\(\int \frac {(a+b (c+d x)^2)^p}{c+d x} \, dx\) [2852]

Optimal result

Integrand size = 21, antiderivative size = 52 \[ \int \frac {\left (a+b (c+d x)^2\right )^p}{c+d x} \, dx=-\frac {\left (a+b (c+d x)^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c+d x)^2}{a}\right )}{2 a d (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {379, 272, 67} \[ \int \frac {\left (a+b (c+d x)^2\right )^p}{c+d x} \, dx=-\frac {\left (a+b (c+d x)^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]

[In]

Int[(a + b*(c + d*x)^2)^p/(c + d*x),x]

[Out]

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

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 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 379

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b (c+d x)^2\right )^p}{c+d x} \, dx=-\frac {\left (a+b (c+d x)^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c+d x)^2}{a}\right )}{2 a d (1+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p/(c + d*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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