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

Optimal. Leaf size=68 \[ \frac{\left (c x^2\right )^p (d x)^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{d (m+2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0219284, antiderivative size = 64, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 20, 66, 64} \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

Antiderivative was successfully verified.

[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] :> 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])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), 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])))

Rubi steps

\begin{align*} \int (d x)^m \left (c x^2\right )^p (a+b x)^n \, dx &=\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]  time = 0.0084658, size = 64, normalized size = 0.94 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{n} \left (d x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p} \left (d x\right )^{m} \left (a + b x\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)