\(\int \frac {(d+e x)^2 (a+b x^2)^p}{x} \, dx\) [396]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x} \, dx=\frac {e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+2 d e x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )-\frac {d^2 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )}{2 a (1+p)} \]

[Out]

1/2*e^2*(b*x^2+a)^(p+1)/b/(p+1)+2*d*e*x*(b*x^2+a)^p*hypergeom([1/2, -p],[3/2],-b*x^2/a)/((1+b*x^2/a)^p)-1/2*d^
2*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],1+b*x^2/a)/a/(p+1)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1666, 457, 81, 67, 12, 252, 251} \[ \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x} \, dx=-\frac {d^2 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{2 a (p+1)}+2 d e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )+\frac {e^2 \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]

[In]

Int[((d + e*x)^2*(a + b*x^2)^p)/x,x]

[Out]

(e^2*(a + b*x^2)^(1 + p))/(2*b*(1 + p)) + (2*d*e*x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)]
)/(1 + (b*x^2)/a)^p - (d^2*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a*(1 + p)
)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 251

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

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1666

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2] && IGtQ[m, -2] &&  !
IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \int 2 d e \left (a+b x^2\right )^p \, dx+\int \frac {\left (a+b x^2\right )^p \left (d^2+e^2 x^2\right )}{x} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^p \left (d^2+e^2 x\right )}{x} \, dx,x,x^2\right )+(2 d e) \int \left (a+b x^2\right )^p \, dx \\ & = \frac {e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {1}{2} d^2 \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^2\right )+\left (2 d e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx \\ & = \frac {e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+2 d e x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )-\frac {d^2 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x} \, dx=\frac {1}{2} \left (a+b x^2\right )^p \left (4 d e x \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )+\frac {\left (a+b x^2\right ) \left (a e^2-b d^2 \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )\right )}{a b (1+p)}\right ) \]

[In]

Integrate[((d + e*x)^2*(a + b*x^2)^p)/x,x]

[Out]

((a + b*x^2)^p*((4*d*e*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + ((a + b*x^2)*(a*e^
2 - b*d^2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a]))/(a*b*(1 + p))))/2

Maple [F]

\[\int \frac {\left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}}{x}d x\]

[In]

int((e*x+d)^2*(b*x^2+a)^p/x,x)

[Out]

int((e*x+d)^2*(b*x^2+a)^p/x,x)

Fricas [F]

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

[In]

integrate((e*x+d)^2*(b*x^2+a)^p/x,x, algorithm="fricas")

[Out]

integral((e^2*x^2 + 2*d*e*x + d^2)*(b*x^2 + a)^p/x, x)

Sympy [A] (verification not implemented)

Time = 4.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x} \, dx=2 a^{p} d e x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} - \frac {b^{p} d^{2} x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + e^{2} \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((e*x+d)**2*(b*x**2+a)**p/x,x)

[Out]

2*a**p*d*e*x*hyper((1/2, -p), (3/2,), b*x**2*exp_polar(I*pi)/a) - b**p*d**2*x**(2*p)*gamma(-p)*hyper((-p, -p),
(1 - p,), a*exp_polar(I*pi)/(b*x**2))/(2*gamma(1 - p)) + e**2*Piecewise((a**p*x**2/2, Eq(b, 0)), (Piecewise((
(a + b*x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(a + b*x**2), True))/(2*b), True))

Maxima [F]

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

[In]

integrate((e*x+d)^2*(b*x^2+a)^p/x,x, algorithm="maxima")

[Out]

integrate((e*x + d)^2*(b*x^2 + a)^p/x, x)

Giac [F]

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

[In]

integrate((e*x+d)^2*(b*x^2+a)^p/x,x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(b*x^2 + a)^p/x, x)

Mupad [F(-1)]

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

[In]

int(((a + b*x^2)^p*(d + e*x)^2)/x,x)

[Out]

int(((a + b*x^2)^p*(d + e*x)^2)/x, x)