Integrand size = 29, antiderivative size = 88 \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {2 (B c+A d) (c+d x)}{d^2 \sqrt {c^2-d^2 x^2}}+\frac {B \sqrt {c^2-d^2 x^2}}{d^2}-\frac {(2 B c+A d) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^2} \] Output:
2*(A*d+B*c)*(d*x+c)/d^2/(-d^2*x^2+c^2)^(1/2)+B*(-d^2*x^2+c^2)^(1/2)/d^2-(A *d+2*B*c)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^2
Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {(-3 B c-2 A d+B d x) \sqrt {c^2-d^2 x^2}}{d^2 (-c+d x)}-\frac {d (2 B c+A d) \log \left (-\sqrt {-d^2} x+\sqrt {c^2-d^2 x^2}\right )}{\left (-d^2\right )^{3/2}} \] Input:
Integrate[((A + B*x)*(c + d*x)^2)/(c^2 - d^2*x^2)^(3/2),x]
Output:
((-3*B*c - 2*A*d + B*d*x)*Sqrt[c^2 - d^2*x^2])/(d^2*(-c + d*x)) - (d*(2*B* c + A*d)*Log[-(Sqrt[-d^2]*x) + Sqrt[c^2 - d^2*x^2]])/(-d^2)^(3/2)
Time = 0.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {665, 455, 224, 216}
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.
\(\displaystyle \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 665 |
\(\displaystyle \frac {2 (c+d x) (A d+B c)}{d^2 \sqrt {c^2-d^2 x^2}}-\frac {\int \frac {2 B c+A d+B d x}{\sqrt {c^2-d^2 x^2}}dx}{d}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {2 (c+d x) (A d+B c)}{d^2 \sqrt {c^2-d^2 x^2}}-\frac {(A d+2 B c) \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx-\frac {B \sqrt {c^2-d^2 x^2}}{d}}{d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {2 (c+d x) (A d+B c)}{d^2 \sqrt {c^2-d^2 x^2}}-\frac {(A d+2 B c) \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}-\frac {B \sqrt {c^2-d^2 x^2}}{d}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 (c+d x) (A d+B c)}{d^2 \sqrt {c^2-d^2 x^2}}-\frac {\frac {(A d+2 B c) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d}-\frac {B \sqrt {c^2-d^2 x^2}}{d}}{d}\) |
Input:
Int[((A + B*x)*(c + d*x)^2)/(c^2 - d^2*x^2)^(3/2),x]
Output:
(2*(B*c + A*d)*(c + d*x))/(d^2*Sqrt[c^2 - d^2*x^2]) - (-((B*Sqrt[c^2 - d^2 *x^2])/d) + ((2*B*c + A*d)*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d)/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2)^(3/2), x_Symbol] :> Simp[(-2^(m - 1))*d^(m - 2)*(e*f + d*g)^n*((d + e*x)/(c*e^(n - 1)*Sqrt[a + c*x^2])), x] + Simp[1/(c*e^(n - 2)) Int[Expand ToSum[(2^(m - 1)*d^(m - 1)*(e*f + d*g)^n - e^n*(d + e*x)^(m - 1)*(f + g*x)^ n)/(d - e*x), x]/Sqrt[a + c*x^2], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 0.40 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {B \sqrt {-d^{2} x^{2}+c^{2}}}{d^{2}}-\frac {\left (A d +2 B c \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d \sqrt {d^{2}}}-\frac {2 \left (A d +B c \right ) \sqrt {-d^{2} \left (x -\frac {c}{d}\right )^{2}-2 c d \left (x -\frac {c}{d}\right )}}{d^{3} \left (x -\frac {c}{d}\right )}\) | \(113\) |
default | \(\frac {A x}{\sqrt {-d^{2} x^{2}+c^{2}}}+d \left (A d +2 B c \right ) \left (\frac {x}{\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}-\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{2} \sqrt {d^{2}}}\right )+\frac {c \left (2 A d +B c \right )}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+B \,d^{2} \left (-\frac {x^{2}}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {2 c^{2}}{d^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )\) | \(158\) |
Input:
int((B*x+A)*(d*x+c)^2/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
B*(-d^2*x^2+c^2)^(1/2)/d^2-(A*d+2*B*c)/d/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/ (-d^2*x^2+c^2)^(1/2))-2*(A*d+B*c)/d^3/(x-c/d)*(-d^2*(x-c/d)^2-2*c*d*(x-c/d ))^(1/2)
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {3 \, B c^{2} + 2 \, A c d - {\left (3 \, B c d + 2 \, A d^{2}\right )} x + 2 \, {\left (2 \, B c^{2} + A c d - {\left (2 \, B c d + A d^{2}\right )} x\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - \sqrt {-d^{2} x^{2} + c^{2}} {\left (B d x - 3 \, B c - 2 \, A d\right )}}{d^{3} x - c d^{2}} \] Input:
integrate((B*x+A)*(d*x+c)^2/(-d^2*x^2+c^2)^(3/2),x, algorithm="fricas")
Output:
-(3*B*c^2 + 2*A*c*d - (3*B*c*d + 2*A*d^2)*x + 2*(2*B*c^2 + A*c*d - (2*B*c* d + A*d^2)*x)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - sqrt(-d^2*x^2 + c^2)*(B*d*x - 3*B*c - 2*A*d))/(d^3*x - c*d^2)
\[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (c + d x\right )^{2}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)*(d*x+c)**2/(-d**2*x**2+c**2)**(3/2),x)
Output:
Integral((A + B*x)*(c + d*x)**2/(-(-c + d*x)*(c + d*x))**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {B x^{2}}{\sqrt {-d^{2} x^{2} + c^{2}}} + \frac {A x}{\sqrt {-d^{2} x^{2} + c^{2}}} + \frac {3 \, B c^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} + \frac {2 \, A c}{\sqrt {-d^{2} x^{2} + c^{2}} d} + \frac {{\left (2 \, B c d + A d^{2}\right )} x}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} - \frac {{\left (2 \, B c d + A d^{2}\right )} \arcsin \left (\frac {d x}{c}\right )}{d^{3}} \] Input:
integrate((B*x+A)*(d*x+c)^2/(-d^2*x^2+c^2)^(3/2),x, algorithm="maxima")
Output:
-B*x^2/sqrt(-d^2*x^2 + c^2) + A*x/sqrt(-d^2*x^2 + c^2) + 3*B*c^2/(sqrt(-d^ 2*x^2 + c^2)*d^2) + 2*A*c/(sqrt(-d^2*x^2 + c^2)*d) + (2*B*c*d + A*d^2)*x/( sqrt(-d^2*x^2 + c^2)*d^2) - (2*B*c*d + A*d^2)*arcsin(d*x/c)/d^3
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, B c + A d\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{d {\left | d \right |}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} B}{d^{2}} + \frac {4 \, {\left (B c + A d\right )}}{d {\left (\frac {c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}}{d^{2} x} - 1\right )} {\left | d \right |}} \] Input:
integrate((B*x+A)*(d*x+c)^2/(-d^2*x^2+c^2)^(3/2),x, algorithm="giac")
Output:
-(2*B*c + A*d)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d*abs(d)) + sqrt(-d^2*x^2 + c^ 2)*B/d^2 + 4*(B*c + A*d)/(d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d))/(d^2*x) - 1)*abs(d))
Timed out. \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c+d\,x\right )}^2}{{\left (c^2-d^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((A + B*x)*(c + d*x)^2)/(c^2 - d^2*x^2)^(3/2),x)
Output:
int(((A + B*x)*(c + d*x)^2)/(c^2 - d^2*x^2)^(3/2), x)
Time = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.30 \[ \int \frac {(A+B x) (c+d x)^2}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {-\sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) a d -2 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) b c +\mathit {asin} \left (\frac {d x}{c}\right ) a c d -\mathit {asin} \left (\frac {d x}{c}\right ) a \,d^{2} x +2 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{2}-2 \mathit {asin} \left (\frac {d x}{c}\right ) b c d x -4 \sqrt {-d^{2} x^{2}+c^{2}}\, a d -4 \sqrt {-d^{2} x^{2}+c^{2}}\, b c +\sqrt {-d^{2} x^{2}+c^{2}}\, b d x +4 a c d +4 b \,c^{2}+b c d x -b \,d^{2} x^{2}}{d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}-c +d x \right )} \] Input:
int((B*x+A)*(d*x+c)^2/(-d^2*x^2+c^2)^(3/2),x)
Output:
( - sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*a*d - 2*sqrt(c**2 - d**2*x**2)*as in((d*x)/c)*b*c + asin((d*x)/c)*a*c*d - asin((d*x)/c)*a*d**2*x + 2*asin((d *x)/c)*b*c**2 - 2*asin((d*x)/c)*b*c*d*x - 4*sqrt(c**2 - d**2*x**2)*a*d - 4 *sqrt(c**2 - d**2*x**2)*b*c + sqrt(c**2 - d**2*x**2)*b*d*x + 4*a*c*d + 4*b *c**2 + b*c*d*x - b*d**2*x**2)/(d**2*(sqrt(c**2 - d**2*x**2) - c + d*x))