\(\int \frac {(A+B x) (c^2-d^2 x^2)^{3/2}}{(c+d x)^6} \, dx\) [21]

Optimal result

Integrand size = 29, antiderivative size = 83 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{5/2}}{7 c d^2 (c+d x)^6}-\frac {(6 B c+A d) \left (c^2-d^2 x^2\right )^{5/2}}{35 c^2 d^2 (c+d x)^5} \] Output:

1/7*(-A*d+B*c)*(-d^2*x^2+c^2)^(5/2)/c/d^2/(d*x+c)^6-1/35*(A*d+6*B*c)*(-d^2 
*x^2+c^2)^(5/2)/c^2/d^2/(d*x+c)^5
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=-\frac {(c-d x)^2 \sqrt {c^2-d^2 x^2} (A d (6 c+d x)+B c (c+6 d x))}{35 c^2 d^2 (c+d x)^4} \] Input:

Integrate[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^6,x]
 

Output:

-1/35*((c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(A*d*(6*c + d*x) + B*c*(c + 6*d*x)) 
)/(c^2*d^2*(c + d*x)^4)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 460}

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.

\(\displaystyle \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {(A d+6 B c) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^5}dx}{7 c d}+\frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{7 c d^2 (c+d x)^6}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{5/2} (B c-A d)}{7 c d^2 (c+d x)^6}-\frac {\left (c^2-d^2 x^2\right )^{5/2} (A d+6 B c)}{35 c^2 d^2 (c+d x)^5}\)

Input:

Int[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^6,x]
 

Output:

((B*c - A*d)*(c^2 - d^2*x^2)^(5/2))/(7*c*d^2*(c + d*x)^6) - ((6*B*c + A*d) 
*(c^2 - d^2*x^2)^(5/2))/(35*c^2*d^2*(c + d*x)^5)
 

Defintions of rubi rules used

rule 460

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

rule 671

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

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.71

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (A \,d^{2} x +6 B c d x +6 A c d +B \,c^{2}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{35 \left (d x +c \right )^{5} c^{2} d^{2}}\)	\(59\)
orering	\(-\frac {\left (-d x +c \right ) \left (A \,d^{2} x +6 B c d x +6 A c d +B \,c^{2}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{35 \left (d x +c \right )^{5} c^{2} d^{2}}\)	\(59\)
trager	\(-\frac {\left (A \,d^{4} x^{3}+6 B c \,d^{3} x^{3}+4 A c \,d^{3} x^{2}-11 x^{2} c^{2} B \,d^{2}-11 A \,c^{2} d^{2} x +4 B \,c^{3} d x +6 A \,c^{3} d +B \,c^{4}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{35 c^{2} \left (d x +c \right )^{4} d^{2}}\)	\(101\)
default	\(-\frac {B \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5 d^{7} c \left (x +\frac {c}{d}\right )^{5}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{35 c^{2} \left (x +\frac {c}{d}\right )^{5}}\right )}{d^{7}}\)	\(148\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^6,x,method=_RETURNVERBOSE)
 

Output:

-1/35*(-d*x+c)*(A*d^2*x+6*B*c*d*x+6*A*c*d+B*c^2)*(-d^2*x^2+c^2)^(3/2)/(d*x 
+c)^5/c^2/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (75) = 150\).

Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.80 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=-\frac {B c^{5} + 6 \, A c^{4} d + {\left (B c d^{4} + 6 \, A d^{5}\right )} x^{4} + 4 \, {\left (B c^{2} d^{3} + 6 \, A c d^{4}\right )} x^{3} + 6 \, {\left (B c^{3} d^{2} + 6 \, A c^{2} d^{3}\right )} x^{2} + 4 \, {\left (B c^{4} d + 6 \, A c^{3} d^{2}\right )} x + {\left (B c^{4} + 6 \, A c^{3} d + {\left (6 \, B c d^{3} + A d^{4}\right )} x^{3} - {\left (11 \, B c^{2} d^{2} - 4 \, A c d^{3}\right )} x^{2} + {\left (4 \, B c^{3} d - 11 \, A c^{2} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{35 \, {\left (c^{2} d^{6} x^{4} + 4 \, c^{3} d^{5} x^{3} + 6 \, c^{4} d^{4} x^{2} + 4 \, c^{5} d^{3} x + c^{6} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^6,x, algorithm="fricas")
 

Output:

-1/35*(B*c^5 + 6*A*c^4*d + (B*c*d^4 + 6*A*d^5)*x^4 + 4*(B*c^2*d^3 + 6*A*c* 
d^4)*x^3 + 6*(B*c^3*d^2 + 6*A*c^2*d^3)*x^2 + 4*(B*c^4*d + 6*A*c^3*d^2)*x + 
 (B*c^4 + 6*A*c^3*d + (6*B*c*d^3 + A*d^4)*x^3 - (11*B*c^2*d^2 - 4*A*c*d^3) 
*x^2 + (4*B*c^3*d - 11*A*c^2*d^2)*x)*sqrt(-d^2*x^2 + c^2))/(c^2*d^6*x^4 + 
4*c^3*d^5*x^3 + 6*c^4*d^4*x^2 + 4*c^5*d^3*x + c^6*d^2)
 

Sympy [F]

\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (c + d x\right )^{6}}\, dx \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(3/2)/(d*x+c)**6,x)
                                                                                    
                                                                                    
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(3/2)*(A + B*x)/(c + d*x)**6, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (75) = 150\).

Time = 0.05 (sec) , antiderivative size = 672, normalized size of antiderivative = 8.10 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=\frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} B c}{2 \, {\left (d^{7} x^{5} + 5 \, c d^{6} x^{4} + 10 \, c^{2} d^{5} x^{3} + 10 \, c^{3} d^{4} x^{2} + 5 \, c^{4} d^{3} x + c^{5} d^{2}\right )}} - \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} B c^{2}}{7 \, {\left (d^{6} x^{4} + 4 \, c d^{5} x^{3} + 6 \, c^{2} d^{4} x^{2} + 4 \, c^{3} d^{3} x + c^{4} d^{2}\right )}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A}{2 \, {\left (d^{6} x^{5} + 5 \, c d^{5} x^{4} + 10 \, c^{2} d^{4} x^{3} + 10 \, c^{3} d^{3} x^{2} + 5 \, c^{4} d^{2} x + c^{5} d\right )}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} B}{d^{6} x^{4} + 4 \, c d^{5} x^{3} + 6 \, c^{2} d^{4} x^{2} + 4 \, c^{3} d^{3} x + c^{4} d^{2}} + \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} A c}{7 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 6 \, c^{2} d^{3} x^{2} + 4 \, c^{3} d^{2} x + c^{4} d\right )}} + \frac {87 \, \sqrt {-d^{2} x^{2} + c^{2}} B c}{70 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} B c}{35 \, {\left (c d^{4} x^{2} + 2 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} B c}{35 \, {\left (c^{2} d^{3} x + c^{3} d^{2}\right )}} - \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} A}{70 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} A}{35 \, {\left (c d^{3} x^{2} + 2 \, c^{2} d^{2} x + c^{3} d\right )}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} A}{35 \, {\left (c^{2} d^{2} x + c^{3} d\right )}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} B}{5 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} B}{5 \, {\left (c d^{3} x + c^{2} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^6,x, algorithm="maxima")
 

Output:

1/2*(-d^2*x^2 + c^2)^(3/2)*B*c/(d^7*x^5 + 5*c*d^6*x^4 + 10*c^2*d^5*x^3 + 1 
0*c^3*d^4*x^2 + 5*c^4*d^3*x + c^5*d^2) - 3/7*sqrt(-d^2*x^2 + c^2)*B*c^2/(d 
^6*x^4 + 4*c*d^5*x^3 + 6*c^2*d^4*x^2 + 4*c^3*d^3*x + c^4*d^2) - 1/2*(-d^2* 
x^2 + c^2)^(3/2)*A/(d^6*x^5 + 5*c*d^5*x^4 + 10*c^2*d^4*x^3 + 10*c^3*d^3*x^ 
2 + 5*c^4*d^2*x + c^5*d) - (-d^2*x^2 + c^2)^(3/2)*B/(d^6*x^4 + 4*c*d^5*x^3 
 + 6*c^2*d^4*x^2 + 4*c^3*d^3*x + c^4*d^2) + 3/7*sqrt(-d^2*x^2 + c^2)*A*c/( 
d^5*x^4 + 4*c*d^4*x^3 + 6*c^2*d^3*x^2 + 4*c^3*d^2*x + c^4*d) + 87/70*sqrt( 
-d^2*x^2 + c^2)*B*c/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2) + 1/35 
*sqrt(-d^2*x^2 + c^2)*B*c/(c*d^4*x^2 + 2*c^2*d^3*x + c^3*d^2) + 1/35*sqrt( 
-d^2*x^2 + c^2)*B*c/(c^2*d^3*x + c^3*d^2) - 3/70*sqrt(-d^2*x^2 + c^2)*A/(d 
^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d) - 1/35*sqrt(-d^2*x^2 + c^2)*A/ 
(c*d^3*x^2 + 2*c^2*d^2*x + c^3*d) - 1/35*sqrt(-d^2*x^2 + c^2)*A/(c^2*d^2*x 
 + c^3*d) - 1/5*sqrt(-d^2*x^2 + c^2)*B/(d^4*x^2 + 2*c*d^3*x + c^2*d^2) - 1 
/5*sqrt(-d^2*x^2 + c^2)*B/(c*d^3*x + c^2*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (75) = 150\).

Time = 0.14 (sec) , antiderivative size = 405, normalized size of antiderivative = 4.88 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=\frac {2 \, {\left (B c + 6 \, A d + \frac {7 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} B c}{d^{2} x} + \frac {7 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} A}{d x} - \frac {14 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} B c}{d^{4} x^{2}} + \frac {91 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} A}{d^{3} x^{2}} + \frac {70 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} B c}{d^{6} x^{3}} + \frac {70 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} A}{d^{5} x^{3}} - \frac {35 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} B c}{d^{8} x^{4}} + \frac {140 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} A}{d^{7} x^{4}} + \frac {35 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{5} B c}{d^{10} x^{5}} + \frac {35 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{5} A}{d^{9} x^{5}} + \frac {35 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{6} A}{d^{11} x^{6}}\right )}}{35 \, c^{2} d {\left (\frac {c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}}{d^{2} x} + 1\right )}^{7} {\left | d \right |}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^6,x, algorithm="giac")
 

Output:

2/35*(B*c + 6*A*d + 7*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/(d^2*x) + 7* 
(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) - 14*(c*d + sqrt(-d^2*x^2 + c^ 
2)*abs(d))^2*B*c/(d^4*x^2) + 91*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*A/(d 
^3*x^2) + 70*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c/(d^6*x^3) + 70*(c*d 
 + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^3) - 35*(c*d + sqrt(-d^2*x^2 + 
c^2)*abs(d))^4*B*c/(d^8*x^4) + 140*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*A 
/(d^7*x^4) + 35*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*B*c/(d^10*x^5) + 35* 
(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A/(d^9*x^5) + 35*(c*d + sqrt(-d^2*x^ 
2 + c^2)*abs(d))^6*A/(d^11*x^6))/(c^2*d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d 
))/(d^2*x) + 1)^7*abs(d))
 

Mupad [B] (verification not implemented)

Time = 10.27 (sec) , antiderivative size = 397, normalized size of antiderivative = 4.78 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=\frac {16\,A\,\sqrt {c^2-d^2\,x^2}}{35\,\left (c^3\,d+3\,c^2\,d^2\,x+3\,c\,d^3\,x^2+d^4\,x^3\right )}-\frac {A\,\sqrt {c^2-d^2\,x^2}}{35\,\left (c^3\,d+2\,c^2\,d^2\,x+c\,d^3\,x^2\right )}+\frac {29\,B\,\sqrt {c^2-d^2\,x^2}}{35\,\left (c^2\,d^2+2\,c\,d^3\,x+d^4\,x^2\right )}-\frac {A\,\sqrt {c^2-d^2\,x^2}}{35\,\left (c^3\,d+x\,c^2\,d^2\right )}-\frac {6\,B\,\sqrt {c^2-d^2\,x^2}}{35\,\left (c^2\,d^2+x\,c\,d^3\right )}-\frac {4\,A\,c\,\sqrt {c^2-d^2\,x^2}}{7\,\left (c^4\,d+4\,c^3\,d^2\,x+6\,c^2\,d^3\,x^2+4\,c\,d^4\,x^3+d^5\,x^4\right )}-\frac {44\,B\,c\,\sqrt {c^2-d^2\,x^2}}{35\,\left (c^3\,d^2+3\,c^2\,d^3\,x+3\,c\,d^4\,x^2+d^5\,x^3\right )}+\frac {4\,B\,c^2\,\sqrt {c^2-d^2\,x^2}}{7\,\left (c^4\,d^2+4\,c^3\,d^3\,x+6\,c^2\,d^4\,x^2+4\,c\,d^5\,x^3+d^6\,x^4\right )} \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x))/(c + d*x)^6,x)
 

Output:

(16*A*(c^2 - d^2*x^2)^(1/2))/(35*(c^3*d + d^4*x^3 + 3*c^2*d^2*x + 3*c*d^3* 
x^2)) - (A*(c^2 - d^2*x^2)^(1/2))/(35*(c^3*d + 2*c^2*d^2*x + c*d^3*x^2)) + 
 (29*B*(c^2 - d^2*x^2)^(1/2))/(35*(c^2*d^2 + d^4*x^2 + 2*c*d^3*x)) - (A*(c 
^2 - d^2*x^2)^(1/2))/(35*(c^3*d + c^2*d^2*x)) - (6*B*(c^2 - d^2*x^2)^(1/2) 
)/(35*(c^2*d^2 + c*d^3*x)) - (4*A*c*(c^2 - d^2*x^2)^(1/2))/(7*(c^4*d + d^5 
*x^4 + 4*c^3*d^2*x + 4*c*d^4*x^3 + 6*c^2*d^3*x^2)) - (44*B*c*(c^2 - d^2*x^ 
2)^(1/2))/(35*(c^3*d^2 + d^5*x^3 + 3*c^2*d^3*x + 3*c*d^4*x^2)) + (4*B*c^2* 
(c^2 - d^2*x^2)^(1/2))/(7*(c^4*d^2 + d^6*x^4 + 4*c^3*d^3*x + 4*c*d^5*x^3 + 
 6*c^2*d^4*x^2))
 

Reduce [F]

\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^6} \, dx=\int \frac {\left (B x +A \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{\left (d x +c \right )^{6}}d x \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^6,x)
 

Output:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^6,x)