Integrand size = 27, antiderivative size = 74 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {(B c+A d) (c+d x)}{3 c d^2 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {(B c-2 A d) x}{3 c^3 d \sqrt {c^2-d^2 x^2}} \] Output:
1/3*(A*d+B*c)*(d*x+c)/c/d^2/(-d^2*x^2+c^2)^(3/2)-1/3*(-2*A*d+B*c)*x/c^3/d/ (-d^2*x^2+c^2)^(1/2)
Time = 0.46 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (A d \left (c^2+2 c d x-2 d^2 x^2\right )+B c \left (c^2-c d x+d^2 x^2\right )\right )}{3 c^3 d^2 (c-d x)^2 (c+d x)} \] Input:
Integrate[((A + B*x)*(c + d*x))/(c^2 - d^2*x^2)^(5/2),x]
Output:
(Sqrt[c^2 - d^2*x^2]*(A*d*(c^2 + 2*c*d*x - 2*d^2*x^2) + B*c*(c^2 - c*d*x + d^2*x^2)))/(3*c^3*d^2*(c - d*x)^2*(c + d*x))
Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {669, 208}
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.
\(\displaystyle \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 669 |
\(\displaystyle \frac {(c+d x) (A d+B c)}{3 c d^2 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {(B c-2 A d) \int \frac {1}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c d}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {(c+d x) (A d+B c)}{3 c d^2 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {x (B c-2 A d)}{3 c^3 d \sqrt {c^2-d^2 x^2}}\) |
Input:
Int[((A + B*x)*(c + d*x))/(c^2 - d^2*x^2)^(5/2),x]
Output:
((B*c + A*d)*(c + d*x))/(3*c*d^2*(c^2 - d^2*x^2)^(3/2)) - ((B*c - 2*A*d)*x )/(3*c^3*d*Sqrt[c^2 - d^2*x^2])
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
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* (p + 1))), x] - Simp[e*((m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11
method | result | size |
gosper | \(\frac {\left (d x +c \right )^{2} \left (-d x +c \right ) \left (-2 A \,d^{3} x^{2}+B c \,d^{2} x^{2}+2 A c \,d^{2} x -B \,c^{2} d x +A \,c^{2} d +B \,c^{3}\right )}{3 c^{3} d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}\) | \(82\) |
orering | \(\frac {\left (d x +c \right )^{2} \left (-d x +c \right ) \left (-2 A \,d^{3} x^{2}+B c \,d^{2} x^{2}+2 A c \,d^{2} x -B \,c^{2} d x +A \,c^{2} d +B \,c^{3}\right )}{3 c^{3} d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}\) | \(82\) |
trager | \(\frac {\left (-2 A \,d^{3} x^{2}+B c \,d^{2} x^{2}+2 A c \,d^{2} x -B \,c^{2} d x +A \,c^{2} d +B \,c^{3}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3 d^{2} c^{3} \left (-d x +c \right )^{2} \left (d x +c \right )}\) | \(84\) |
default | \(A c \left (\frac {x}{3 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )+\frac {A d +B c}{3 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+B d \left (\frac {x}{2 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}-\frac {c^{2} \left (\frac {x}{3 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{2}}\right )\) | \(145\) |
Input:
int((B*x+A)*(d*x+c)/(-d^2*x^2+c^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*(d*x+c)^2*(-d*x+c)*(-2*A*d^3*x^2+B*c*d^2*x^2+2*A*c*d^2*x-B*c^2*d*x+A*c ^2*d+B*c^3)/c^3/d^2/(-d^2*x^2+c^2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (66) = 132\).
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {B c^{4} + A c^{3} d + {\left (B c d^{3} + A d^{4}\right )} x^{3} - {\left (B c^{2} d^{2} + A c d^{3}\right )} x^{2} - {\left (B c^{3} d + A c^{2} d^{2}\right )} x + {\left (B c^{3} + A c^{2} d + {\left (B c d^{2} - 2 \, A d^{3}\right )} x^{2} - {\left (B c^{2} d - 2 \, A c d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{3 \, {\left (c^{3} d^{5} x^{3} - c^{4} d^{4} x^{2} - c^{5} d^{3} x + c^{6} d^{2}\right )}} \] Input:
integrate((B*x+A)*(d*x+c)/(-d^2*x^2+c^2)^(5/2),x, algorithm="fricas")
Output:
1/3*(B*c^4 + A*c^3*d + (B*c*d^3 + A*d^4)*x^3 - (B*c^2*d^2 + A*c*d^3)*x^2 - (B*c^3*d + A*c^2*d^2)*x + (B*c^3 + A*c^2*d + (B*c*d^2 - 2*A*d^3)*x^2 - (B *c^2*d - 2*A*c*d^2)*x)*sqrt(-d^2*x^2 + c^2))/(c^3*d^5*x^3 - c^4*d^4*x^2 - c^5*d^3*x + c^6*d^2)
Result contains complex when optimal does not.
Time = 5.20 (sec) , antiderivative size = 481, normalized size of antiderivative = 6.50 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=A c \left (\begin {cases} \frac {3 i c^{2} x}{- 3 c^{7} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}} + 3 c^{5} d^{2} x^{2} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}}} - \frac {2 i d^{2} x^{3}}{- 3 c^{7} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}} + 3 c^{5} d^{2} x^{2} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}}} & \text {for}\: \left |{\frac {d^{2} x^{2}}{c^{2}}}\right | > 1 \\- \frac {3 c^{2} x}{- 3 c^{7} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}} + 3 c^{5} d^{2} x^{2} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}}} + \frac {2 d^{2} x^{3}}{- 3 c^{7} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}} + 3 c^{5} d^{2} x^{2} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}}} & \text {otherwise} \end {cases}\right ) + A d \left (\begin {cases} - \frac {1}{- 3 c^{2} d^{2} \sqrt {c^{2} - d^{2} x^{2}} + 3 d^{4} x^{2} \sqrt {c^{2} - d^{2} x^{2}}} & \text {for}\: d \neq 0 \\\frac {x^{2}}{2 \left (c^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + B c \left (\begin {cases} - \frac {1}{- 3 c^{2} d^{2} \sqrt {c^{2} - d^{2} x^{2}} + 3 d^{4} x^{2} \sqrt {c^{2} - d^{2} x^{2}}} & \text {for}\: d \neq 0 \\\frac {x^{2}}{2 \left (c^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + B d \left (\begin {cases} \frac {i x^{3}}{- 3 c^{5} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}} + 3 c^{3} d^{2} x^{2} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}}} & \text {for}\: \left |{\frac {d^{2} x^{2}}{c^{2}}}\right | > 1 \\- \frac {x^{3}}{- 3 c^{5} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}} + 3 c^{3} d^{2} x^{2} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((B*x+A)*(d*x+c)/(-d**2*x**2+c**2)**(5/2),x)
Output:
A*c*Piecewise((3*I*c**2*x/(-3*c**7*sqrt(-1 + d**2*x**2/c**2) + 3*c**5*d**2 *x**2*sqrt(-1 + d**2*x**2/c**2)) - 2*I*d**2*x**3/(-3*c**7*sqrt(-1 + d**2*x **2/c**2) + 3*c**5*d**2*x**2*sqrt(-1 + d**2*x**2/c**2)), Abs(d**2*x**2/c** 2) > 1), (-3*c**2*x/(-3*c**7*sqrt(1 - d**2*x**2/c**2) + 3*c**5*d**2*x**2*s qrt(1 - d**2*x**2/c**2)) + 2*d**2*x**3/(-3*c**7*sqrt(1 - d**2*x**2/c**2) + 3*c**5*d**2*x**2*sqrt(1 - d**2*x**2/c**2)), True)) + A*d*Piecewise((-1/(- 3*c**2*d**2*sqrt(c**2 - d**2*x**2) + 3*d**4*x**2*sqrt(c**2 - d**2*x**2)), Ne(d, 0)), (x**2/(2*(c**2)**(5/2)), True)) + B*c*Piecewise((-1/(-3*c**2*d* *2*sqrt(c**2 - d**2*x**2) + 3*d**4*x**2*sqrt(c**2 - d**2*x**2)), Ne(d, 0)) , (x**2/(2*(c**2)**(5/2)), True)) + B*d*Piecewise((I*x**3/(-3*c**5*sqrt(-1 + d**2*x**2/c**2) + 3*c**3*d**2*x**2*sqrt(-1 + d**2*x**2/c**2)), Abs(d**2 *x**2/c**2) > 1), (-x**3/(-3*c**5*sqrt(1 - d**2*x**2/c**2) + 3*c**3*d**2*x **2*sqrt(1 - d**2*x**2/c**2)), True))
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {A x}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} c} + \frac {B x}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d} + \frac {B c}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {A}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d} + \frac {2 \, A x}{3 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{3}} - \frac {B x}{3 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2} d} \] Input:
integrate((B*x+A)*(d*x+c)/(-d^2*x^2+c^2)^(5/2),x, algorithm="maxima")
Output:
1/3*A*x/((-d^2*x^2 + c^2)^(3/2)*c) + 1/3*B*x/((-d^2*x^2 + c^2)^(3/2)*d) + 1/3*B*c/((-d^2*x^2 + c^2)^(3/2)*d^2) + 1/3*A/((-d^2*x^2 + c^2)^(3/2)*d) + 2/3*A*x/(sqrt(-d^2*x^2 + c^2)*c^3) - 1/3*B*x/(sqrt(-d^2*x^2 + c^2)*c^2*d)
\[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (d x + c\right )}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)*(d*x+c)/(-d^2*x^2+c^2)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(d*x + c)/(-d^2*x^2 + c^2)^(5/2), x)
Time = 9.78 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (B\,c^3-B\,c^2\,d\,x+A\,c^2\,d+B\,c\,d^2\,x^2+2\,A\,c\,d^2\,x-2\,A\,d^3\,x^2\right )}{3\,c^3\,d^2\,\left (c+d\,x\right )\,{\left (c-d\,x\right )}^2} \] Input:
int(((A + B*x)*(c + d*x))/(c^2 - d^2*x^2)^(5/2),x)
Output:
((c^2 - d^2*x^2)^(1/2)*(B*c^3 - 2*A*d^3*x^2 + A*c^2*d + 2*A*c*d^2*x - B*c^ 2*d*x + B*c*d^2*x^2))/(3*c^3*d^2*(c + d*x)*(c - d*x)^2)
Time = 0.21 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.05 \[ \int \frac {(A+B x) (c+d x)}{\left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {-2 \sqrt {-d^{2} x^{2}+c^{2}}\, a c d +2 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{2} x +\sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2}-\sqrt {-d^{2} x^{2}+c^{2}}\, b c d x +a \,c^{2} d +2 a c \,d^{2} x -2 a \,d^{3} x^{2}+b \,c^{3}-b \,c^{2} d x +b c \,d^{2} x^{2}}{3 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} \left (-d x +c \right )} \] Input:
int((B*x+A)*(d*x+c)/(-d^2*x^2+c^2)^(5/2),x)
Output:
( - 2*sqrt(c**2 - d**2*x**2)*a*c*d + 2*sqrt(c**2 - d**2*x**2)*a*d**2*x + s qrt(c**2 - d**2*x**2)*b*c**2 - sqrt(c**2 - d**2*x**2)*b*c*d*x + a*c**2*d + 2*a*c*d**2*x - 2*a*d**3*x**2 + b*c**3 - b*c**2*d*x + b*c*d**2*x**2)/(3*sq rt(c**2 - d**2*x**2)*c**3*d**2*(c - d*x))