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

Optimal. Leaf size=40 \[ \frac {2 a (a+b x)^{m+2}}{b (m+2)}-\frac {(a+b x)^{m+3}}{b (m+3)} \]

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {627, 43} \begin {gather*} \frac {2 a (a+b x)^{m+2}}{b (m+2)}-\frac {(a+b x)^{m+3}}{b (m+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(a^2 - b^2*x^2),x]

[Out]

(2*a*(a + b*x)^(2 + m))/(b*(2 + m)) - (a + b*x)^(3 + m)/(b*(3 + m))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 627

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

Rubi steps

\begin {align*} \int (a+b x)^m \left (a^2-b^2 x^2\right ) \, dx &=\int (a-b x) (a+b x)^{1+m} \, dx\\ &=\int \left (2 a (a+b x)^{1+m}-(a+b x)^{2+m}\right ) \, dx\\ &=\frac {2 a (a+b x)^{2+m}}{b (2+m)}-\frac {(a+b x)^{3+m}}{b (3+m)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 36, normalized size = 0.90 \begin {gather*} \frac {(a+b x)^{m+2} (a (m+4)-b (m+2) x)}{b (m+2) (m+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(2 + m)*(a*(4 + m) - b*(2 + m)*x))/(b*(2 + m)*(3 + m))

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^m \left (a^2-b^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(a + b*x)^m*(a^2 - b^2*x^2), x]

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fricas [A]  time = 0.41, size = 76, normalized size = 1.90 \begin {gather*} -\frac {{\left (a b^{2} m x^{2} - a^{3} m + {\left (b^{3} m + 2 \, b^{3}\right )} x^{3} - 4 \, a^{3} - {\left (a^{2} b m + 6 \, a^{2} b\right )} x\right )} {\left (b x + a\right )}^{m}}{b m^{2} + 5 \, b m + 6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b^2*m*x^2 - a^3*m + (b^3*m + 2*b^3)*x^3 - 4*a^3 - (a^2*b*m + 6*a^2*b)*x)*(b*x + a)^m/(b*m^2 + 5*b*m + 6*b)

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giac [B]  time = 0.19, size = 118, normalized size = 2.95 \begin {gather*} -\frac {{\left (b x + a\right )}^{m} b^{3} m x^{3} + {\left (b x + a\right )}^{m} a b^{2} m x^{2} + 2 \, {\left (b x + a\right )}^{m} b^{3} x^{3} - {\left (b x + a\right )}^{m} a^{2} b m x - {\left (b x + a\right )}^{m} a^{3} m - 6 \, {\left (b x + a\right )}^{m} a^{2} b x - 4 \, {\left (b x + a\right )}^{m} a^{3}}{b m^{2} + 5 \, b m + 6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b*x + a)^m*b^3*m*x^3 + (b*x + a)^m*a*b^2*m*x^2 + 2*(b*x + a)^m*b^3*x^3 - (b*x + a)^m*a^2*b*m*x - (b*x + a)^
m*a^3*m - 6*(b*x + a)^m*a^2*b*x - 4*(b*x + a)^m*a^3)/(b*m^2 + 5*b*m + 6*b)

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maple [A]  time = 0.05, size = 40, normalized size = 1.00 \begin {gather*} \frac {\left (-b m x +a m -2 b x +4 a \right ) \left (b x +a \right )^{m +2}}{\left (m^{2}+5 m +6\right ) b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(m+2)*(-b*m*x+a*m-2*b*x+4*a)/b/(m^2+5*m+6)

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maxima [B]  time = 1.35, size = 91, normalized size = 2.28 \begin {gather*} \frac {{\left (b x + a\right )}^{m + 1} a^{2}}{b {\left (m + 1\right )}} - \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)^(m + 1)*a^2/(b*(m + 1)) - ((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2*x^2 - 2*a^2*b*m*x + 2*a^3)*(b*x
+ a)^m/((m^3 + 6*m^2 + 11*m + 6)*b)

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mupad [B]  time = 0.46, size = 86, normalized size = 2.15 \begin {gather*} {\left (a+b\,x\right )}^m\,\left (\frac {a^3\,\left (m+4\right )}{b\,\left (m^2+5\,m+6\right )}-\frac {b^2\,x^3\,\left (m+2\right )}{m^2+5\,m+6}+\frac {a^2\,x\,\left (m+6\right )}{m^2+5\,m+6}-\frac {a\,b\,m\,x^2}{m^2+5\,m+6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 - b^2*x^2)*(a + b*x)^m,x)

[Out]

(a + b*x)^m*((a^3*(m + 4))/(b*(5*m + m^2 + 6)) - (b^2*x^3*(m + 2))/(5*m + m^2 + 6) + (a^2*x*(m + 6))/(5*m + m^
2 + 6) - (a*b*m*x^2)/(5*m + m^2 + 6))

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sympy [A]  time = 1.09, size = 267, normalized size = 6.68 \begin {gather*} \begin {cases} a^{2} a^{m} x & \text {for}\: b = 0 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {2 a}{a b + b^{2} x} - \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} & \text {for}\: m = -3 \\\frac {2 a \log {\left (\frac {a}{b} + x \right )}}{b} - x & \text {for}\: m = -2 \\\frac {a^{3} m \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac {4 a^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac {a^{2} b m x \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} + \frac {6 a^{2} b x \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac {a b^{2} m x^{2} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac {b^{3} m x^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} - \frac {2 b^{3} x^{3} \left (a + b x\right )^{m}}{b m^{2} + 5 b m + 6 b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*a**m*x, Eq(b, 0)), (-a*log(a/b + x)/(a*b + b**2*x) - 2*a/(a*b + b**2*x) - b*x*log(a/b + x)/(a*
b + b**2*x), Eq(m, -3)), (2*a*log(a/b + x)/b - x, Eq(m, -2)), (a**3*m*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) + 4*
a**3*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) + a**2*b*m*x*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) + 6*a**2*b*x*(a + b*
x)**m/(b*m**2 + 5*b*m + 6*b) - a*b**2*m*x**2*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b) - b**3*m*x**3*(a + b*x)**m/(b
*m**2 + 5*b*m + 6*b) - 2*b**3*x**3*(a + b*x)**m/(b*m**2 + 5*b*m + 6*b), True))

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