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]
((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))
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Rubi [A] time = 0.0318711, antiderivative size = 78, normalized size of antiderivative = 1.,
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*}
Mathematica [A] time = 0.073742, 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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Maple [B] time = 0.006, size = 159, normalized size = 2. \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+m} \left ({b}^{2}{d}^{2}{m}^{2}{x}^{2}+2\,{b}^{2}cd{m}^{2}x+3\,{b}^{2}{d}^{2}m{x}^{2}-2\,ab{d}^{2}mx+{b}^{2}{c}^{2}{m}^{2}+8\,{b}^{2}cdmx+2\,{b}^{2}{d}^{2}{x}^{2}-2\,abcdm-2\,ab{d}^{2}x+5\,{b}^{2}{c}^{2}m+6\,{b}^{2}cdx+2\,{a}^{2}{d}^{2}-6\,abcd+6\,{b}^{2}{c}^{2} \right ) }{{b}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \end{align*}
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)^(1+m)*(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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^m*(d*x+c)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.84051, size = 478, normalized size = 6.13 \begin{align*} \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}} \end{align*}
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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Sympy [A] time = 1.87896, size = 1504, normalized size = 19.28 \begin{align*} \text{result too large to display} \end{align*}
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**3*d**2*log(a/b + x)/(2*a**3*b**3 + 4*a**2*
b**4*x + 2*a*b**5*x**2) + a**3*d**2/(2*a**3*b**3 + 4*a**2*b**4*x + 2*a*b**5*x**2) + 4*a**2*b*d**2*x*log(a/b +
x)/(2*a**3*b**3 + 4*a**2*b**4*x + 2*a*b**5*x**2) - a*b**2*c**2/(2*a**3*b**3 + 4*a**2*b**4*x + 2*a*b**5*x**2) +
2*a*b**2*d**2*x**2*log(a/b + x)/(2*a**3*b**3 + 4*a**2*b**4*x + 2*a*b**5*x**2) - 2*a*b**2*d**2*x**2/(2*a**3*b*
*3 + 4*a**2*b**4*x + 2*a*b**5*x**2) + 2*b**3*c*d*x**2/(2*a**3*b**3 + 4*a**2*b**4*x + 2*a*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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Giac [B] time = 1.05417, size = 520, normalized size = 6.67 \begin{align*} \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}} \end{align*}
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)