\(\int \frac {x^2 (a+b x^2)^p}{d+e x} \, dx\) [410]

Optimal result

Integrand size = 20, antiderivative size = 161 \[ \int \frac {x^2 \left (a+b x^2\right )^p}{d+e x} \, dx=\frac {\left (a+b x^2\right )^{1+p}}{2 b e (1+p)}+\frac {x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,1,\frac {5}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d}-\frac {d^2 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e \left (b d^2+a e^2\right ) (1+p)} \]

[Out]

1/2*(b*x^2+a)^(p+1)/b/e/(p+1)+1/3*x^3*(b*x^2+a)^p*AppellF1(3/2,1,-p,5/2,e^2*x^2/d^2,-b*x^2/a)/d/((1+b*x^2/a)^p
)-1/2*d^2*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],e^2*(b*x^2+a)/(a*e^2+b*d^2))/e/(a*e^2+b*d^2)/(p+1)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {973, 525, 524, 457, 81, 70} \[ \int \frac {x^2 \left (a+b x^2\right )^p}{d+e x} \, dx=\frac {x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,1,\frac {5}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d}-\frac {d^2 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e (p+1) \left (a e^2+b d^2\right )}+\frac {\left (a+b x^2\right )^{p+1}}{2 b e (p+1)} \]

[In]

Int[(x^2*(a + b*x^2)^p)/(d + e*x),x]

[Out]

(a + b*x^2)^(1 + p)/(2*b*e*(1 + p)) + (x^3*(a + b*x^2)^p*AppellF1[3/2, -p, 1, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2
])/(3*d*(1 + (b*x^2)/a)^p) - (d^2*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*
d^2 + a*e^2)])/(2*e*(b*d^2 + a*e^2)*(1 + p))

Rule 70

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

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 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 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

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

Rule 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

Rubi steps \begin{align*} \text {integral}& = d \int \frac {x^2 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx-e \int \frac {x^3 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx \\ & = -\left (\frac {1}{2} e \text {Subst}\left (\int \frac {x (a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )\right )+\left (d \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {x^2 \left (1+\frac {b x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx \\ & = \frac {\left (a+b x^2\right )^{1+p}}{2 b e (1+p)}+\frac {x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {3}{2};-p,1;\frac {5}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d}-\frac {d^2 \text {Subst}\left (\int \frac {(a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )}{2 e} \\ & = \frac {\left (a+b x^2\right )^{1+p}}{2 b e (1+p)}+\frac {x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {3}{2};-p,1;\frac {5}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{3 d}-\frac {d^2 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e \left (b d^2+a e^2\right ) (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.41 \[ \int \frac {x^2 \left (a+b x^2\right )^p}{d+e x} \, dx=\frac {\left (a+b x^2\right )^p \left (a e^2 p+b e^2 p x^2-a e^2 p \left (1+\frac {b x^2}{a}\right )^{-p}+b d^2 (1+p) \left (\frac {e \left (-\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )-2 b d e p (1+p) x \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{2 b e^3 p (1+p)} \]

[In]

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

[Out]

((a + b*x^2)^p*(a*e^2*p + b*e^2*p*x^2 - (a*e^2*p)/(1 + (b*x^2)/a)^p + (b*d^2*(1 + p)*AppellF1[-2*p, -p, -p, 1
- 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p
*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) - (2*b*d*e*p*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(
1 + (b*x^2)/a)^p))/(2*b*e^3*p*(1 + p))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((x^2*(a + b*x^2)^p)/(d + e*x),x)

[Out]

int((x^2*(a + b*x^2)^p)/(d + e*x), x)