Integrand size = 27, antiderivative size = 108 \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\frac {2 c^2 \sqrt {c^2-d^2 x^2}}{3 d^3}-\frac {c x \sqrt {c^2-d^2 x^2}}{2 d^2}+\frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}+\frac {c^3 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d^3} \] Output:
2/3*c^2*(-d^2*x^2+c^2)^(1/2)/d^3-1/2*c*x*(-d^2*x^2+c^2)^(1/2)/d^2+1/3*x^2* (-d^2*x^2+c^2)^(1/2)/d+1/2*c^3*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^3
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (4 c^2-3 c d x+2 d^2 x^2\right )}{6 d^3}-\frac {c^3 \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{d^3} \] Input:
Integrate[(x^2*Sqrt[c^2 - d^2*x^2])/(c + d*x),x]
Output:
(Sqrt[c^2 - d^2*x^2]*(4*c^2 - 3*c*d*x + 2*d^2*x^2))/(6*d^3) - (c^3*ArcTan[ (d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^3
Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {566, 533, 25, 27, 533, 25, 27, 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.
\(\displaystyle \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx\) |
\(\Big \downarrow \) 566 |
\(\displaystyle \int \frac {x^2 (c-d x)}{\sqrt {c^2-d^2 x^2}}dx\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {\int -\frac {c d x (2 c-3 d x)}{\sqrt {c^2-d^2 x^2}}dx}{3 d^2}+\frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {\int \frac {c d x (2 c-3 d x)}{\sqrt {c^2-d^2 x^2}}dx}{3 d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \int \frac {x (2 c-3 d x)}{\sqrt {c^2-d^2 x^2}}dx}{3 d}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \left (\frac {\int -\frac {c d (3 c-4 d x)}{\sqrt {c^2-d^2 x^2}}dx}{2 d^2}+\frac {3 x \sqrt {c^2-d^2 x^2}}{2 d}\right )}{3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \left (\frac {3 x \sqrt {c^2-d^2 x^2}}{2 d}-\frac {\int \frac {c d (3 c-4 d x)}{\sqrt {c^2-d^2 x^2}}dx}{2 d^2}\right )}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \left (\frac {3 x \sqrt {c^2-d^2 x^2}}{2 d}-\frac {c \int \frac {3 c-4 d x}{\sqrt {c^2-d^2 x^2}}dx}{2 d}\right )}{3 d}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \left (\frac {3 x \sqrt {c^2-d^2 x^2}}{2 d}-\frac {c \left (3 c \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {4 \sqrt {c^2-d^2 x^2}}{d}\right )}{2 d}\right )}{3 d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \left (\frac {3 x \sqrt {c^2-d^2 x^2}}{2 d}-\frac {c \left (3 c \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {4 \sqrt {c^2-d^2 x^2}}{d}\right )}{2 d}\right )}{3 d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {x^2 \sqrt {c^2-d^2 x^2}}{3 d}-\frac {c \left (\frac {3 x \sqrt {c^2-d^2 x^2}}{2 d}-\frac {c \left (\frac {3 c \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d}+\frac {4 \sqrt {c^2-d^2 x^2}}{d}\right )}{2 d}\right )}{3 d}\) |
Input:
Int[(x^2*Sqrt[c^2 - d^2*x^2])/(c + d*x),x]
Output:
(x^2*Sqrt[c^2 - d^2*x^2])/(3*d) - (c*((3*x*Sqrt[c^2 - d^2*x^2])/(2*d) - (c *((4*Sqrt[c^2 - d^2*x^2])/d + (3*c*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d))/ (2*d)))/(3*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Int[x^m*(a/c + b*(x/d))*(a + b*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0]
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {\left (2 d^{2} x^{2}-3 c d x +4 c^{2}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{6 d^{3}}+\frac {c^{3} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{2} \sqrt {d^{2}}}\) | \(75\) |
default | \(-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3 d^{3}}+\frac {c^{2} \left (\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}+\frac {c d \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{\sqrt {d^{2}}}\right )}{d^{3}}-\frac {c \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{d^{2}}\) | \(157\) |
Input:
int(x^2*(-d^2*x^2+c^2)^(1/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/6*(2*d^2*x^2-3*c*d*x+4*c^2)/d^3*(-d^2*x^2+c^2)^(1/2)+1/2*c^3/d^2/(d^2)^( 1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=-\frac {6 \, c^{3} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (2 \, d^{2} x^{2} - 3 \, c d x + 4 \, c^{2}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{6 \, d^{3}} \] Input:
integrate(x^2*(-d^2*x^2+c^2)^(1/2)/(d*x+c),x, algorithm="fricas")
Output:
-1/6*(6*c^3*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - (2*d^2*x^2 - 3*c*d *x + 4*c^2)*sqrt(-d^2*x^2 + c^2))/d^3
\[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\int \frac {x^{2} \sqrt {- \left (- c + d x\right ) \left (c + d x\right )}}{c + d x}\, dx \] Input:
integrate(x**2*(-d**2*x**2+c**2)**(1/2)/(d*x+c),x)
Output:
Integral(x**2*sqrt(-(-c + d*x)*(c + d*x))/(c + d*x), x)
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\frac {c^{3} \arcsin \left (\frac {d x}{c}\right )}{2 \, d^{3}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} c x}{2 \, d^{2}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} c^{2}}{d^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}}{3 \, d^{3}} \] Input:
integrate(x^2*(-d^2*x^2+c^2)^(1/2)/(d*x+c),x, algorithm="maxima")
Output:
1/2*c^3*arcsin(d*x/c)/d^3 - 1/2*sqrt(-d^2*x^2 + c^2)*c*x/d^2 + sqrt(-d^2*x ^2 + c^2)*c^2/d^3 - 1/3*(-d^2*x^2 + c^2)^(3/2)/d^3
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\frac {c^{3} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{2 \, d^{2} {\left | d \right |}} + \frac {1}{6} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (x {\left (\frac {2 \, x}{d} - \frac {3 \, c}{d^{2}}\right )} + \frac {4 \, c^{2}}{d^{3}}\right )} \] Input:
integrate(x^2*(-d^2*x^2+c^2)^(1/2)/(d*x+c),x, algorithm="giac")
Output:
1/2*c^3*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^2*abs(d)) + 1/6*sqrt(-d^2*x^2 + c^2 )*(x*(2*x/d - 3*c/d^2) + 4*c^2/d^3)
Timed out. \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\int \frac {x^2\,\sqrt {c^2-d^2\,x^2}}{c+d\,x} \,d x \] Input:
int((x^2*(c^2 - d^2*x^2)^(1/2))/(c + d*x),x)
Output:
int((x^2*(c^2 - d^2*x^2)^(1/2))/(c + d*x), x)
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \sqrt {c^2-d^2 x^2}}{c+d x} \, dx=\frac {3 \mathit {asin} \left (\frac {d x}{c}\right ) c^{3}+4 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-3 \sqrt {-d^{2} x^{2}+c^{2}}\, c d x +2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}-4 c^{3}}{6 d^{3}} \] Input:
int(x^2*(-d^2*x^2+c^2)^(1/2)/(d*x+c),x)
Output:
(3*asin((d*x)/c)*c**3 + 4*sqrt(c**2 - d**2*x**2)*c**2 - 3*sqrt(c**2 - d**2 *x**2)*c*d*x + 2*sqrt(c**2 - d**2*x**2)*d**2*x**2 - 4*c**3)/(6*d**3)