3.3.69 \(\int \frac {x^2 (d^2-e^2 x^2)^p}{d+e x} \, dx\) [269]
Optimal. Leaf size=119 \[ \frac {d^2 \left (d^2-e^2 x^2\right )^p}{2 e^3 p}-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e^3 (1+p)}+\frac {x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d} \]
[Out]
1/2*d^2*(-e^2*x^2+d^2)^p/e^3/p-1/2*(-e^2*x^2+d^2)^(1+p)/e^3/(1+p)+1/3*x^3*(-e^2*x^2+d^2)^p*hypergeom([3/2, 1-p
],[5/2],e^2*x^2/d^2)/d/((1-e^2*x^2/d^2)^p)
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Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {864, 778, 372,
371, 272, 45} \begin {gather*} \frac {x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d}+\frac {d^2 \left (d^2-e^2 x^2\right )^p}{2 e^3 p}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
(d^2*(d^2 - e^2*x^2)^p)/(2*e^3*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^3*(1 + p)) + (x^3*(d^2 - e^2*x^2)^p*Hypergeom
etric2F1[3/2, 1 - p, 5/2, (e^2*x^2)/d^2])/(3*d*(1 - (e^2*x^2)/d^2)^p)
Rule 45
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 272
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 372
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[
p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Rule 778
Int[(x_)^(m_.)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[f, Int[x^m*(a + c*x^2)^p, x]
, x] + Dist[g, Int[x^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] && !IntegerQ[2
*p]
Rule 864
Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + c*(x/e))*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))
Rubi steps
\begin {align*} \int \frac {x^2 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\int x^2 (d-e x) \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=d \int x^2 \left (d^2-e^2 x^2\right )^{-1+p} \, dx-e \int x^3 \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=-\left (\frac {1}{2} e \text {Subst}\left (\int x \left (d^2-e^2 x\right )^{-1+p} \, dx,x,x^2\right )\right )+\frac {\left (\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d}\\ &=\frac {x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d}-\frac {1}{2} e \text {Subst}\left (\int \left (\frac {d^2 \left (d^2-e^2 x\right )^{-1+p}}{e^2}-\frac {\left (d^2-e^2 x\right )^p}{e^2}\right ) \, dx,x,x^2\right )\\ &=\frac {d^2 \left (d^2-e^2 x^2\right )^p}{2 e^3 p}-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e^3 (1+p)}+\frac {x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},1-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 198, normalized size = 1.66 \begin {gather*} -\frac {\left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (\left (1+\frac {e x}{d}\right )^p \left (-e^2 x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^p+d^2 \left (-1+\left (1-\frac {e^2 x^2}{d^2}\right )^p\right )\right )+2 d e (1+p) x \left (1+\frac {e x}{d}\right )^p \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )+d (d-e x) \left (2-\frac {2 e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,1+p;2+p;\frac {d-e x}{2 d}\right )\right )}{2 e^3 (1+p)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
-1/2*((d^2 - e^2*x^2)^p*((1 + (e*x)/d)^p*(-(e^2*x^2*(1 - (e^2*x^2)/d^2)^p) + d^2*(-1 + (1 - (e^2*x^2)/d^2)^p))
+ 2*d*e*(1 + p)*x*(1 + (e*x)/d)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + d*(d - e*x)*(2 - (2*e^2*x^
2)/d^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(e^3*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2
)/d^2)^p)
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x)
[Out]
int(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x, algorithm="maxima")
[Out]
integrate((-x^2*e^2 + d^2)^p*x^2/(x*e + d), x)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x, algorithm="fricas")
[Out]
integral((-x^2*e^2 + d^2)^p*x^2/(x*e + d), x)
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Sympy [C] Result contains complex when optimal does not.
time = 12.62, size = 4124, normalized size = 34.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(-e**2*x**2+d**2)**p/(e*x+d),x)
[Out]
Piecewise((-0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*ga
mma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(1/2 -
p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(
2*p)*p*acoth(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p
)*gamma(p + 1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p
)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2) - 1)*gamma(1/2
- p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d*
*(2*p)*acoth(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p
)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) +
2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 -
p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(1/2 - p)*gamma(p + 1)/
(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(1/
2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**2*d**(2*p
)*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*
gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2*p)*p**2*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p,
-p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(
p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,),
d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**(2*p)*e**2*
p*x**2*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*
e**3*gamma(1/2 - p)*gamma(p + 1)), (Abs(e**2*x**2/d**2) > 1) & (Abs(d**2/(e**2*x**2)) > 1)), (-0**p*d**2*d**(2
*p)*p*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1
/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*g
amma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*p*acoth(d/(e*x))*gamma
(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 0**p*d**2
*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gam
ma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2) - 1)*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p
*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*acoth(d/(e*x))*gamma
(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 2*0**p*d*
d**(2*p)*e*p*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma
(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*ga
mma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*ga
mma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(1/2 - p)*gamma(p + 1)/(2*e**3
*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**2*d**(2*p)*(1 - e**2*x**2/d**2)**p*g
amma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2
*p)*p**2*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**2)
)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(
I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 -
p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**(2*p)*e**2*p*x**2*(1 - e**2*x**2/d**2)**p*gamma(p)*
gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)), Abs(d**2/(e**2*x**
2)) > 1), (-0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*ga
mma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(-d**2/(e**2*x**2) + 1)*gamma(1/2 -
p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**
(2*p)*p*atanh(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 -
p)*gamma(p + 1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 -
p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(-d**2/(e**2*x**2) + 1)*gamma(1/
2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*ga...
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x, algorithm="giac")
[Out]
integrate((-x^2*e^2 + d^2)^p*x^2/(x*e + d), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (d^2-e^2\,x^2\right )}^p}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2*(d^2 - e^2*x^2)^p)/(d + e*x),x)
[Out]
int((x^2*(d^2 - e^2*x^2)^p)/(d + e*x), x)
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