3.1847 \(\int (a+b x)^m (c+d x)^2 \, dx\)

Optimal. Leaf size=78 \[ \frac {(b c-a d)^2 (a+b x)^{m+1}}{b^3 (m+1)}+\frac {2 d (b c-a d) (a+b x)^{m+2}}{b^3 (m+2)}+\frac {d^2 (a+b x)^{m+3}}{b^3 (m+3)} \]

[Out]

(-a*d+b*c)^2*(b*x+a)^(1+m)/b^3/(1+m)+2*d*(-a*d+b*c)*(b*x+a)^(2+m)/b^3/(2+m)+d^2*(b*x+a)^(3+m)/b^3/(3+m)

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Rubi [A]  time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {(b c-a d)^2 (a+b x)^{m+1}}{b^3 (m+1)}+\frac {2 d (b c-a d) (a+b x)^{m+2}}{b^3 (m+2)}+\frac {d^2 (a+b x)^{m+3}}{b^3 (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^2,x]

[Out]

((b*c - a*d)^2*(a + b*x)^(1 + m))/(b^3*(1 + m)) + (2*d*(b*c - a*d)*(a + b*x)^(2 + m))/(b^3*(2 + m)) + (d^2*(a
+ b*x)^(3 + m))/(b^3*(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])

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 67, normalized size = 0.86 \[ \frac {(a+b x)^{m+1} \left (\frac {2 d (a+b x) (b c-a d)}{m+2}+\frac {(b c-a d)^2}{m+1}+\frac {d^2 (a+b x)^2}{m+3}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^2,x]

[Out]

((a + b*x)^(1 + m)*((b*c - a*d)^2/(1 + m) + (2*d*(b*c - a*d)*(a + b*x))/(2 + m) + (d^2*(a + b*x)^2)/(3 + m)))/
b^3

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fricas [B]  time = 0.94, size = 235, normalized size = 3.01 \[ \frac {{\left (a b^{2} c^{2} m^{2} + 6 \, a b^{2} c^{2} - 6 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (b^{3} d^{2} m^{2} + 3 \, b^{3} d^{2} m + 2 \, b^{3} d^{2}\right )} x^{3} + {\left (6 \, b^{3} c d + {\left (2 \, b^{3} c d + a b^{2} d^{2}\right )} m^{2} + {\left (8 \, b^{3} c d + a b^{2} d^{2}\right )} m\right )} x^{2} + {\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d\right )} m + {\left (6 \, b^{3} c^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d\right )} m^{2} + {\left (5 \, b^{3} c^{2} + 6 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m}}{b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^2*c^2*m^2 + 6*a*b^2*c^2 - 6*a^2*b*c*d + 2*a^3*d^2 + (b^3*d^2*m^2 + 3*b^3*d^2*m + 2*b^3*d^2)*x^3 + (6*b^3*
c*d + (2*b^3*c*d + a*b^2*d^2)*m^2 + (8*b^3*c*d + a*b^2*d^2)*m)*x^2 + (5*a*b^2*c^2 - 2*a^2*b*c*d)*m + (6*b^3*c^
2 + (b^3*c^2 + 2*a*b^2*c*d)*m^2 + (5*b^3*c^2 + 6*a*b^2*c*d - 2*a^2*b*d^2)*m)*x)*(b*x + a)^m/(b^3*m^3 + 6*b^3*m
^2 + 11*b^3*m + 6*b^3)

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giac [B]  time = 0.97, size = 385, normalized size = 4.94 \[ \frac {{\left (b x + a\right )}^{m} b^{3} d^{2} m^{2} x^{3} + 2 \, {\left (b x + a\right )}^{m} b^{3} c d m^{2} x^{2} + {\left (b x + a\right )}^{m} a b^{2} d^{2} m^{2} x^{2} + 3 \, {\left (b x + a\right )}^{m} b^{3} d^{2} m x^{3} + {\left (b x + a\right )}^{m} b^{3} c^{2} m^{2} x + 2 \, {\left (b x + a\right )}^{m} a b^{2} c d m^{2} x + 8 \, {\left (b x + a\right )}^{m} b^{3} c d m x^{2} + {\left (b x + a\right )}^{m} a b^{2} d^{2} m x^{2} + 2 \, {\left (b x + a\right )}^{m} b^{3} d^{2} x^{3} + {\left (b x + a\right )}^{m} a b^{2} c^{2} m^{2} + 5 \, {\left (b x + a\right )}^{m} b^{3} c^{2} m x + 6 \, {\left (b x + a\right )}^{m} a b^{2} c d m x - 2 \, {\left (b x + a\right )}^{m} a^{2} b d^{2} m x + 6 \, {\left (b x + a\right )}^{m} b^{3} c d x^{2} + 5 \, {\left (b x + a\right )}^{m} a b^{2} c^{2} m - 2 \, {\left (b x + a\right )}^{m} a^{2} b c d m + 6 \, {\left (b x + a\right )}^{m} b^{3} c^{2} x + 6 \, {\left (b x + a\right )}^{m} a b^{2} c^{2} - 6 \, {\left (b x + a\right )}^{m} a^{2} b c d + 2 \, {\left (b x + a\right )}^{m} a^{3} d^{2}}{b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^m*b^3*d^2*m^2*x^3 + 2*(b*x + a)^m*b^3*c*d*m^2*x^2 + (b*x + a)^m*a*b^2*d^2*m^2*x^2 + 3*(b*x + a)^m*b
^3*d^2*m*x^3 + (b*x + a)^m*b^3*c^2*m^2*x + 2*(b*x + a)^m*a*b^2*c*d*m^2*x + 8*(b*x + a)^m*b^3*c*d*m*x^2 + (b*x
+ a)^m*a*b^2*d^2*m*x^2 + 2*(b*x + a)^m*b^3*d^2*x^3 + (b*x + a)^m*a*b^2*c^2*m^2 + 5*(b*x + a)^m*b^3*c^2*m*x + 6
*(b*x + a)^m*a*b^2*c*d*m*x - 2*(b*x + a)^m*a^2*b*d^2*m*x + 6*(b*x + a)^m*b^3*c*d*x^2 + 5*(b*x + a)^m*a*b^2*c^2
*m - 2*(b*x + a)^m*a^2*b*c*d*m + 6*(b*x + a)^m*b^3*c^2*x + 6*(b*x + a)^m*a*b^2*c^2 - 6*(b*x + a)^m*a^2*b*c*d +
2*(b*x + a)^m*a^3*d^2)/(b^3*m^3 + 6*b^3*m^2 + 11*b^3*m + 6*b^3)

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maple [B]  time = 0.01, size = 159, normalized size = 2.04 \[ \frac {\left (b^{2} d^{2} m^{2} x^{2}+2 b^{2} c d \,m^{2} x +3 b^{2} d^{2} m \,x^{2}-2 a b \,d^{2} m x +b^{2} c^{2} m^{2}+8 b^{2} c d m x +2 b^{2} x^{2} d^{2}-2 a b c d m -2 a b \,d^{2} x +5 b^{2} c^{2} m +6 b^{2} c d x +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right ) \left (b x +a \right )^{m +1}}{\left (m^{3}+6 m^{2}+11 m +6\right ) b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(m+1)*(b^2*d^2*m^2*x^2+2*b^2*c*d*m^2*x+3*b^2*d^2*m*x^2-2*a*b*d^2*m*x+b^2*c^2*m^2+8*b^2*c*d*m*x+2*b^2*d
^2*x^2-2*a*b*c*d*m-2*a*b*d^2*x+5*b^2*c^2*m+6*b^2*c*d*x+2*a^2*d^2-6*a*b*c*d+6*b^2*c^2)/b^3/(m^3+6*m^2+11*m+6)

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maxima [A]  time = 1.15, size = 138, normalized size = 1.77 \[ \frac {2 \, {\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{m + 1} c^{2}}{b {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.66, size = 226, normalized size = 2.90 \[ {\left (a+b\,x\right )}^m\,\left (\frac {a\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d\,m-6\,a\,b\,c\,d+b^2\,c^2\,m^2+5\,b^2\,c^2\,m+6\,b^2\,c^2\right )}{b^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d^2\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {x\,\left (-2\,a^2\,b\,d^2\,m+2\,a\,b^2\,c\,d\,m^2+6\,a\,b^2\,c\,d\,m+b^3\,c^2\,m^2+5\,b^3\,c^2\,m+6\,b^3\,c^2\right )}{b^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,x^2\,\left (m+1\right )\,\left (6\,b\,c+a\,d\,m+2\,b\,c\,m\right )}{b\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^m*(c + d*x)^2,x)

[Out]

(a + b*x)^m*((a*(2*a^2*d^2 + 6*b^2*c^2 + 5*b^2*c^2*m + b^2*c^2*m^2 - 6*a*b*c*d - 2*a*b*c*d*m))/(b^3*(11*m + 6*
m^2 + m^3 + 6)) + (d^2*x^3*(3*m + m^2 + 2))/(11*m + 6*m^2 + m^3 + 6) + (x*(6*b^3*c^2 + 5*b^3*c^2*m + b^3*c^2*m
^2 - 2*a^2*b*d^2*m + 2*a*b^2*c*d*m^2 + 6*a*b^2*c*d*m))/(b^3*(11*m + 6*m^2 + m^3 + 6)) + (d*x^2*(m + 1)*(6*b*c
+ a*d*m + 2*b*c*m))/(b*(11*m + 6*m^2 + m^3 + 6)))

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sympy [A]  time = 2.14, size = 1506, normalized size = 19.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**2,x)

[Out]

Piecewise((a**m*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(b, 0)), (2*a**2*d**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**
4*x + 2*b**5*x**2) + 3*a**2*d**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 2*a*b*c*d/(2*a**2*b**3 + 4*a*b**4*
x + 2*b**5*x**2) + 4*a*b*d**2*x*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*d**2*x/(2*a**2*b
**3 + 4*a*b**4*x + 2*b**5*x**2) - b**2*c**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 4*b**2*c*d*x/(2*a**2*b*
*3 + 4*a*b**4*x + 2*b**5*x**2) + 2*b**2*d**2*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2), Eq(m,
-3)), (-2*a**2*d**2*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2*d**2/(a*b**3 + b**4*x) + 2*a*b*c*d*log(a/b + x)/(
a*b**3 + b**4*x) + 2*a*b*c*d/(a*b**3 + b**4*x) - 2*a*b*d**2*x*log(a/b + x)/(a*b**3 + b**4*x) - b**2*c**2/(a*b*
*3 + b**4*x) + 2*b**2*c*d*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*d**2*x**2/(a*b**3 + b**4*x), Eq(m, -2)), (a*
*2*d**2*log(a/b + x)/b**3 - 2*a*c*d*log(a/b + x)/b**2 - a*d**2*x/b**2 + c**2*log(a/b + x)/b + 2*c*d*x/b + d**2
*x**2/(2*b), Eq(m, -1)), (2*a**3*d**2*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) - 2*a**2*b*c
*d*m*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) - 6*a**2*b*c*d*(a + b*x)**m/(b**3*m**3 + 6*b*
*3*m**2 + 11*b**3*m + 6*b**3) - 2*a**2*b*d**2*m*x*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3)
+ a*b**2*c**2*m**2*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 5*a*b**2*c**2*m*(a + b*x)**m/
(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 6*a*b**2*c**2*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3
*m + 6*b**3) + 2*a*b**2*c*d*m**2*x*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 6*a*b**2*c*d*
m*x*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + a*b**2*d**2*m**2*x**2*(a + b*x)**m/(b**3*m**
3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + a*b**2*d**2*m*x**2*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m +
6*b**3) + b**3*c**2*m**2*x*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 5*b**3*c**2*m*x*(a +
b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 6*b**3*c**2*x*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2
+ 11*b**3*m + 6*b**3) + 2*b**3*c*d*m**2*x**2*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 8*b
**3*c*d*m*x**2*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 6*b**3*c*d*x**2*(a + b*x)**m/(b**
3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + b**3*d**2*m**2*x**3*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b*
*3*m + 6*b**3) + 3*b**3*d**2*m*x**3*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3) + 2*b**3*d**2*
x**3*(a + b*x)**m/(b**3*m**3 + 6*b**3*m**2 + 11*b**3*m + 6*b**3), True))

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