3.15.35 \(\int (d+e x)^m (a^2+2 a b x+b^2 x^2) \, dx\)

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

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Rubi [A]  time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \begin {gather*} \frac {(b d-a e)^2 (d+e x)^{m+1}}{e^3 (m+1)}-\frac {2 b (b d-a e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac {b^2 (d+e x)^{m+3}}{e^3 (m+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

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Mathematica [A]  time = 0.07, size = 67, normalized size = 0.86 \begin {gather*} \frac {(d+e x)^{m+1} \left (-\frac {2 b (d+e x) (b d-a e)}{m+2}+\frac {(b d-a e)^2}{m+1}+\frac {b^2 (d+e x)^2}{m+3}\right )}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 237, normalized size = 3.04 \begin {gather*} \frac {{\left (a^{2} d e^{2} m^{2} + 2 \, b^{2} d^{3} - 6 \, a b d^{2} e + 6 \, a^{2} d e^{2} + {\left (b^{2} e^{3} m^{2} + 3 \, b^{2} e^{3} m + 2 \, b^{2} e^{3}\right )} x^{3} + {\left (6 \, a b e^{3} + {\left (b^{2} d e^{2} + 2 \, a b e^{3}\right )} m^{2} + {\left (b^{2} d e^{2} + 8 \, a b e^{3}\right )} m\right )} x^{2} - {\left (2 \, a b d^{2} e - 5 \, a^{2} d e^{2}\right )} m + {\left (6 \, a^{2} e^{3} + {\left (2 \, a b d e^{2} + a^{2} e^{3}\right )} m^{2} - {\left (2 \, b^{2} d^{2} e - 6 \, a b d e^{2} - 5 \, a^{2} e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.18, size = 388, normalized size = 4.97 \begin {gather*} \frac {{\left (x e + d\right )}^{m} b^{2} m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} b^{2} d m^{2} x^{2} e^{2} + 2 \, {\left (x e + d\right )}^{m} a b m^{2} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} b^{2} m x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} a b d m^{2} x e^{2} + {\left (x e + d\right )}^{m} b^{2} d m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m x e + {\left (x e + d\right )}^{m} a^{2} m^{2} x e^{3} + 8 \, {\left (x e + d\right )}^{m} a b m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} b^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} a^{2} d m^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} a b d m x e^{2} - 2 \, {\left (x e + d\right )}^{m} a b d^{2} m e + 2 \, {\left (x e + d\right )}^{m} b^{2} d^{3} + 5 \, {\left (x e + d\right )}^{m} a^{2} m x e^{3} + 6 \, {\left (x e + d\right )}^{m} a b x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} a^{2} d m e^{2} - 6 \, {\left (x e + d\right )}^{m} a b d^{2} e + 6 \, {\left (x e + d\right )}^{m} a^{2} x e^{3} + 6 \, {\left (x e + d\right )}^{m} a^{2} d e^{2}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 159, normalized size = 2.04 \begin {gather*} \frac {\left (b^{2} e^{2} m^{2} x^{2}+2 a b \,e^{2} m^{2} x +3 b^{2} e^{2} m \,x^{2}+a^{2} e^{2} m^{2}+8 a b \,e^{2} m x -2 b^{2} d e m x +2 b^{2} x^{2} e^{2}+5 a^{2} e^{2} m -2 a b d e m +6 a b \,e^{2} x -2 b^{2} d e x +6 a^{2} e^{2}-6 a b d e +2 b^{2} d^{2}\right ) \left (e x +d \right )^{m +1}}{\left (m^{3}+6 m^{2}+11 m +6\right ) e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.11, size = 138, normalized size = 1.77 \begin {gather*} \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{2}}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.72, size = 226, normalized size = 2.90 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {b^2\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {d\,\left (a^2\,e^2\,m^2+5\,a^2\,e^2\,m+6\,a^2\,e^2-2\,a\,b\,d\,e\,m-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {x\,\left (a^2\,e^3\,m^2+5\,a^2\,e^3\,m+6\,a^2\,e^3+2\,a\,b\,d\,e^2\,m^2+6\,a\,b\,d\,e^2\,m-2\,b^2\,d^2\,e\,m\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {b\,x^2\,\left (m+1\right )\,\left (6\,a\,e+2\,a\,e\,m+b\,d\,m\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.07, size = 1506, normalized size = 19.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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