3.10.46 \(\int (a+b x)^m (a^2-b^2 x^2)^2 \, dx\) [946]

Optimal. Leaf size=61 \[ \frac {4 a^2 (a+b x)^{3+m}}{b (3+m)}-\frac {4 a (a+b x)^{4+m}}{b (4+m)}+\frac {(a+b x)^{5+m}}{b (5+m)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45} \begin {gather*} \frac {4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac {4 a (a+b x)^{m+4}}{b (m+4)}+\frac {(a+b x)^{m+5}}{b (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(a^2 - b^2*x^2)^2,x]

[Out]

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

Rule 45

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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
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 )^2 \, dx &=\int (a-b x)^2 (a+b x)^{2+m} \, dx\\ &=\int \left (4 a^2 (a+b x)^{2+m}-4 a (a+b x)^{3+m}+(a+b x)^{4+m}\right ) \, dx\\ &=\frac {4 a^2 (a+b x)^{3+m}}{b (3+m)}-\frac {4 a (a+b x)^{4+m}}{b (4+m)}+\frac {(a+b x)^{5+m}}{b (5+m)}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 69, normalized size = 1.13 \begin {gather*} \frac {(a+b x)^{3+m} \left (a^2 \left (32+11 m+m^2\right )-2 a b \left (18+9 m+m^2\right ) x+b^2 \left (12+7 m+m^2\right ) x^2\right )}{b (3+m) (4+m) (5+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(3 + m)*(a^2*(32 + 11*m + m^2) - 2*a*b*(18 + 9*m + m^2)*x + b^2*(12 + 7*m + m^2)*x^2))/(b*(3 + m)*(
4 + m)*(5 + m))

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Maple [A]
time = 0.47, size = 94, normalized size = 1.54

method	result	size
gosper	\(\frac {\left (b x +a \right )^{3+m} \left (b^{2} m^{2} x^{2}-2 a b \,m^{2} x +7 b^{2} m \,x^{2}+a^{2} m^{2}-18 a b m x +12 b^{2} x^{2}+11 m \,a^{2}-36 a b x +32 a^{2}\right )}{b \left (m^{3}+12 m^{2}+47 m +60\right )}\)	\(94\)
risch	\(\frac {\left (b^{5} m^{2} x^{5}+m^{2} b^{4} a \,x^{4}+7 m \,b^{5} x^{5}-2 a^{2} b^{3} m^{2} x^{3}+3 a \,b^{4} m \,x^{4}+12 b^{5} x^{5}-2 a^{3} b^{2} m^{2} x^{2}-22 a^{2} b^{3} m \,x^{3}+a^{4} b \,m^{2} x -14 a^{3} b^{2} m \,x^{2}-40 a^{2} b^{3} x^{3}+a^{5} m^{2}+15 a^{4} b m x +11 a^{5} m +60 a^{4} b x +32 a^{5}\right ) \left (b x +a \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) b}\)	\(181\)
norman	\(\frac {b^{4} x^{5} {\mathrm e}^{m \ln \left (b x +a \right )}}{5+m}+\frac {a^{4} \left (m^{2}+15 m +60\right ) x \,{\mathrm e}^{m \ln \left (b x +a \right )}}{m^{3}+12 m^{2}+47 m +60}+\frac {a^{5} \left (m^{2}+11 m +32\right ) {\mathrm e}^{m \ln \left (b x +a \right )}}{b \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {a m \,b^{3} x^{4} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{2}+9 m +20}-\frac {2 a^{2} b^{2} \left (m^{2}+11 m +20\right ) x^{3} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{3}+12 m^{2}+47 m +60}-\frac {2 b m \,a^{3} \left (7+m \right ) x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{3}+12 m^{2}+47 m +60}\)	\(207\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(3+m)*(b^2*m^2*x^2-2*a*b*m^2*x+7*b^2*m*x^2+a^2*m^2-18*a*b*m*x+12*b^2*x^2+11*a^2*m-36*a*b*x+32*a^2)/b/(
m^3+12*m^2+47*m+60)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (61) = 122\).
time = 0.28, size = 233, normalized size = 3.82 \begin {gather*} \frac {{\left (b x + a\right )}^{m + 1} a^{4}}{b {\left (m + 1\right )}} - \frac {2 \, {\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} a^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a b^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{2} b^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b m x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\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)^2,x, algorithm="maxima")

[Out]

(b*x + a)^(m + 1)*a^4/(b*(m + 1)) - 2*((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*a^2/((m^3 + 6*m^2 + 11*m + 6)*b) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^5*x^5 + (m^4 + 6*m^3 + 11*
m^2 + 6*m)*a*b^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*a^2*b^3*x^3 + 12*(m^2 + m)*a^3*b^2*x^2 - 24*a^4*b*m*x + 24*a^5)*(
b*x + a)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (61) = 122\).
time = 2.55, size = 173, normalized size = 2.84 \begin {gather*} \frac {{\left (a^{5} m^{2} + 11 \, a^{5} m + {\left (b^{5} m^{2} + 7 \, b^{5} m + 12 \, b^{5}\right )} x^{5} + 32 \, a^{5} + {\left (a b^{4} m^{2} + 3 \, a b^{4} m\right )} x^{4} - 2 \, {\left (a^{2} b^{3} m^{2} + 11 \, a^{2} b^{3} m + 20 \, a^{2} b^{3}\right )} x^{3} - 2 \, {\left (a^{3} b^{2} m^{2} + 7 \, a^{3} b^{2} m\right )} x^{2} + {\left (a^{4} b m^{2} + 15 \, a^{4} b m + 60 \, a^{4} b\right )} x\right )} {\left (b x + a\right )}^{m}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, 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)^2,x, algorithm="fricas")

[Out]

(a^5*m^2 + 11*a^5*m + (b^5*m^2 + 7*b^5*m + 12*b^5)*x^5 + 32*a^5 + (a*b^4*m^2 + 3*a*b^4*m)*x^4 - 2*(a^2*b^3*m^2
+ 11*a^2*b^3*m + 20*a^2*b^3)*x^3 - 2*(a^3*b^2*m^2 + 7*a^3*b^2*m)*x^2 + (a^4*b*m^2 + 15*a^4*b*m + 60*a^4*b)*x)
*(b*x + a)^m/(b*m^3 + 12*b*m^2 + 47*b*m + 60*b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 876 vs. \(2 (48) = 96\).
time = 0.65, size = 876, normalized size = 14.36 \begin {gather*} \begin {cases} a^{4} a^{m} x & \text {for}\: b = 0 \\\frac {a^{2} \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {2 a^{2}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {2 a b x \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {4 a b x}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} & \text {for}\: m = -5 \\- \frac {4 a^{2} \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {7 a^{2}}{a b + b^{2} x} - \frac {4 a b x \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {2 a b x}{a b + b^{2} x} + \frac {b^{2} x^{2}}{a b + b^{2} x} & \text {for}\: m = -4 \\\frac {4 a^{2} \log {\left (\frac {a}{b} + x \right )}}{b} - 3 a x + \frac {b x^{2}}{2} & \text {for}\: m = -3 \\\frac {a^{5} m^{2} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {11 a^{5} m \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {32 a^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {a^{4} b m^{2} x \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {15 a^{4} b m x \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {60 a^{4} b x \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {2 a^{3} b^{2} m^{2} x^{2} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {14 a^{3} b^{2} m x^{2} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {2 a^{2} b^{3} m^{2} x^{3} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {22 a^{2} b^{3} m x^{3} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {40 a^{2} b^{3} x^{3} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {a b^{4} m^{2} x^{4} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {3 a b^{4} m x^{4} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {b^{5} m^{2} x^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {7 b^{5} m x^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {12 b^{5} x^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 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)**2,x)

[Out]

Piecewise((a**4*a**m*x, Eq(b, 0)), (a**2*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2) + 2*a**2/(a**2*b + 2*a
*b**2*x + b**3*x**2) + 2*a*b*x*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2) + 4*a*b*x/(a**2*b + 2*a*b**2*x +
b**3*x**2) + b**2*x**2*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2), Eq(m, -5)), (-4*a**2*log(a/b + x)/(a*b
+ b**2*x) - 7*a**2/(a*b + b**2*x) - 4*a*b*x*log(a/b + x)/(a*b + b**2*x) - 2*a*b*x/(a*b + b**2*x) + b**2*x**2/
(a*b + b**2*x), Eq(m, -4)), (4*a**2*log(a/b + x)/b - 3*a*x + b*x**2/2, Eq(m, -3)), (a**5*m**2*(a + b*x)**m/(b*
m**3 + 12*b*m**2 + 47*b*m + 60*b) + 11*a**5*m*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 32*a**5*(a +
b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + a**4*b*m**2*x*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b
) + 15*a**4*b*m*x*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 60*a**4*b*x*(a + b*x)**m/(b*m**3 + 12*b*
m**2 + 47*b*m + 60*b) - 2*a**3*b**2*m**2*x**2*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) - 14*a**3*b**2
*m*x**2*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) - 2*a**2*b**3*m**2*x**3*(a + b*x)**m/(b*m**3 + 12*b*
m**2 + 47*b*m + 60*b) - 22*a**2*b**3*m*x**3*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) - 40*a**2*b**3*x
**3*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + a*b**4*m**2*x**4*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47
*b*m + 60*b) + 3*a*b**4*m*x**4*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + b**5*m**2*x**5*(a + b*x)**m
/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 7*b**5*m*x**5*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b) + 12*b
**5*x**5*(a + b*x)**m/(b*m**3 + 12*b*m**2 + 47*b*m + 60*b), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (61) = 122\).
time = 1.58, size = 288, normalized size = 4.72 \begin {gather*} \frac {{\left (b x + a\right )}^{m} b^{5} m^{2} x^{5} + {\left (b x + a\right )}^{m} a b^{4} m^{2} x^{4} + 7 \, {\left (b x + a\right )}^{m} b^{5} m x^{5} - 2 \, {\left (b x + a\right )}^{m} a^{2} b^{3} m^{2} x^{3} + 3 \, {\left (b x + a\right )}^{m} a b^{4} m x^{4} + 12 \, {\left (b x + a\right )}^{m} b^{5} x^{5} - 2 \, {\left (b x + a\right )}^{m} a^{3} b^{2} m^{2} x^{2} - 22 \, {\left (b x + a\right )}^{m} a^{2} b^{3} m x^{3} + {\left (b x + a\right )}^{m} a^{4} b m^{2} x - 14 \, {\left (b x + a\right )}^{m} a^{3} b^{2} m x^{2} - 40 \, {\left (b x + a\right )}^{m} a^{2} b^{3} x^{3} + {\left (b x + a\right )}^{m} a^{5} m^{2} + 15 \, {\left (b x + a\right )}^{m} a^{4} b m x + 11 \, {\left (b x + a\right )}^{m} a^{5} m + 60 \, {\left (b x + a\right )}^{m} a^{4} b x + 32 \, {\left (b x + a\right )}^{m} a^{5}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, 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)^2,x, algorithm="giac")

[Out]

((b*x + a)^m*b^5*m^2*x^5 + (b*x + a)^m*a*b^4*m^2*x^4 + 7*(b*x + a)^m*b^5*m*x^5 - 2*(b*x + a)^m*a^2*b^3*m^2*x^3
+ 3*(b*x + a)^m*a*b^4*m*x^4 + 12*(b*x + a)^m*b^5*x^5 - 2*(b*x + a)^m*a^3*b^2*m^2*x^2 - 22*(b*x + a)^m*a^2*b^3
*m*x^3 + (b*x + a)^m*a^4*b*m^2*x - 14*(b*x + a)^m*a^3*b^2*m*x^2 - 40*(b*x + a)^m*a^2*b^3*x^3 + (b*x + a)^m*a^5
*m^2 + 15*(b*x + a)^m*a^4*b*m*x + 11*(b*x + a)^m*a^5*m + 60*(b*x + a)^m*a^4*b*x + 32*(b*x + a)^m*a^5)/(b*m^3 +
12*b*m^2 + 47*b*m + 60*b)

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Mupad [B]
time = 0.55, size = 186, normalized size = 3.05 \begin {gather*} {\left (a+b\,x\right )}^m\,\left (\frac {a^4\,x\,\left (m^2+15\,m+60\right )}{m^3+12\,m^2+47\,m+60}+\frac {a^5\,\left (m^2+11\,m+32\right )}{b\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {b^4\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}-\frac {2\,a^2\,b^2\,x^3\,\left (m^2+11\,m+20\right )}{m^3+12\,m^2+47\,m+60}+\frac {a\,b^3\,m\,x^4\,\left (m+3\right )}{m^3+12\,m^2+47\,m+60}-\frac {2\,a^3\,b\,m\,x^2\,\left (m+7\right )}{m^3+12\,m^2+47\,m+60}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 - b^2*x^2)^2*(a + b*x)^m,x)

[Out]

(a + b*x)^m*((a^4*x*(15*m + m^2 + 60))/(47*m + 12*m^2 + m^3 + 60) + (a^5*(11*m + m^2 + 32))/(b*(47*m + 12*m^2
+ m^3 + 60)) + (b^4*x^5*(7*m + m^2 + 12))/(47*m + 12*m^2 + m^3 + 60) - (2*a^2*b^2*x^3*(11*m + m^2 + 20))/(47*m
+ 12*m^2 + m^3 + 60) + (a*b^3*m*x^4*(m + 3))/(47*m + 12*m^2 + m^3 + 60) - (2*a^3*b*m*x^2*(m + 7))/(47*m + 12*
m^2 + m^3 + 60))

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