3.3.42 \(\int x^3 (d+e x) (d^2-e^2 x^2)^p \, dx\) [242]

3.3.42.1 Optimal result

Integrand size = 23, antiderivative size = 120 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=-\frac {d^3 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (1+p)}+\frac {d \left (d^2-e^2 x^2\right )^{2+p}}{2 e^4 (2+p)}+\frac {1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right ) \]

output

-1/2*d^3*(-e^2*x^2+d^2)^(p+1)/e^4/(p+1)+1/2*d*(-e^2*x^2+d^2)^(2+p)/e^4/(2+ 
p)+1/5*e*x^5*(-e^2*x^2+d^2)^p*hypergeom([5/2, -p],[7/2],e^2*x^2/d^2)/((1-e 
^2*x^2/d^2)^p)
 

3.3.42.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {5 d \left (d^2-e^2 x^2\right ) \left (d^2+e^2 (1+p) x^2\right )}{(1+p) (2+p)}+2 e^5 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )\right )}{10 e^4} \]

input

Integrate[x^3*(d + e*x)*(d^2 - e^2*x^2)^p,x]
 

output

((d^2 - e^2*x^2)^p*((-5*d*(d^2 - e^2*x^2)*(d^2 + e^2*(1 + p)*x^2))/((1 + p 
)*(2 + p)) + (2*e^5*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 
 - (e^2*x^2)/d^2)^p))/(10*e^4)
 

3.3.42.3 Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {542, 243, 53, 279, 278, 2009}

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.

\(\displaystyle \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx\)

\(\Big \downarrow \) 542

\(\displaystyle e \int x^4 \left (d^2-e^2 x^2\right )^pdx+d \int x^3 \left (d^2-e^2 x^2\right )^pdx\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{2} d \int x^2 \left (d^2-e^2 x^2\right )^pdx^2+e \int x^4 \left (d^2-e^2 x^2\right )^pdx\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {1}{2} d \int \left (\frac {d^2 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^2}\right )dx^2+e \int x^4 \left (d^2-e^2 x^2\right )^pdx\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {1}{2} d \int \left (\frac {d^2 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^2}\right )dx^2+e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^pdx\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {1}{2} d \int \left (\frac {d^2 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^2}\right )dx^2+\frac {1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )+\frac {1}{2} d \left (\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^4 (p+2)}-\frac {d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}\right )\)

input

Int[x^3*(d + e*x)*(d^2 - e^2*x^2)^p,x]
 

output

(d*(-((d^2*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 + p))) + (d^2 - e^2*x^2)^(2 + 
p)/(e^4*(2 + p))))/2 + (e*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, -p, 
 7/2, (e^2*x^2)/d^2])/(5*(1 - (e^2*x^2)/d^2)^p)
 

3.3.42.3.1 Defintions of rubi rules used

rule 53

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] && 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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 542

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[c   Int[x^m*(a + b*x^2)^p, x], x] + Simp[d   Int[x^(m + 1)*(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] &&  !IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.3.42.4 Maple [F]

\[\int x^{3} \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]

input

int(x^3*(e*x+d)*(-e^2*x^2+d^2)^p,x)
 

output

int(x^3*(e*x+d)*(-e^2*x^2+d^2)^p,x)
 

3.3.42.5 Fricas [F]

\[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3} \,d x } \]

input

integrate(x^3*(e*x+d)*(-e^2*x^2+d^2)^p,x, algorithm="fricas")
 

output

integral((e*x^4 + d*x^3)*(-e^2*x^2 + d^2)^p, x)
 

3.3.42.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (97) = 194\).

Time = 1.59 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.18 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=d \left (\begin {cases} \frac {x^{4} \left (d^{2}\right )^{p}}{4} & \text {for}\: e = 0 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{4}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 e^{4}} - \frac {x^{2}}{2 e^{2}} & \text {for}\: p = -1 \\- \frac {d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac {d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text {otherwise} \end {cases}\right ) + \frac {d^{2 p} e x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - p \\ \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{5} \]

input

integrate(x**3*(e*x+d)*(-e**2*x**2+d**2)**p,x)
 

output

d*Piecewise((x**4*(d**2)**p/4, Eq(e, 0)), (-d**2*log(-d/e + x)/(-2*d**2*e* 
*4 + 2*e**6*x**2) - d**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**2/ 
(-2*d**2*e**4 + 2*e**6*x**2) + e**2*x**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e 
**6*x**2) + e**2*x**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2) 
), (-d**2*log(-d/e + x)/(2*e**4) - d**2*log(d/e + x)/(2*e**4) - x**2/(2*e* 
*2), Eq(p, -1)), (-d**4*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4* 
e**4) - d**2*e**2*p*x**2*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4 
*e**4) + e**4*p*x**4*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e** 
4) + e**4*x**4*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4), Tr 
ue)) + d**(2*p)*e*x**5*hyper((5/2, -p), (7/2,), e**2*x**2*exp_polar(2*I*pi 
)/d**2)/5
 

3.3.42.7 Maxima [F]

\[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3} \,d x } \]

input

integrate(x^3*(e*x+d)*(-e^2*x^2+d^2)^p,x, algorithm="maxima")
 

output

e*integrate(x^4*e^(p*log(e*x + d) + p*log(-e*x + d)), x) + 1/2*(e^4*(p + 1 
)*x^4 - d^2*e^2*p*x^2 - d^4)*(-e^2*x^2 + d^2)^p*d/((p^2 + 3*p + 2)*e^4)
 

3.3.42.8 Giac [F]

\[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3} \,d x } \]

input

integrate(x^3*(e*x+d)*(-e^2*x^2+d^2)^p,x, algorithm="giac")
 

output

integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^3, x)
 

3.3.42.9 Mupad [F(-1)]

Timed out. \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int x^3\,{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right ) \,d x \]

input

int(x^3*(d^2 - e^2*x^2)^p*(d + e*x),x)
 

output

int(x^3*(d^2 - e^2*x^2)^p*(d + e*x), x)