3.19.45 \(\int (a+b x)^m (c+d x)^n \, dx\) [1845]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {72, 71} \begin {gather*} \frac {(a+b x)^{m+1} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^n,x]

[Out]

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

Rule 71

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

Rule 72

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])

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 73, normalized size = 1.20 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;\frac {d (a+b x)}{-b c+a d}\right )}{b (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^n,x]

[Out]

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

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^m*(d*x + c)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^m*(d*x + c)^n, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^m*(d*x + c)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^m*(c + d*x)^n,x)

[Out]

int((a + b*x)^m*(c + d*x)^n, x)

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