\(\int \frac {x (d+e x)^n}{a+c x^2} \, dx\) [366]

Optimal result

Integrand size = 18, antiderivative size = 163 \[ \int \frac {x (d+e x)^n}{a+c x^2} \, dx=-\frac {(d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 \sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {(d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {c} \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)} \]

[Out]

-1/2*(e*x+d)^(1+n)*hypergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2)))/(1+n)/c^(1/2)/(-e*(-a)^
(1/2)+d*c^(1/2))-1/2*(e*x+d)^(1+n)*hypergeom([1, 1+n],[2+n],(e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))/(1+n)/c^
(1/2)/(e*(-a)^(1/2)+d*c^(1/2))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {845, 70} \[ \int \frac {x (d+e x)^n}{a+c x^2} \, dx=-\frac {(d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 \sqrt {c} (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {(d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 \sqrt {c} (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]

[In]

Int[(x*(d + e*x)^n)/(a + c*x^2),x]

[Out]

-1/2*((d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqr
t[c]*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - ((d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e
*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*Sqrt[c]*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))

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 845

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !Ration
alQ[m]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int \frac {x (d+e x)^n}{a+c x^2} \, dx=-\frac {(d+e x)^{1+n} \left (\left (\sqrt {c} d+\sqrt {-a} e\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )+\left (\sqrt {c} d-\sqrt {-a} e\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )\right )}{2 \sqrt {c} \left (c d^2+a e^2\right ) (1+n)} \]

[In]

Integrate[(x*(d + e*x)^n)/(a + c*x^2),x]

[Out]

-1/2*((d + e*x)^(1 + n)*((Sqrt[c]*d + Sqrt[-a]*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt
[c]*d - Sqrt[-a]*e)] + (Sqrt[c]*d - Sqrt[-a]*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c
]*d + Sqrt[-a]*e)]))/(Sqrt[c]*(c*d^2 + a*e^2)*(1 + n))

Maple [F]

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

[In]

int(x*(e*x+d)^n/(c*x^2+a),x)

[Out]

int(x*(e*x+d)^n/(c*x^2+a),x)

Fricas [F]

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

[In]

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

[Out]

integral((e*x + d)^n*x/(c*x^2 + a), x)

Sympy [F]

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

[In]

integrate(x*(e*x+d)**n/(c*x**2+a),x)

[Out]

Integral(x*(d + e*x)**n/(a + c*x**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((e*x + d)^n*x/(c*x^2 + a), x)

Giac [F]

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

[In]

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

[Out]

integrate((e*x + d)^n*x/(c*x^2 + a), x)

Mupad [F(-1)]

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

[In]

int((x*(d + e*x)^n)/(a + c*x^2),x)

[Out]

int((x*(d + e*x)^n)/(a + c*x^2), x)