3.960 \(\int (a+b x)^n (c+d x)^p \, dx\)

Optimal. Leaf size=61 \[ -\frac{(a+b x)^{n+1} (c+d x)^{p+1} \, _2F_1\left (1,n+p+2;p+2;\frac{b (c+d x)}{b c-a d}\right )}{(p+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0228391, antiderivative size = 74, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {70, 69} \[ \frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x)^p,x]

[Out]

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

Rule 70

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

Rule 69

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

Rubi steps

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

Mathematica [A]  time = 0.0178497, size = 73, normalized size = 1.2 \[ \frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;\frac{d (a+b x)}{a d-b c}\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n*(c + d*x)^p,x]

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n*(d*x + c)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^n*(d*x + c)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n*(d*x + c)^p, x)