\(\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^p \, dx\) [1744]

Optimal result

Integrand size = 26, antiderivative size = 181 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(b d-a e)^3 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (1+2 p)}+\frac {3 e (b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (1+p)}+\frac {3 e^2 (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (3+2 p)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (2+p)} \]

[Out]

(-a*e+b*d)^3*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^p/b^4/(1+2*p)+3/2*e*(-a*e+b*d)^2*(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2)^p/
b^4/(p+1)+3*e^2*(-a*e+b*d)*(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2)^p/b^4/(3+2*p)+1/2*e^3*(b*x+a)^4*(b^2*x^2+2*a*b*x+a^
2)^p/b^4/(2+p)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {660, 45} \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {3 e^2 (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac {3 e (a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac {(a+b x) (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+1)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)} \]

[In]

Int[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((b*d - a*e)^3*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(1 + 2*p)) + (3*e*(b*d - a*e)^2*(a + b*x)^2*(a^2 +
2*a*b*x + b^2*x^2)^p)/(2*b^4*(1 + p)) + (3*e^2*(b*d - a*e)*(a + b*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(3 +
2*p)) + (e^3*(a + b*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(2 + p))

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 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^3 \, dx \\ & = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (\frac {(b d-a e)^3 \left (a b+b^2 x\right )^{2 p}}{b^3}+\frac {3 e (b d-a e)^2 \left (a b+b^2 x\right )^{1+2 p}}{b^4}+\frac {3 e^2 (b d-a e) \left (a b+b^2 x\right )^{2+2 p}}{b^5}+\frac {e^3 \left (a b+b^2 x\right )^{3+2 p}}{b^6}\right ) \, dx \\ & = \frac {(b d-a e)^3 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (1+2 p)}+\frac {3 e (b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (1+p)}+\frac {3 e^2 (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (3+2 p)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (2+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.59 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(a+b x) \left ((a+b x)^2\right )^p \left (\frac {2 (b d-a e)^3}{1+2 p}+\frac {3 e (b d-a e)^2 (a+b x)}{1+p}+\frac {6 e^2 (b d-a e) (a+b x)^2}{3+2 p}+\frac {e^3 (a+b x)^3}{2+p}\right )}{2 b^4} \]

[In]

Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((a + b*x)*((a + b*x)^2)^p*((2*(b*d - a*e)^3)/(1 + 2*p) + (3*e*(b*d - a*e)^2*(a + b*x))/(1 + p) + (6*e^2*(b*d
- a*e)*(a + b*x)^2)/(3 + 2*p) + (e^3*(a + b*x)^3)/(2 + p)))/(2*b^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(404\) vs. \(2(177)=354\).

Time = 2.34 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.24

method	result	size
gosper	\(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} \left (-4 b^{3} e^{3} p^{3} x^{3}-12 b^{3} d \,e^{2} p^{3} x^{2}-12 b^{3} e^{3} p^{2} x^{3}+6 a \,b^{2} e^{3} p^{2} x^{2}-12 b^{3} d^{2} e \,p^{3} x -42 b^{3} d \,e^{2} p^{2} x^{2}-11 b^{3} e^{3} p \,x^{3}+12 a \,b^{2} d \,e^{2} p^{2} x +9 a \,b^{2} e^{3} p \,x^{2}-4 b^{3} d^{3} p^{3}-48 b^{3} d^{2} e \,p^{2} x -42 b^{3} d \,e^{2} p \,x^{2}-3 e^{3} x^{3} b^{3}-6 a^{2} b \,e^{3} p x +6 a \,b^{2} d^{2} e \,p^{2}+30 a \,b^{2} d \,e^{2} p x +3 x^{2} a \,b^{2} e^{3}-18 b^{3} d^{3} p^{2}-57 b^{3} d^{2} e p x -12 x^{2} b^{3} d \,e^{2}-6 a^{2} b d \,e^{2} p -3 a^{2} b \,e^{3} x +21 a \,b^{2} d^{2} e p +12 x a \,b^{2} d \,e^{2}-26 b^{3} d^{3} p -18 b^{3} d^{2} e x +3 a^{3} e^{3}-12 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -12 b^{3} d^{3}\right ) \left (b x +a \right )}{2 b^{4} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}\)	\(405\)
norman	\(\frac {\left (6 a \,b^{2} d^{2} e \,p^{3}+2 b^{3} d^{3} p^{3}-6 a^{2} b d \,e^{2} p^{2}+21 a \,b^{2} d^{2} e \,p^{2}+9 b^{3} d^{3} p^{2}+3 a^{3} e^{3} p -12 a^{2} b d \,e^{2} p +18 a \,b^{2} d^{2} e p +13 b^{3} d^{3} p +6 b^{3} d^{3}\right ) x \,{\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b^{3} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}+\frac {e^{2} \left (a e p +3 b d p +6 b d \right ) x^{3} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b \left (2 p^{2}+7 p +6\right )}+\frac {e^{3} x^{4} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{4+2 p}-\frac {a \left (-4 b^{3} d^{3} p^{3}+6 a \,b^{2} d^{2} e \,p^{2}-18 b^{3} d^{3} p^{2}-6 a^{2} b d \,e^{2} p +21 a \,b^{2} d^{2} e p -26 b^{3} d^{3} p +3 a^{3} e^{3}-12 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -12 b^{3} d^{3}\right ) {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{4} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}-\frac {3 \left (-2 a b d e \,p^{2}-2 b^{2} d^{2} p^{2}+a^{2} e^{2} p -4 a b d e p -7 b^{2} d^{2} p -6 b^{2} d^{2}\right ) e \,x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{2} \left (2 p^{3}+9 p^{2}+13 p +6\right )}\)	\(498\)
risch	\(-\frac {\left (-4 b^{4} e^{3} p^{3} x^{4}-4 a \,b^{3} e^{3} p^{3} x^{3}-12 b^{4} d \,e^{2} p^{3} x^{3}-12 b^{4} e^{3} p^{2} x^{4}-12 a \,b^{3} d \,e^{2} p^{3} x^{2}-6 a \,b^{3} e^{3} p^{2} x^{3}-12 b^{4} d^{2} e \,p^{3} x^{2}-42 b^{4} d \,e^{2} p^{2} x^{3}-11 b^{4} e^{3} p \,x^{4}+6 a^{2} b^{2} e^{3} p^{2} x^{2}-12 a \,b^{3} d^{2} e \,p^{3} x -30 a \,b^{3} d \,e^{2} p^{2} x^{2}-2 a \,b^{3} e^{3} p \,x^{3}-4 b^{4} d^{3} p^{3} x -48 b^{4} d^{2} e \,p^{2} x^{2}-42 b^{4} d \,e^{2} p \,x^{3}-3 e^{3} x^{4} b^{4}+12 a^{2} b^{2} d \,e^{2} p^{2} x +3 a^{2} b^{2} e^{3} p \,x^{2}-4 a \,b^{3} d^{3} p^{3}-42 a \,b^{3} d^{2} e \,p^{2} x -12 a \,b^{3} d \,e^{2} p \,x^{2}-18 b^{4} d^{3} p^{2} x -57 b^{4} d^{2} e p \,x^{2}-12 b^{4} d \,e^{2} x^{3}-6 a^{3} b \,e^{3} p x +6 a^{2} b^{2} d^{2} e \,p^{2}+24 a^{2} b^{2} d \,e^{2} p x -18 a \,b^{3} d^{3} p^{2}-36 a \,b^{3} d^{2} e p x -26 b^{4} d^{3} p x -18 b^{4} d^{2} e \,x^{2}-6 a^{3} b d \,e^{2} p +21 a^{2} b^{2} d^{2} e p -26 a \,b^{3} d^{3} p -12 b^{4} d^{3} x +3 a^{4} e^{3}-12 a^{3} b d \,e^{2}+18 a^{2} b^{2} d^{2} e -12 a \,b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{p}}{2 \left (3+2 p \right ) \left (2+p \right ) \left (1+p \right ) \left (1+2 p \right ) b^{4}}\)	\(558\)
parallelrisch	\(\text {Expression too large to display}\)	\(1314\)

[In]

int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^p,x,method=_RETURNVERBOSE)

[Out]

-1/2*(b^2*x^2+2*a*b*x+a^2)^p*(-4*b^3*e^3*p^3*x^3-12*b^3*d*e^2*p^3*x^2-12*b^3*e^3*p^2*x^3+6*a*b^2*e^3*p^2*x^2-1
2*b^3*d^2*e*p^3*x-42*b^3*d*e^2*p^2*x^2-11*b^3*e^3*p*x^3+12*a*b^2*d*e^2*p^2*x+9*a*b^2*e^3*p*x^2-4*b^3*d^3*p^3-4
8*b^3*d^2*e*p^2*x-42*b^3*d*e^2*p*x^2-3*b^3*e^3*x^3-6*a^2*b*e^3*p*x+6*a*b^2*d^2*e*p^2+30*a*b^2*d*e^2*p*x+3*a*b^
2*e^3*x^2-18*b^3*d^3*p^2-57*b^3*d^2*e*p*x-12*b^3*d*e^2*x^2-6*a^2*b*d*e^2*p-3*a^2*b*e^3*x+21*a*b^2*d^2*e*p+12*a
*b^2*d*e^2*x-26*b^3*d^3*p-18*b^3*d^2*e*x+3*a^3*e^3-12*a^2*b*d*e^2+18*a*b^2*d^2*e-12*b^3*d^3)*(b*x+a)/b^4/(4*p^
4+20*p^3+35*p^2+25*p+6)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (177) = 354\).

Time = 0.38 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.85 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (4 \, a b^{3} d^{3} p^{3} + 12 \, a b^{3} d^{3} - 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} - 3 \, a^{4} e^{3} + {\left (4 \, b^{4} e^{3} p^{3} + 12 \, b^{4} e^{3} p^{2} + 11 \, b^{4} e^{3} p + 3 \, b^{4} e^{3}\right )} x^{4} + 2 \, {\left (6 \, b^{4} d e^{2} + 2 \, {\left (3 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{3} + 3 \, {\left (7 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{2} + {\left (21 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p\right )} x^{3} + 6 \, {\left (3 \, a b^{3} d^{3} - a^{2} b^{2} d^{2} e\right )} p^{2} + 3 \, {\left (6 \, b^{4} d^{2} e + 4 \, {\left (b^{4} d^{2} e + a b^{3} d e^{2}\right )} p^{3} + 2 \, {\left (8 \, b^{4} d^{2} e + 5 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p^{2} + {\left (19 \, b^{4} d^{2} e + 4 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p\right )} x^{2} + {\left (26 \, a b^{3} d^{3} - 21 \, a^{2} b^{2} d^{2} e + 6 \, a^{3} b d e^{2}\right )} p + 2 \, {\left (6 \, b^{4} d^{3} + 2 \, {\left (b^{4} d^{3} + 3 \, a b^{3} d^{2} e\right )} p^{3} + 3 \, {\left (3 \, b^{4} d^{3} + 7 \, a b^{3} d^{2} e - 2 \, a^{2} b^{2} d e^{2}\right )} p^{2} + {\left (13 \, b^{4} d^{3} + 18 \, a b^{3} d^{2} e - 12 \, a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3}\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, {\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]

[In]

integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="fricas")

[Out]

1/2*(4*a*b^3*d^3*p^3 + 12*a*b^3*d^3 - 18*a^2*b^2*d^2*e + 12*a^3*b*d*e^2 - 3*a^4*e^3 + (4*b^4*e^3*p^3 + 12*b^4*
e^3*p^2 + 11*b^4*e^3*p + 3*b^4*e^3)*x^4 + 2*(6*b^4*d*e^2 + 2*(3*b^4*d*e^2 + a*b^3*e^3)*p^3 + 3*(7*b^4*d*e^2 +
a*b^3*e^3)*p^2 + (21*b^4*d*e^2 + a*b^3*e^3)*p)*x^3 + 6*(3*a*b^3*d^3 - a^2*b^2*d^2*e)*p^2 + 3*(6*b^4*d^2*e + 4*
(b^4*d^2*e + a*b^3*d*e^2)*p^3 + 2*(8*b^4*d^2*e + 5*a*b^3*d*e^2 - a^2*b^2*e^3)*p^2 + (19*b^4*d^2*e + 4*a*b^3*d*
e^2 - a^2*b^2*e^3)*p)*x^2 + (26*a*b^3*d^3 - 21*a^2*b^2*d^2*e + 6*a^3*b*d*e^2)*p + 2*(6*b^4*d^3 + 2*(b^4*d^3 +
3*a*b^3*d^2*e)*p^3 + 3*(3*b^4*d^3 + 7*a*b^3*d^2*e - 2*a^2*b^2*d*e^2)*p^2 + (13*b^4*d^3 + 18*a*b^3*d^2*e - 12*a
^2*b^2*d*e^2 + 3*a^3*b*e^3)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p/(4*b^4*p^4 + 20*b^4*p^3 + 35*b^4*p^2 + 25*b^4*p
+ 6*b^4)

Sympy [F]

\[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Piecewise(((d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4)*(a**2)**p, Eq(b, 0)), (6*a**3*e**3*log(a/b +
x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 11*a**3*e**3/(6*a**3*b**4 + 18*a**2*b**5*x
+ 18*a*b**6*x**2 + 6*b**7*x**3) - 6*a**2*b*d*e**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**
3) + 18*a**2*b*e**3*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*e
**3*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*a*b**2*d**2*e/(6*a**3*b**4 + 18*a**2*b
**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 18*a*b**2*d*e**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*
b**7*x**3) + 18*a*b**2*e**3*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) +
18*a*b**2*e**3*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*d**3/(6*a**3*b**4 +
18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 9*b**3*d**2*e*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x*
*2 + 6*b**7*x**3) - 18*b**3*d*e**2*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3
*e**3*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(p, -2)), (Integral((
d + e*x)**3/((a + b*x)**2)**(3/2), x), Eq(p, -3/2)), (6*a**3*e**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*
e**3/(2*a*b**4 + 2*b**5*x) - 12*a**2*b*d*e**2*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 12*a**2*b*d*e**2/(2*a*b**4
+ 2*b**5*x) + 6*a**2*b*e**3*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a*b**2*d**2*e*log(a/b + x)/(2*a*b**4 + 2*
b**5*x) + 6*a*b**2*d**2*e/(2*a*b**4 + 2*b**5*x) - 12*a*b**2*d*e**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*
b**2*e**3*x**2/(2*a*b**4 + 2*b**5*x) - 2*b**3*d**3/(2*a*b**4 + 2*b**5*x) + 6*b**3*d**2*e*x*log(a/b + x)/(2*a*b
**4 + 2*b**5*x) + 6*b**3*d*e**2*x**2/(2*a*b**4 + 2*b**5*x) + b**3*e**3*x**3/(2*a*b**4 + 2*b**5*x), Eq(p, -1)),
(Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(e**3*x**2/(3*b**2) + x*(-5*a*e**3/(3*b) + 3*d*e**2)/(2*b**2) +
(-2*a**2*e**3/(3*b**2) - 3*a*(-5*a*e**3/(3*b) + 3*d*e**2)/(2*b) + 3*d**2*e)/b**2) + (a/b + x)*(-a**2*(-5*a*e**
3/(3*b) + 3*d*e**2)/(2*b**2) - a*(-2*a**2*e**3/(3*b**2) - 3*a*(-5*a*e**3/(3*b) + 3*d*e**2)/(2*b) + 3*d**2*e)/b
+ d**3)*log(a/b + x)/sqrt(b**2*(a/b + x)**2), Ne(b**2, 0)), ((sqrt(a**2 + 2*a*b*x)*(-a**3*e**3 + 6*a**2*b*d*e
**2 - 12*a*b**2*d**2*e + 8*b**3*d**3)/(8*b**3) + (a**2 + 2*a*b*x)**(3/2)*(3*a**2*e**3 - 12*a*b*d*e**2 + 12*b**
2*d**2*e)/(24*a*b**3) + (a**2 + 2*a*b*x)**(5/2)*(-3*a*e**3 + 6*b*d*e**2)/(40*a**2*b**3) + e**3*(a**2 + 2*a*b*x
)**(7/2)/(56*a**3*b**3))/(a*b), Ne(a*b, 0)), (Piecewise((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True))/sqrt(a
**2), True)), Eq(p, -1/2)), (-3*a**4*e**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**
4*p**2 + 50*b**4*p + 12*b**4) + 6*a**3*b*d*e**2*p*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3
+ 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 12*a**3*b*d*e**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**
4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 6*a**3*b*e**3*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4
+ 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) - 6*a**2*b**2*d**2*e*p**2*(a**2 + 2*a*b*x + b**2*x**2)**p
/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) - 21*a**2*b**2*d**2*e*p*(a**2 + 2*a*b*x + b
**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) - 18*a**2*b**2*d**2*e*(a**2 + 2
*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) - 12*a**2*b**2*d*e**2
*p**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) - 24
*a**2*b**2*d*e**2*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p +
12*b**4) - 6*a**2*b**2*e**3*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p
**2 + 50*b**4*p + 12*b**4) - 3*a**2*b**2*e**3*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p*
*3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 4*a*b**3*d**3*p**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 4
0*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 18*a*b**3*d**3*p**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**
4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 26*a*b**3*d**3*p*(a**2 + 2*a*b*x + b**2*x**2)**p
/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 12*a*b**3*d**3*(a**2 + 2*a*b*x + b**2*x**
2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 12*a*b**3*d**2*e*p**3*x*(a**2 + 2*a*
b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 42*a*b**3*d**2*e*p**2*
x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 36*a*b**
3*d**2*e*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4)
+ 12*a*b**3*d*e**2*p**3*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*
b**4*p + 12*b**4) + 30*a*b**3*d*e**2*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 7
0*b**4*p**2 + 50*b**4*p + 12*b**4) + 12*a*b**3*d*e**2*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40
*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 4*a*b**3*e**3*p**3*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*
b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 6*a*b**3*e**3*p**2*x**3*(a**2 + 2*a*b*x + b**
2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 2*a*b**3*e**3*p*x**3*(a**2 + 2*
a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 4*b**4*d**3*p**3*x*(
a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 18*b**4*d**
3*p**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 2
6*b**4*d**3*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b*
*4) + 12*b**4*d**3*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p +
12*b**4) + 12*b**4*d**2*e*p**3*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2
+ 50*b**4*p + 12*b**4) + 48*b**4*d**2*e*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3
+ 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 57*b**4*d**2*e*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 +
40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 18*b**4*d**2*e*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b*
*4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 12*b**4*d*e**2*p**3*x**3*(a**2 + 2*a*b*x + b**2
*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 42*b**4*d*e**2*p**2*x**3*(a**2 +
2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 42*b**4*d*e**2*p*
x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 12*b*
*4*d*e**2*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**
4) + 4*b**4*e**3*p**3*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*p**2 + 50*b**
4*p + 12*b**4) + 12*b**4*e**3*p**2*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3 + 70*b**4*
p**2 + 50*b**4*p + 12*b**4) + 11*b**4*e**3*p*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**4*p**3
+ 70*b**4*p**2 + 50*b**4*p + 12*b**4) + 3*b**4*e**3*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 40*b**
4*p**3 + 70*b**4*p**2 + 50*b**4*p + 12*b**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.52 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} d^{3}}{b {\left (2 \, p + 1\right )}} + \frac {3 \, {\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} {\left (b x + a\right )}^{2 \, p} d^{2} e}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac {3 \, {\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} + {\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )} {\left (b x + a\right )}^{2 \, p} d e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac {{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \, {\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \, {\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )} {\left (b x + a\right )}^{2 \, p} e^{3}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \]

[In]

integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="maxima")

[Out]

(b*x + a)*(b*x + a)^(2*p)*d^3/(b*(2*p + 1)) + 3/2*(b^2*(2*p + 1)*x^2 + 2*a*b*p*x - a^2)*(b*x + a)^(2*p)*d^2*e/
((2*p^2 + 3*p + 1)*b^2) + 3*((2*p^2 + 3*p + 1)*b^3*x^3 + (2*p^2 + p)*a*b^2*x^2 - 2*a^2*b*p*x + a^3)*(b*x + a)^
(2*p)*d*e^2/((4*p^3 + 12*p^2 + 11*p + 3)*b^3) + 1/2*((4*p^3 + 12*p^2 + 11*p + 3)*b^4*x^4 + 2*(2*p^3 + 3*p^2 +
p)*a*b^3*x^3 - 3*(2*p^2 + p)*a^2*b^2*x^2 + 6*a^3*b*p*x - 3*a^4)*(b*x + a)^(2*p)*e^3/((4*p^4 + 20*p^3 + 35*p^2
+ 25*p + 6)*b^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1279 vs. \(2 (177) = 354\).

Time = 0.30 (sec) , antiderivative size = 1279, normalized size of antiderivative = 7.07 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="giac")

[Out]

1/2*(4*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*e^3*p^3*x^4 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*e^2*p^3*x^3 + 4*(b^2
*x^2 + 2*a*b*x + a^2)^p*a*b^3*e^3*p^3*x^3 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*e^3*p^2*x^4 + 12*(b^2*x^2 + 2*a
*b*x + a^2)^p*b^4*d^2*e*p^3*x^2 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*e^2*p^3*x^2 + 42*(b^2*x^2 + 2*a*b*x +
a^2)^p*b^4*d*e^2*p^2*x^3 + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*e^3*p^2*x^3 + 11*(b^2*x^2 + 2*a*b*x + a^2)^p*b
^4*e^3*p*x^4 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*p^3*x + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*e*p^3*x
+ 48*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^2*e*p^2*x^2 + 30*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*e^2*p^2*x^2 - 6*(b
^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*e^3*p^2*x^2 + 42*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*e^2*p*x^3 + 2*(b^2*x^2 +
2*a*b*x + a^2)^p*a*b^3*e^3*p*x^3 + 3*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*e^3*x^4 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a
*b^3*d^3*p^3 + 18*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*p^2*x + 42*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*e*p^2*x
- 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d*e^2*p^2*x + 57*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^2*e*p*x^2 + 12*(b
^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*e^2*p*x^2 - 3*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*e^3*p*x^2 + 12*(b^2*x^2 +
2*a*b*x + a^2)^p*b^4*d*e^2*x^3 + 18*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^3*p^2 - 6*(b^2*x^2 + 2*a*b*x + a^2)^p*
a^2*b^2*d^2*e*p^2 + 26*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*p*x + 36*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*e*p*
x - 24*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d*e^2*p*x + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b*e^3*p*x + 18*(b^2*x
^2 + 2*a*b*x + a^2)^p*b^4*d^2*e*x^2 + 26*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^3*p - 21*(b^2*x^2 + 2*a*b*x + a^2
)^p*a^2*b^2*d^2*e*p + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b*d*e^2*p + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*x +
12*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^3 - 18*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d^2*e + 12*(b^2*x^2 + 2*a*b
*x + a^2)^p*a^3*b*d*e^2 - 3*(b^2*x^2 + 2*a*b*x + a^2)^p*a^4*e^3)/(4*b^4*p^4 + 20*b^4*p^3 + 35*b^4*p^2 + 25*b^4
*p + 6*b^4)

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.67 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx={\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {a\,\left (-3\,a^3\,e^3+6\,a^2\,b\,d\,e^2\,p+12\,a^2\,b\,d\,e^2-6\,a\,b^2\,d^2\,e\,p^2-21\,a\,b^2\,d^2\,e\,p-18\,a\,b^2\,d^2\,e+4\,b^3\,d^3\,p^3+18\,b^3\,d^3\,p^2+26\,b^3\,d^3\,p+12\,b^3\,d^3\right )}{2\,b^4\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {e^3\,x^4\,\left (4\,p^3+12\,p^2+11\,p+3\right )}{2\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {x\,\left (6\,a^3\,b\,e^3\,p-12\,a^2\,b^2\,d\,e^2\,p^2-24\,a^2\,b^2\,d\,e^2\,p+12\,a\,b^3\,d^2\,e\,p^3+42\,a\,b^3\,d^2\,e\,p^2+36\,a\,b^3\,d^2\,e\,p+4\,b^4\,d^3\,p^3+18\,b^4\,d^3\,p^2+26\,b^4\,d^3\,p+12\,b^4\,d^3\right )}{2\,b^4\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {3\,e\,x^2\,\left (2\,p+1\right )\,\left (-a^2\,e^2\,p+2\,a\,b\,d\,e\,p^2+4\,a\,b\,d\,e\,p+2\,b^2\,d^2\,p^2+7\,b^2\,d^2\,p+6\,b^2\,d^2\right )}{2\,b^2\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {e^2\,x^3\,\left (2\,p^2+3\,p+1\right )\,\left (6\,b\,d+a\,e\,p+3\,b\,d\,p\right )}{b\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}\right ) \]

[In]

int((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^p,x)

[Out]

(a^2 + b^2*x^2 + 2*a*b*x)^p*((a*(12*b^3*d^3 - 3*a^3*e^3 + 26*b^3*d^3*p + 18*b^3*d^3*p^2 + 4*b^3*d^3*p^3 - 18*a
*b^2*d^2*e + 12*a^2*b*d*e^2 - 21*a*b^2*d^2*e*p + 6*a^2*b*d*e^2*p - 6*a*b^2*d^2*e*p^2))/(2*b^4*(25*p + 35*p^2 +
20*p^3 + 4*p^4 + 6)) + (e^3*x^4*(11*p + 12*p^2 + 4*p^3 + 3))/(2*(25*p + 35*p^2 + 20*p^3 + 4*p^4 + 6)) + (x*(1
2*b^4*d^3 + 26*b^4*d^3*p + 18*b^4*d^3*p^2 + 4*b^4*d^3*p^3 + 6*a^3*b*e^3*p + 36*a*b^3*d^2*e*p - 24*a^2*b^2*d*e^
2*p + 42*a*b^3*d^2*e*p^2 + 12*a*b^3*d^2*e*p^3 - 12*a^2*b^2*d*e^2*p^2))/(2*b^4*(25*p + 35*p^2 + 20*p^3 + 4*p^4
+ 6)) + (3*e*x^2*(2*p + 1)*(6*b^2*d^2 - a^2*e^2*p + 7*b^2*d^2*p + 2*b^2*d^2*p^2 + 4*a*b*d*e*p + 2*a*b*d*e*p^2)
)/(2*b^2*(25*p + 35*p^2 + 20*p^3 + 4*p^4 + 6)) + (e^2*x^3*(3*p + 2*p^2 + 1)*(6*b*d + a*e*p + 3*b*d*p))/(b*(25*
p + 35*p^2 + 20*p^3 + 4*p^4 + 6)))