Integrand size = 32, antiderivative size = 95 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {B d^2+c^2 D+\left (\frac {A}{c^2}+\frac {C}{d^2}\right ) d^4 x}{d^4 \sqrt {c^2-d^2 x^2}}+\frac {D \sqrt {c^2-d^2 x^2}}{d^4}-\frac {C \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^3} \] Output:
(B*d^2+D*c^2+(A/c^2+C/d^2)*d^4*x)/d^4/(-d^2*x^2+c^2)^(1/2)+D*(-d^2*x^2+c^2 )^(1/2)/d^4-C*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^3
Time = 0.68 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {c^2-d^2 x^2} \left (B c^2 d^2+2 c^4 D+A d^4 x+c^2 d^2 x (C-D x)\right )}{c^2 (c-d x) (c+d x)}+2 C d \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{d^4} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(c^2 - d^2*x^2)^(3/2),x]
Output:
((Sqrt[c^2 - d^2*x^2]*(B*c^2*d^2 + 2*c^4*D + A*d^4*x + c^2*d^2*x*(C - D*x) ))/(c^2*(c - d*x)*(c + d*x)) + 2*C*d*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
Time = 0.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2345, 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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {x \left (A d^2+c^2 C\right )+c^2 \left (B+\frac {c^2 D}{d^2}\right )}{c^2 d^2 \sqrt {c^2-d^2 x^2}}-\frac {\int \frac {c^2 (C+D x)}{d^2 \sqrt {c^2-d^2 x^2}}dx}{c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x \left (A d^2+c^2 C\right )+c^2 \left (B+\frac {c^2 D}{d^2}\right )}{c^2 d^2 \sqrt {c^2-d^2 x^2}}-\frac {\int \frac {C+D x}{\sqrt {c^2-d^2 x^2}}dx}{d^2}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {x \left (A d^2+c^2 C\right )+c^2 \left (B+\frac {c^2 D}{d^2}\right )}{c^2 d^2 \sqrt {c^2-d^2 x^2}}-\frac {C \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx-\frac {D \sqrt {c^2-d^2 x^2}}{d^2}}{d^2}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {x \left (A d^2+c^2 C\right )+c^2 \left (B+\frac {c^2 D}{d^2}\right )}{c^2 d^2 \sqrt {c^2-d^2 x^2}}-\frac {C \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}-\frac {D \sqrt {c^2-d^2 x^2}}{d^2}}{d^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {x \left (A d^2+c^2 C\right )+c^2 \left (B+\frac {c^2 D}{d^2}\right )}{c^2 d^2 \sqrt {c^2-d^2 x^2}}-\frac {\frac {C \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d}-\frac {D \sqrt {c^2-d^2 x^2}}{d^2}}{d^2}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(c^2 - d^2*x^2)^(3/2),x]
Output:
(c^2*(B + (c^2*D)/d^2) + (c^2*C + A*d^2)*x)/(c^2*d^2*Sqrt[c^2 - d^2*x^2]) - (-((D*Sqrt[c^2 - d^2*x^2])/d^2) + (C*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/ d)/d^2
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[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.32 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {A x}{c^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {B}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+C \left (\frac {x}{\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}-\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{2} \sqrt {d^{2}}}\right )+D \left (-\frac {x^{2}}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {2 c^{2}}{d^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )\) | \(142\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
A*x/c^2/(-d^2*x^2+c^2)^(1/2)+B/d^2/(-d^2*x^2+c^2)^(1/2)+C*(1/(-d^2*x^2+c^2 )^(1/2)/d^2*x-1/d^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2)) )+D*(-x^2/d^2/(-d^2*x^2+c^2)^(1/2)+2*c^2/d^4/(-d^2*x^2+c^2)^(1/2))
Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.76 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, D c^{5} + B c^{3} d^{2} - {\left (2 \, D c^{3} d^{2} + B c d^{4}\right )} x^{2} - 2 \, {\left (C c^{2} d^{3} x^{2} - C c^{4} d\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (D c^{2} d^{2} x^{2} - 2 \, D c^{4} - B c^{2} d^{2} - {\left (C c^{2} d^{2} + A d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{c^{2} d^{6} x^{2} - c^{4} d^{4}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(-d^2*x^2+c^2)^(3/2),x, algorithm="fricas")
Output:
-(2*D*c^5 + B*c^3*d^2 - (2*D*c^3*d^2 + B*c*d^4)*x^2 - 2*(C*c^2*d^3*x^2 - C *c^4*d)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - (D*c^2*d^2*x^2 - 2*D*c ^4 - B*c^2*d^2 - (C*c^2*d^2 + A*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(c^2*d^6*x^2 - c^4*d^4)
Result contains complex when optimal does not.
Time = 6.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.35 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=A \left (\begin {cases} - \frac {i x}{c^{3} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}}} & \text {for}\: \left |{\frac {d^{2} x^{2}}{c^{2}}}\right | > 1 \\\frac {x}{c^{3} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}}} & \text {otherwise} \end {cases}\right ) + B \left (\begin {cases} \frac {1}{d^{2} \sqrt {c^{2} - d^{2} x^{2}}} & \text {for}\: d \neq 0 \\\frac {x^{2}}{2 \left (c^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + C \left (\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {d x}{c} \right )}}{d^{3}} - \frac {i x}{c d^{2} \sqrt {-1 + \frac {d^{2} x^{2}}{c^{2}}}} & \text {for}\: \left |{\frac {d^{2} x^{2}}{c^{2}}}\right | > 1 \\- \frac {\operatorname {asin}{\left (\frac {d x}{c} \right )}}{d^{3}} + \frac {x}{c d^{2} \sqrt {1 - \frac {d^{2} x^{2}}{c^{2}}}} & \text {otherwise} \end {cases}\right ) + D \left (\begin {cases} \frac {2 c^{2}}{d^{4} \sqrt {c^{2} - d^{2} x^{2}}} - \frac {x^{2}}{d^{2} \sqrt {c^{2} - d^{2} x^{2}}} & \text {for}\: d \neq 0 \\\frac {x^{4}}{4 \left (c^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(-d**2*x**2+c**2)**(3/2),x)
Output:
A*Piecewise((-I*x/(c**3*sqrt(-1 + d**2*x**2/c**2)), Abs(d**2*x**2/c**2) > 1), (x/(c**3*sqrt(1 - d**2*x**2/c**2)), True)) + B*Piecewise((1/(d**2*sqrt (c**2 - d**2*x**2)), Ne(d, 0)), (x**2/(2*(c**2)**(3/2)), True)) + C*Piecew ise((I*acosh(d*x/c)/d**3 - I*x/(c*d**2*sqrt(-1 + d**2*x**2/c**2)), Abs(d** 2*x**2/c**2) > 1), (-asin(d*x/c)/d**3 + x/(c*d**2*sqrt(1 - d**2*x**2/c**2) ), True)) + D*Piecewise((2*c**2/(d**4*sqrt(c**2 - d**2*x**2)) - x**2/(d**2 *sqrt(c**2 - d**2*x**2)), Ne(d, 0)), (x**4/(4*(c**2)**(3/2)), True))
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {D x^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} + \frac {A x}{\sqrt {-d^{2} x^{2} + c^{2}} c^{2}} + \frac {C x}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} - \frac {C \arcsin \left (\frac {d x}{c}\right )}{d^{3}} + \frac {2 \, D c^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} d^{4}} + \frac {B}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(-d^2*x^2+c^2)^(3/2),x, algorithm="maxima")
Output:
-D*x^2/(sqrt(-d^2*x^2 + c^2)*d^2) + A*x/(sqrt(-d^2*x^2 + c^2)*c^2) + C*x/( sqrt(-d^2*x^2 + c^2)*d^2) - C*arcsin(d*x/c)/d^3 + 2*D*c^2/(sqrt(-d^2*x^2 + c^2)*d^4) + B/(sqrt(-d^2*x^2 + c^2)*d^2)
Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {C \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{d^{2} {\left | d \right |}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} {\left (x {\left (\frac {D x}{d^{2}} - \frac {C c^{2} d^{4} + A d^{6}}{c^{2} d^{6}}\right )} - \frac {2 \, D c^{4} d^{2} + B c^{2} d^{4}}{c^{2} d^{6}}\right )}}{d^{2} x^{2} - c^{2}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(-d^2*x^2+c^2)^(3/2),x, algorithm="giac")
Output:
-C*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^2*abs(d)) + sqrt(-d^2*x^2 + c^2)*(x*(D*x /d^2 - (C*c^2*d^4 + A*d^6)/(c^2*d^6)) - (2*D*c^4*d^2 + B*c^2*d^4)/(c^2*d^6 ))/(d^2*x^2 - c^2)
Time = 9.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {C\,\ln \left (x\,\sqrt {-d^2}+\sqrt {c^2-d^2\,x^2}\right )}{{\left (-d^2\right )}^{3/2}}+\frac {B}{d^2\,\sqrt {c^2-d^2\,x^2}}+\frac {A\,x}{c^2\,\sqrt {c^2-d^2\,x^2}}+\frac {C\,x}{d^2\,\sqrt {c^2-d^2\,x^2}}+\frac {\left (2\,c^2-d^2\,x^2\right )\,D}{d^4\,\sqrt {c^2-d^2\,x^2}} \] Input:
int((A + B*x + C*x^2 + x^3*D)/(c^2 - d^2*x^2)^(3/2),x)
Output:
(C*log(x*(-d^2)^(1/2) + (c^2 - d^2*x^2)^(1/2)))/(-d^2)^(3/2) + B/(d^2*(c^2 - d^2*x^2)^(1/2)) + (A*x)/(c^2*(c^2 - d^2*x^2)^(1/2)) + (C*x)/(d^2*(c^2 - d^2*x^2)^(1/2)) + ((2*c^2 - d^2*x^2)*D)/(d^4*(c^2 - d^2*x^2)^(1/2))
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x+C x^2+D x^3}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {-\sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) c^{3}+a \,d^{3} x +b \,c^{2} d +2 c^{4}+c^{3} d x -c^{2} d^{2} x^{2}}{\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3}} \] Input:
int((D*x^3+C*x^2+B*x+A)/(-d^2*x^2+c^2)^(3/2),x)
Output:
( - sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**3 + a*d**3*x + b*c**2*d + 2*c* *4 + c**3*d*x - c**2*d**2*x**2)/(sqrt(c**2 - d**2*x**2)*c**2*d**3)