3.959 \(\int (a+b x)^m (a^2-b^2 x^2)^p \, dx\)

Optimal. Leaf size=63 \[ \frac {(a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,m+2 p+2;m+p+2;\frac {a+b x}{2 a}\right )}{2 a b (m+p+1)} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {680, 678, 69} \[ -\frac {2^{m+p} (a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \left (\frac {b x}{a}+1\right )^{-m-p-1} \, _2F_1\left (-m-p,p+1;p+2;\frac {a-b x}{2 a}\right )}{a b (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(a^2 - b^2*x^2)^p,x]

[Out]

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

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]))

Rule 678

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^(m - 1)*(a + c*x^2)^(p + 1))/((1
+ (e*x)/d)^(p + 1)*(a/d + (c*x)/e)^(p + 1)), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a,
c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (
IntegerQ[3*p] || IntegerQ[4*p]))

Rule 680

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^IntPart[m]*(d + e*x)^FracPart[m]
)/(1 + (e*x)/d)^FracPart[m], Int[(1 + (e*x)/d)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && EqQ[c*d
^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] || GtQ[d, 0])

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 85, normalized size = 1.35 \[ \frac {2^{m+p} (b x-a) (a+b x)^m \left (a^2-b^2 x^2\right )^p \left (\frac {b x}{a}+1\right )^{-m-p} \, _2F_1\left (-m-p,p+1;p+2;\frac {a-b x}{2 a}\right )}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(a^2 - b^2*x^2)^p,x]

[Out]

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

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fricas [F]  time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b^{2} x^{2} + a^{2}\right )}^{p} {\left (b x + a\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-b^2*x^2 + a^2)^p*(b*x + a)^m, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-b^{2} x^{2} + a^{2}\right )}^{p} {\left (b x + a\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-b^2*x^2 + a^2)^p*(b*x + a)^m, x)

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maple [F]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{m} \left (-b^{2} x^{2}+a^{2}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(-b^2*x^2+a^2)^p,x)

[Out]

int((b*x+a)^m*(-b^2*x^2+a^2)^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-b^{2} x^{2} + a^{2}\right )}^{p} {\left (b x + a\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-b^2*x^2 + a^2)^p*(b*x + a)^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a^2-b^2\,x^2\right )}^p\,{\left (a+b\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 - b^2*x^2)^p*(a + b*x)^m,x)

[Out]

int((a^2 - b^2*x^2)^p*(a + b*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p} \left (a + b x\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(-b**2*x**2+a**2)**p,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**p*(a + b*x)**m, x)

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