^(1/2))/(1155*b^6*c - 1155*a*b^5*d) + (5775*a*b^4*x^4*(a*d - b 
*c)*(a + b*x)^(1/2))/(1155*b^6*c - 1155*a*b^5*d) + (11550*a^3*b^2*x^2*(a*d 
 - b*c)*(a + b*x)^(1/2))/(1155*b^6*c - 1155*a*b^5*d) + (11550*a^2*b^3*x^3* 
(a*d - b*c)*(a + b*x)^(1/2))/(1155*b^6*c - 1155*a*b^5*d))
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 603, normalized size of antiderivative = 4.86 \[ \int \frac {(c+d x)^{3/2} (b c-2 a d-b d x)^3}{(a+b x)^{13/2}} \, dx=\frac {\frac {646 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{5} d^{5}}{1155}+\frac {646 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{4} b \,d^{5} x}{231}+\frac {1292 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{3} b^{2} d^{5} x^{2}}{231}+\frac {1292 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{3} d^{5} x^{3}}{231}+\frac {646 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{4} d^{5} x^{4}}{231}+\frac {646 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{5} d^{5} x^{5}}{1155}-\frac {16 \sqrt {d x +c}\, a^{3} b^{3} c^{2} d^{3}}{5}-\frac {32 \sqrt {d x +c}\, a^{3} b^{3} c \,d^{4} x}{5}-\frac {16 \sqrt {d x +c}\, a^{3} b^{3} d^{5} x^{2}}{5}+\frac {24 \sqrt {d x +c}\, a^{2} b^{4} c^{3} d^{2}}{7}+\frac {24 \sqrt {d x +c}\, a^{2} b^{4} c^{2} d^{3} x}{35}-\frac {312 \sqrt {d x +c}\, a^{2} b^{4} c \,d^{4} x^{2}}{35}-\frac {216 \sqrt {d x +c}\, a^{2} b^{4} d^{5} x^{3}}{35}-\frac {4 \sqrt {d x +c}\, a \,b^{5} c^{4} d}{3}+\frac {32 \sqrt {d x +c}\, a \,b^{5} c^{3} d^{2} x}{21}+\frac {104 \sqrt {d x +c}\, a \,b^{5} c^{2} d^{3} x^{2}}{35}-\frac {416 \sqrt {d x +c}\, a \,b^{5} c \,d^{4} x^{3}}{105}-\frac {428 \sqrt {d x +c}\, a \,b^{5} d^{5} x^{4}}{105}+\frac {2 \sqrt {d x +c}\, b^{6} c^{5}}{11}-\frac {14 \sqrt {d x +c}\, b^{6} c^{4} d x}{33}-\frac {20 \sqrt {d x +c}\, b^{6} c^{3} d^{2} x^{2}}{231}+\frac {348 \sqrt {d x +c}\, b^{6} c^{2} d^{3} x^{3}}{385}-\frac {622 \sqrt {d x +c}\, b^{6} c \,d^{4} x^{4}}{1155}-\frac {1066 \sqrt {d x +c}\, b^{6} d^{5} x^{5}}{1155}}{\sqrt {b x +a}\, b^{3} \left (a \,b^{5} d \,x^{5}-b^{6} c \,x^{5}+5 a^{2} b^{4} d \,x^{4}-5 a \,b^{5} c \,x^{4}+10 a^{3} b^{3} d \,x^{3}-10 a^{2} b^{4} c \,x^{3}+10 a^{4} b^{2} d \,x^{2}-10 a^{3} b^{3} c \,x^{2}+5 a^{5} b d x -5 a^{4} b^{2} c x +a^{6} d -a^{5} b c \right )} \] Input:

int((d*x+c)^(3/2)*(-b*d*x-2*a*d+b*c)^3/(b*x+a)^(13/2),x)
 

Output:

(2*(323*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*d**5 + 1615*sqrt(d)*sqrt(b)*sqr 
t(a + b*x)*a**4*b*d**5*x + 3230*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**2*d* 
*5*x**2 + 3230*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**3*d**5*x**3 + 1615*sq 
rt(d)*sqrt(b)*sqrt(a + b*x)*a*b**4*d**5*x**4 + 323*sqrt(d)*sqrt(b)*sqrt(a 
+ b*x)*b**5*d**5*x**5 - 1848*sqrt(c + d*x)*a**3*b**3*c**2*d**3 - 3696*sqrt 
(c + d*x)*a**3*b**3*c*d**4*x - 1848*sqrt(c + d*x)*a**3*b**3*d**5*x**2 + 19 
80*sqrt(c + d*x)*a**2*b**4*c**3*d**2 + 396*sqrt(c + d*x)*a**2*b**4*c**2*d* 
*3*x - 5148*sqrt(c + d*x)*a**2*b**4*c*d**4*x**2 - 3564*sqrt(c + d*x)*a**2* 
b**4*d**5*x**3 - 770*sqrt(c + d*x)*a*b**5*c**4*d + 880*sqrt(c + d*x)*a*b** 
5*c**3*d**2*x + 1716*sqrt(c + d*x)*a*b**5*c**2*d**3*x**2 - 2288*sqrt(c + d 
*x)*a*b**5*c*d**4*x**3 - 2354*sqrt(c + d*x)*a*b**5*d**5*x**4 + 105*sqrt(c 
+ d*x)*b**6*c**5 - 245*sqrt(c + d*x)*b**6*c**4*d*x - 50*sqrt(c + d*x)*b**6 
*c**3*d**2*x**2 + 522*sqrt(c + d*x)*b**6*c**2*d**3*x**3 - 311*sqrt(c + d*x 
)*b**6*c*d**4*x**4 - 533*sqrt(c + d*x)*b**6*d**5*x**5))/(1155*sqrt(a + b*x 
)*b**3*(a**6*d - a**5*b*c + 5*a**5*b*d*x - 5*a**4*b**2*c*x + 10*a**4*b**2* 
d*x**2 - 10*a**3*b**3*c*x**2 + 10*a**3*b**3*d*x**3 - 10*a**2*b**4*c*x**3 + 
 5*a**2*b**4*d*x**4 - 5*a*b**5*c*x**4 + a*b**5*d*x**5 - b**6*c*x**5))