\(\int (d x)^m (c x^2)^p (a+b x)^n \, dx\) [998]

Optimal result

Integrand size = 20, antiderivative size = 68 \[ \int (d x)^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {(d x)^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{d (1+m+2 p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 68, 66} \[ \int (d x)^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,m+2 p+1,m+2 p+2,-\frac {b x}{a}\right )}{m+2 p+1} \]

[In]

Int[(d*x)^m*(c*x^2)^p*(a + b*x)^n,x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int (d x)^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x (d x)^m \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{1+m+2 p} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((d*x)^m*(c*x^2)^p*(a + b*x)^n,x)

[Out]

int((d*x)^m*(c*x^2)^p*(a + b*x)^n, x)