Integrand size = 27, antiderivative size = 80 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\frac {(4 c+d x) \sqrt {-b c^2+b d^2 x^2}}{2 b d}+\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {b} d x}{\sqrt {-b c^2+b d^2 x^2}}\right )}{2 \sqrt {b} d} \] Output:
1/2*(d*x+4*c)*(b*d^2*x^2-b*c^2)^(1/2)/b/d+3/2*c^2*arctanh(b^(1/2)*d*x/(b*d ^2*x^2-b*c^2)^(1/2))/b^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(80)=160\).
Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.10 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\frac {-4 c^3-c^2 d x+4 c d^2 x^2+d^3 x^3+3 c^2 \sqrt {-c^2+d^2 x^2} \log \left (\sqrt {-c^2}-d x-\sqrt {-c^2+d^2 x^2}\right )-3 c^2 \sqrt {-c^2+d^2 x^2} \log \left (d \left (\sqrt {-c^2}+d x-\sqrt {-c^2+d^2 x^2}\right )\right )}{2 d \sqrt {b \left (-c^2+d^2 x^2\right )}} \] Input:
Integrate[(c + d*x)^2/Sqrt[-(b*c^2) + b*d^2*x^2],x]
Output:
(-4*c^3 - c^2*d*x + 4*c*d^2*x^2 + d^3*x^3 + 3*c^2*Sqrt[-c^2 + d^2*x^2]*Log [Sqrt[-c^2] - d*x - Sqrt[-c^2 + d^2*x^2]] - 3*c^2*Sqrt[-c^2 + d^2*x^2]*Log [d*(Sqrt[-c^2] + d*x - Sqrt[-c^2 + d^2*x^2])])/(2*d*Sqrt[b*(-c^2 + d^2*x^2 )])
Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {469, 455, 224, 219}
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.
\(\displaystyle \int \frac {(c+d x)^2}{\sqrt {b d^2 x^2-b c^2}} \, dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle \frac {3}{2} c \int \frac {c+d x}{\sqrt {b d^2 x^2-b c^2}}dx+\frac {(c+d x) \sqrt {b d^2 x^2-b c^2}}{2 b d}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {3}{2} c \left (c \int \frac {1}{\sqrt {b d^2 x^2-b c^2}}dx+\frac {\sqrt {b d^2 x^2-b c^2}}{b d}\right )+\frac {(c+d x) \sqrt {b d^2 x^2-b c^2}}{2 b d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {3}{2} c \left (c \int \frac {1}{1-\frac {b d^2 x^2}{b d^2 x^2-b c^2}}d\frac {x}{\sqrt {b d^2 x^2-b c^2}}+\frac {\sqrt {b d^2 x^2-b c^2}}{b d}\right )+\frac {(c+d x) \sqrt {b d^2 x^2-b c^2}}{2 b d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{2} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {b} d x}{\sqrt {b d^2 x^2-b c^2}}\right )}{\sqrt {b} d}+\frac {\sqrt {b d^2 x^2-b c^2}}{b d}\right )+\frac {(c+d x) \sqrt {b d^2 x^2-b c^2}}{2 b d}\) |
Input:
Int[(c + d*x)^2/Sqrt[-(b*c^2) + b*d^2*x^2],x]
Output:
((c + d*x)*Sqrt[-(b*c^2) + b*d^2*x^2])/(2*b*d) + (3*c*(Sqrt[-(b*c^2) + b*d ^2*x^2]/(b*d) + (c*ArcTanh[(Sqrt[b]*d*x)/Sqrt[-(b*c^2) + b*d^2*x^2]])/(Sqr t[b]*d)))/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* ((n + p)/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* p + 1, 0] && IntegerQ[2*p]
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09
method | result | size |
risch | \(-\frac {\left (d x +4 c \right ) \left (-d^{2} x^{2}+c^{2}\right )}{2 d \sqrt {b \left (d^{2} x^{2}-c^{2}\right )}}+\frac {3 c^{2} \ln \left (\frac {b x \,d^{2}}{\sqrt {b \,d^{2}}}+\sqrt {b \,x^{2} d^{2}-b \,c^{2}}\right )}{2 \sqrt {b \,d^{2}}}\) | \(87\) |
default | \(\frac {c^{2} \ln \left (\frac {b x \,d^{2}}{\sqrt {b \,d^{2}}}+\sqrt {b \,x^{2} d^{2}-b \,c^{2}}\right )}{\sqrt {b \,d^{2}}}+d^{2} \left (\frac {x \sqrt {b \,x^{2} d^{2}-b \,c^{2}}}{2 b \,d^{2}}+\frac {c^{2} \ln \left (\frac {b x \,d^{2}}{\sqrt {b \,d^{2}}}+\sqrt {b \,x^{2} d^{2}-b \,c^{2}}\right )}{2 d^{2} \sqrt {b \,d^{2}}}\right )+\frac {2 c \sqrt {b \,x^{2} d^{2}-b \,c^{2}}}{d b}\) | \(149\) |
Input:
int((d*x+c)^2/(b*d^2*x^2-b*c^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(d*x+4*c)*(-d^2*x^2+c^2)/d/(b*(d^2*x^2-c^2))^(1/2)+3/2*c^2*ln(b*x*d^2 /(b*d^2)^(1/2)+(b*d^2*x^2-b*c^2)^(1/2))/(b*d^2)^(1/2)
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.95 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\left [\frac {3 \, \sqrt {b} c^{2} \log \left (2 \, b d^{2} x^{2} - b c^{2} + 2 \, \sqrt {b d^{2} x^{2} - b c^{2}} \sqrt {b} d x\right ) + 2 \, \sqrt {b d^{2} x^{2} - b c^{2}} {\left (d x + 4 \, c\right )}}{4 \, b d}, -\frac {3 \, \sqrt {-b} c^{2} \arctan \left (\frac {\sqrt {-b} d x}{\sqrt {b d^{2} x^{2} - b c^{2}}}\right ) - \sqrt {b d^{2} x^{2} - b c^{2}} {\left (d x + 4 \, c\right )}}{2 \, b d}\right ] \] Input:
integrate((d*x+c)^2/(b*d^2*x^2-b*c^2)^(1/2),x, algorithm="fricas")
Output:
[1/4*(3*sqrt(b)*c^2*log(2*b*d^2*x^2 - b*c^2 + 2*sqrt(b*d^2*x^2 - b*c^2)*sq rt(b)*d*x) + 2*sqrt(b*d^2*x^2 - b*c^2)*(d*x + 4*c))/(b*d), -1/2*(3*sqrt(-b )*c^2*arctan(sqrt(-b)*d*x/sqrt(b*d^2*x^2 - b*c^2)) - sqrt(b*d^2*x^2 - b*c^ 2)*(d*x + 4*c))/(b*d)]
Time = 0.49 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\begin {cases} \frac {3 c^{2} \left (\begin {cases} \frac {\log {\left (2 b d^{2} x + 2 \sqrt {b d^{2}} \sqrt {- b c^{2} + b d^{2} x^{2}} \right )}}{\sqrt {b d^{2}}} & \text {for}\: b c^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b d^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \sqrt {- b c^{2} + b d^{2} x^{2}} \cdot \left (\frac {2 c}{b d} + \frac {x}{2 b}\right ) & \text {for}\: b d^{2} \neq 0 \\\frac {\begin {cases} c^{2} x & \text {for}\: d = 0 \\\frac {\left (c + d x\right )^{3}}{3 d} & \text {otherwise} \end {cases}}{\sqrt {- b c^{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2/(b*d**2*x**2-b*c**2)**(1/2),x)
Output:
Piecewise((3*c**2*Piecewise((log(2*b*d**2*x + 2*sqrt(b*d**2)*sqrt(-b*c**2 + b*d**2*x**2))/sqrt(b*d**2), Ne(b*c**2, 0)), (x*log(x)/sqrt(b*d**2*x**2), True))/2 + sqrt(-b*c**2 + b*d**2*x**2)*(2*c/(b*d) + x/(2*b)), Ne(b*d**2, 0)), (Piecewise((c**2*x, Eq(d, 0)), ((c + d*x)**3/(3*d), True))/sqrt(-b*c* *2), True))
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\frac {3 \, c^{2} \log \left (2 \, b d^{2} x + 2 \, \sqrt {b d^{2} x^{2} - b c^{2}} \sqrt {b} d\right )}{2 \, \sqrt {b} d} + \frac {\sqrt {b d^{2} x^{2} - b c^{2}} x}{2 \, b} + \frac {2 \, \sqrt {b d^{2} x^{2} - b c^{2}} c}{b d} \] Input:
integrate((d*x+c)^2/(b*d^2*x^2-b*c^2)^(1/2),x, algorithm="maxima")
Output:
3/2*c^2*log(2*b*d^2*x + 2*sqrt(b*d^2*x^2 - b*c^2)*sqrt(b)*d)/(sqrt(b)*d) + 1/2*sqrt(b*d^2*x^2 - b*c^2)*x/b + 2*sqrt(b*d^2*x^2 - b*c^2)*c/(b*d)
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\frac {1}{2} \, \sqrt {b d^{2} x^{2} - b c^{2}} {\left (\frac {x}{b} + \frac {4 \, c}{b d}\right )} - \frac {3 \, c^{2} \log \left ({\left | -\sqrt {b d^{2}} x + \sqrt {b d^{2} x^{2} - b c^{2}} \right |}\right )}{2 \, \sqrt {b} {\left | d \right |}} \] Input:
integrate((d*x+c)^2/(b*d^2*x^2-b*c^2)^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(b*d^2*x^2 - b*c^2)*(x/b + 4*c/(b*d)) - 3/2*c^2*log(abs(-sqrt(b*d^ 2)*x + sqrt(b*d^2*x^2 - b*c^2)))/(sqrt(b)*abs(d))
Timed out. \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{\sqrt {b\,d^2\,x^2-b\,c^2}} \,d x \] Input:
int((c + d*x)^2/(b*d^2*x^2 - b*c^2)^(1/2),x)
Output:
int((c + d*x)^2/(b*d^2*x^2 - b*c^2)^(1/2), x)
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x)^2}{\sqrt {-b c^2+b d^2 x^2}} \, dx=\frac {\sqrt {b}\, \left (4 \sqrt {d^{2} x^{2}-c^{2}}\, c +\sqrt {d^{2} x^{2}-c^{2}}\, d x +3 \,\mathrm {log}\left (\frac {\sqrt {d^{2} x^{2}-c^{2}}+d x}{c}\right ) c^{2}\right )}{2 b d} \] Input:
int((d*x+c)^2/(b*d^2*x^2-b*c^2)^(1/2),x)
Output:
(sqrt(b)*(4*sqrt( - c**2 + d**2*x**2)*c + sqrt( - c**2 + d**2*x**2)*d*x + 3*log((sqrt( - c**2 + d**2*x**2) + d*x)/c)*c**2))/(2*b*d)