Integrand size = 26, antiderivative size = 90 \[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=-\frac {(d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^{1+p}}{2 d e (2+p)}-\frac {(d+e x)^{-2 (1+p)} \left (d^2-e^2 x^2\right )^{1+p}}{4 d^2 e (1+p) (2+p)} \] Output:
-1/2*(e*x+d)^(-3-2*p)*(-e^2*x^2+d^2)^(p+1)/d/e/(2+p)-1/4*(-e^2*x^2+d^2)^(p +1)/d^2/e/(p+1)/(2+p)/((e*x+d)^(2*p+2))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=-\frac {2^{-3-p} (d-e x) (d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^p \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1+p,3+p,2+p,\frac {d-e x}{2 d}\right )}{d^3 e (1+p)} \] Input:
Integrate[(d + e*x)^(-3 - 2*p)*(d^2 - e^2*x^2)^p,x]
Output:
-((2^(-3 - p)*(d - e*x)*(1 + (e*x)/d)^p*(d^2 - e^2*x^2)^p*Hypergeometric2F 1[1 + p, 3 + p, 2 + p, (d - e*x)/(2*d)])/(d^3*e*(1 + p)*(d + e*x)^(2*p)))
Time = 0.42 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.62, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {474, 473, 55, 48}
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.
\(\displaystyle \int (d+e x)^{-2 p-3} \left (d^2-e^2 x^2\right )^p \, dx\) |
\(\Big \downarrow \) 474 |
\(\displaystyle \frac {(d+e x)^{-2 p} \left (\frac {e x}{d}+1\right )^{2 p} \int \left (\frac {e x}{d}+1\right )^{-2 p-3} \left (d^2-e^2 x^2\right )^pdx}{d^3}\) |
\(\Big \downarrow \) 473 |
\(\displaystyle \frac {(d+e x)^{-2 p} \left (\frac {e x}{d}+1\right )^{p-1} \left (d^2-d e x\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \int \left (\frac {e x}{d}+1\right )^{-p-3} \left (d^2-d e x\right )^pdx}{d^3}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {(d+e x)^{-2 p} \left (\frac {e x}{d}+1\right )^{p-1} \left (d^2-d e x\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \left (\frac {\int \left (\frac {e x}{d}+1\right )^{-p-2} \left (d^2-d e x\right )^pdx}{2 (p+2)}-\frac {\left (\frac {e x}{d}+1\right )^{-p-2} \left (d^2-d e x\right )^{p+1}}{2 d e (p+2)}\right )}{d^3}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(d+e x)^{-2 p} \left (\frac {e x}{d}+1\right )^{p-1} \left (d^2-d e x\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \left (-\frac {\left (d^2-d e x\right )^{p+1} \left (\frac {e x}{d}+1\right )^{-p-2}}{2 d e (p+2)}-\frac {\left (d^2-d e x\right )^{p+1} \left (\frac {e x}{d}+1\right )^{-p-1}}{4 d e (p+1) (p+2)}\right )}{d^3}\) |
Input:
Int[(d + e*x)^(-3 - 2*p)*(d^2 - e^2*x^2)^p,x]
Output:
((1 + (e*x)/d)^(-1 + p)*(d^2 - d*e*x)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*(-1 /2*((1 + (e*x)/d)^(-2 - p)*(d^2 - d*e*x)^(1 + p))/(d*e*(2 + p)) - ((1 + (e *x)/d)^(-1 - p)*(d^2 - d*e*x)^(1 + p))/(4*d*e*(1 + p)*(2 + p))))/(d^3*(d + e*x)^(2*p))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 1))) Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) && !Gt Q[a, 0] && !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(1 + d *(x/c))^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && !(IntegerQ[n] || GtQ[c, 0])
Time = 0.39 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{-2 p -2} \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (2 d p +e x +3 d \right )}{4 d^{2} e \left (p^{2}+3 p +2\right )}\) | \(61\) |
orering | \(-\frac {\left (e x +d \right ) \left (-e x +d \right ) \left (2 d p +e x +3 d \right ) \left (e x +d \right )^{-3-2 p} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{4 d^{2} \left (p^{2}+3 p +2\right ) e}\) | \(66\) |
parallelrisch | \(\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (e x +d \right )^{-3-2 p} e^{4}+2 x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (e x +d \right )^{-3-2 p} d \,e^{3} p +3 x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (e x +d \right )^{-3-2 p} d \,e^{3}-x \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (e x +d \right )^{-3-2 p} d^{2} e^{2}-2 \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (e x +d \right )^{-3-2 p} d^{3} e p -3 \left (-e^{2} x^{2}+d^{2}\right )^{p} \left (e x +d \right )^{-3-2 p} d^{3} e}{4 d^{2} \left (p^{2}+3 p +2\right ) e^{2}}\) | \(218\) |
Input:
int((e*x+d)^(-3-2*p)*(-e^2*x^2+d^2)^p,x,method=_RETURNVERBOSE)
Output:
-1/4/d^2/e*(-e*x+d)*(e*x+d)^(-2*p-2)/(p^2+3*p+2)*(-e^2*x^2+d^2)^p*(2*d*p+e *x+3*d)
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06 \[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=\frac {{\left (e^{3} x^{3} - 2 \, d^{3} p - d^{2} e x - 3 \, d^{3} + {\left (2 \, d e^{2} p + 3 \, d e^{2}\right )} x^{2}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{4 \, {\left (d^{2} e p^{2} + 3 \, d^{2} e p + 2 \, d^{2} e\right )}} \] Input:
integrate((e*x+d)^(-3-2*p)*(-e^2*x^2+d^2)^p,x, algorithm="fricas")
Output:
1/4*(e^3*x^3 - 2*d^3*p - d^2*e*x - 3*d^3 + (2*d*e^2*p + 3*d*e^2)*x^2)*(-e^ 2*x^2 + d^2)^p*(e*x + d)^(-2*p - 3)/(d^2*e*p^2 + 3*d^2*e*p + 2*d^2*e)
\[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=\int \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p} \left (d + e x\right )^{- 2 p - 3}\, dx \] Input:
integrate((e*x+d)**(-3-2*p)*(-e**2*x**2+d**2)**p,x)
Output:
Integral((-(-d + e*x)*(d + e*x))**p*(d + e*x)**(-2*p - 3), x)
\[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \] Input:
integrate((e*x+d)^(-3-2*p)*(-e^2*x^2+d^2)^p,x, algorithm="maxima")
Output:
integrate((-e^2*x^2 + d^2)^p*(e*x + d)^(-2*p - 3), x)
\[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \] Input:
integrate((e*x+d)^(-3-2*p)*(-e^2*x^2+d^2)^p,x, algorithm="giac")
Output:
integrate((-e^2*x^2 + d^2)^p*(e*x + d)^(-2*p - 3), x)
Time = 6.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.64 \[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=-{\left (d^2-e^2\,x^2\right )}^p\,\left (\frac {x}{4\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}+\frac {d\,\left (2\,p+3\right )}{4\,e\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}-\frac {e^2\,x^3}{4\,d^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}-\frac {e\,x^2\,\left (2\,p+3\right )}{4\,d\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}\right ) \] Input:
int((d^2 - e^2*x^2)^p/(d + e*x)^(2*p + 3),x)
Output:
-(d^2 - e^2*x^2)^p*(x/(4*(d + e*x)^(2*p + 3)*(3*p + p^2 + 2)) + (d*(2*p + 3))/(4*e*(d + e*x)^(2*p + 3)*(3*p + p^2 + 2)) - (e^2*x^3)/(4*d^2*(d + e*x) ^(2*p + 3)*(3*p + p^2 + 2)) - (e*x^2*(2*p + 3))/(4*d*(d + e*x)^(2*p + 3)*( 3*p + p^2 + 2)))
\[ \int (d+e x)^{-3-2 p} \left (d^2-e^2 x^2\right )^p \, dx=\int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2 p} d^{3}+3 \left (e x +d \right )^{2 p} d^{2} e x +3 \left (e x +d \right )^{2 p} d \,e^{2} x^{2}+\left (e x +d \right )^{2 p} e^{3} x^{3}}d x \] Input:
int((e*x+d)^(-3-2*p)*(-e^2*x^2+d^2)^p,x)
Output:
int((d**2 - e**2*x**2)**p/((d + e*x)**(2*p)*d**3 + 3*(d + e*x)**(2*p)*d**2 *e*x + 3*(d + e*x)**(2*p)*d*e**2*x**2 + (d + e*x)**(2*p)*e**3*x**3),x)