Integrand size = 31, antiderivative size = 99 \[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {(B c+A d) \sqrt {c+d x}}{c d^2 \sqrt {c^2-d^2 x^2}}+\frac {(B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {2} c^{3/2} d^2} \] Output:
(A*d+B*c)*(d*x+c)^(1/2)/c/d^2/(-d^2*x^2+c^2)^(1/2)+1/2*(-A*d+B*c)*arctanh( 2^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(1/2))*2^(1/2)/c^(3/2)/d^2
Time = 0.97 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {(B c+A d) \sqrt {c^2-d^2 x^2}}{c d^2 (c-d x) \sqrt {c+d x}}+\frac {(B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {2} c^{3/2} d^2} \] Input:
Integrate[((A + B*x)*Sqrt[c + d*x])/(c^2 - d^2*x^2)^(3/2),x]
Output:
((B*c + A*d)*Sqrt[c^2 - d^2*x^2])/(c*d^2*(c - d*x)*Sqrt[c + d*x]) + ((B*c - A*d)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(Sqrt [2]*c^(3/2)*d^2)
Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {669, 471, 221}
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.
\(\displaystyle \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 669 |
\(\displaystyle \frac {\sqrt {c+d x} (A d+B c)}{c d^2 \sqrt {c^2-d^2 x^2}}-\frac {(B c-A d) \int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{2 c d}\) |
\(\Big \downarrow \) 471 |
\(\displaystyle \frac {\sqrt {c+d x} (A d+B c)}{c d^2 \sqrt {c^2-d^2 x^2}}-\frac {(B c-A d) \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(B c-A d) \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {2} c^{3/2} d^2}+\frac {\sqrt {c+d x} (A d+B c)}{c d^2 \sqrt {c^2-d^2 x^2}}\) |
Input:
Int[((A + B*x)*Sqrt[c + d*x])/(c^2 - d^2*x^2)^(3/2),x]
Output:
((B*c + A*d)*Sqrt[c + d*x])/(c*d^2*Sqrt[c^2 - d^2*x^2]) + ((B*c - A*d)*Arc Tanh[Sqrt[c^2 - d^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])])/(Sqrt[2]*c^(3/2 )*d^2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2*d Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] ], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
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 = 0.36 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16
method | result | size |
default | \(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d \sqrt {-d x +c}-B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \sqrt {-d x +c}-2 A d \sqrt {c}-2 B \,c^{\frac {3}{2}}\right )}{2 c^{\frac {3}{2}} \sqrt {d x +c}\, \left (-d x +c \right ) d^{2}}\) | \(115\) |
Input:
int((B*x+A)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(-d^2*x^2+c^2)^(1/2)/c^(3/2)*(A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^ (1/2)/c^(1/2))*d*(-d*x+c)^(1/2)-B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/ 2)/c^(1/2))*c*(-d*x+c)^(1/2)-2*A*d*c^(1/2)-2*B*c^(3/2))/(d*x+c)^(1/2)/(-d* x+c)/d^2
Time = 0.10 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.06 \[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\left [\frac {\sqrt {2} {\left (B c^{3} - A c^{2} d - {\left (B c d^{2} - A d^{3}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x + 2 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {c} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 4 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (B c^{2} + A c d\right )} \sqrt {d x + c}}{4 \, {\left (c^{2} d^{4} x^{2} - c^{4} d^{2}\right )}}, \frac {\sqrt {2} {\left (B c^{3} - A c^{2} d - {\left (B c d^{2} - A d^{3}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) - 2 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (B c^{2} + A c d\right )} \sqrt {d x + c}}{2 \, {\left (c^{2} d^{4} x^{2} - c^{4} d^{2}\right )}}\right ] \] Input:
integrate((B*x+A)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="fricas" )
Output:
[1/4*(sqrt(2)*(B*c^3 - A*c^2*d - (B*c*d^2 - A*d^3)*x^2)*sqrt(c)*log(-(d^2* x^2 - 2*c*d*x + 2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(c) - 3*c ^2)/(d^2*x^2 + 2*c*d*x + c^2)) - 4*sqrt(-d^2*x^2 + c^2)*(B*c^2 + A*c*d)*sq rt(d*x + c))/(c^2*d^4*x^2 - c^4*d^2), 1/2*(sqrt(2)*(B*c^3 - A*c^2*d - (B*c *d^2 - A*d^3)*x^2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d *x + c)*sqrt(-c)/(c*d*x + c^2)) - 2*sqrt(-d^2*x^2 + c^2)*(B*c^2 + A*c*d)*s qrt(d*x + c))/(c^2*d^4*x^2 - c^4*d^2)]
\[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {c + d x}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)*(d*x+c)**(1/2)/(-d**2*x**2+c**2)**(3/2),x)
Output:
Integral((A + B*x)*sqrt(c + d*x)/(-(-c + d*x)*(c + d*x))**(3/2), x)
\[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {d x + c}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="maxima" )
Output:
integrate((B*x + A)*sqrt(d*x + c)/(-d^2*x^2 + c^2)^(3/2), x)
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {\frac {\sqrt {2} {\left (B c - A d\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c d} - \frac {2 \, {\left (B c + A d\right )}}{\sqrt {-d x + c} c d}}{2 \, d} \] Input:
integrate((B*x+A)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algorithm="giac")
Output:
-1/2*(sqrt(2)*(B*c - A*d)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqr t(-c)*c*d) - 2*(B*c + A*d)/(sqrt(-d*x + c)*c*d))/d
Timed out. \[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c+d\,x}}{{\left (c^2-d^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((A + B*x)*(c + d*x)^(1/2))/(c^2 - d^2*x^2)^(3/2),x)
Output:
int(((A + B*x)*(c + d*x)^(1/2))/(c^2 - d^2*x^2)^(3/2), x)
Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a d -\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b c -\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a d +\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b c +4 a c d +4 b \,c^{2}}{4 \sqrt {-d x +c}\, c^{2} d^{2}} \] Input:
int((B*x+A)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x)
Output:
(sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*d - sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c - s qrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*d + sq rt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c + 4*a *c*d + 4*b*c**2)/(4*sqrt(c - d*x)*c**2*d**2)