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

Optimal. Leaf size=183 \[ \frac {3 e^2 (a+b x)^4 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)}+\frac {3 e (a+b x)^3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac {(a+b x)^2 (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac {e^3 (a+b x)^5 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+5)} \]

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Rubi [A]  time = 0.12, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {770, 21, 43} \begin {gather*} \frac {3 e^2 (a+b x)^4 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)}+\frac {3 e (a+b x)^3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac {(a+b x)^2 (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac {e^3 (a+b x)^5 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(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)^2*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(1 + p)) + (3*e*(b*d - a*e)^2*(a + b*x)^3*(a^2
+ 2*a*b*x + b^2*x^2)^p)/(b^4*(3 + 2*p)) + (3*e^2*(b*d - a*e)*(a + b*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(
2 + p)) + (e^3*(a + b*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(5 + 2*p))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 104, normalized size = 0.57 \begin {gather*} \frac {\left ((a+b x)^2\right )^{p+1} \left (\frac {3 e^2 (a+b x)^2 (b d-a e)}{p+2}+\frac {6 e (a+b x) (b d-a e)^2}{2 p+3}+\frac {(b d-a e)^3}{p+1}+\frac {2 e^3 (a+b x)^3}{2 p+5}\right )}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((a + b*x)^2)^(1 + p)*((b*d - a*e)^3/(1 + p) + (6*e*(b*d - a*e)^2*(a + b*x))/(3 + 2*p) + (3*e^2*(b*d - a*e)*(
a + b*x)^2)/(2 + p) + (2*e^3*(a + b*x)^3)/(5 + 2*p)))/(2*b^4)

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IntegrateAlgebraic [F]  time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.47, size = 715, normalized size = 3.91 \begin {gather*} \frac {{\left (4 \, a^{2} b^{3} d^{3} p^{3} + 30 \, a^{2} b^{3} d^{3} - 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} - 3 \, a^{5} e^{3} + 2 \, {\left (2 \, b^{5} e^{3} p^{3} + 9 \, b^{5} e^{3} p^{2} + 13 \, b^{5} e^{3} p + 6 \, b^{5} e^{3}\right )} x^{5} + {\left (45 \, b^{5} d e^{2} + 15 \, a b^{4} e^{3} + 4 \, {\left (3 \, b^{5} d e^{2} + 2 \, a b^{4} e^{3}\right )} p^{3} + 30 \, {\left (2 \, b^{5} d e^{2} + a b^{4} e^{3}\right )} p^{2} + {\left (93 \, b^{5} d e^{2} + 37 \, a b^{4} e^{3}\right )} p\right )} x^{4} + 2 \, {\left (30 \, b^{5} d^{2} e + 30 \, a b^{4} d e^{2} + 2 \, {\left (3 \, b^{5} d^{2} e + 6 \, a b^{4} d e^{2} + a^{2} b^{3} e^{3}\right )} p^{3} + 3 \, {\left (11 \, b^{5} d^{2} e + 18 \, a b^{4} d e^{2} + a^{2} b^{3} e^{3}\right )} p^{2} + {\left (57 \, b^{5} d^{2} e + 72 \, a b^{4} d e^{2} + a^{2} b^{3} e^{3}\right )} p\right )} x^{3} + 6 \, {\left (4 \, a^{2} b^{3} d^{3} - a^{3} b^{2} d^{2} e\right )} p^{2} + {\left (30 \, b^{5} d^{3} + 90 \, a b^{4} d^{2} e + 4 \, {\left (b^{5} d^{3} + 6 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2}\right )} p^{3} + 6 \, {\left (4 \, b^{5} d^{3} + 21 \, a b^{4} d^{2} e + 6 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} p^{2} + {\left (47 \, b^{5} d^{3} + 201 \, a b^{4} d^{2} e + 15 \, a^{2} b^{3} d e^{2} - 3 \, a^{3} b^{2} e^{3}\right )} p\right )} x^{2} + {\left (47 \, a^{2} b^{3} d^{3} - 27 \, a^{3} b^{2} d^{2} e + 6 \, a^{4} b d e^{2}\right )} p + 2 \, {\left (30 \, a b^{4} d^{3} + 2 \, {\left (2 \, a b^{4} d^{3} + 3 \, a^{2} b^{3} d^{2} e\right )} p^{3} + 3 \, {\left (8 \, a b^{4} d^{3} + 9 \, a^{2} b^{3} d^{2} e - 2 \, a^{3} b^{2} d e^{2}\right )} p^{2} + {\left (47 \, a b^{4} d^{3} + 30 \, a^{2} b^{3} d^{2} e - 15 \, a^{3} b^{2} d e^{2} + 3 \, a^{4} 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} + 28 \, b^{4} p^{3} + 71 \, b^{4} p^{2} + 77 \, b^{4} p + 30 \, b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*a^2*b^3*d^3*p^3 + 30*a^2*b^3*d^3 - 30*a^3*b^2*d^2*e + 15*a^4*b*d*e^2 - 3*a^5*e^3 + 2*(2*b^5*e^3*p^3 + 9
*b^5*e^3*p^2 + 13*b^5*e^3*p + 6*b^5*e^3)*x^5 + (45*b^5*d*e^2 + 15*a*b^4*e^3 + 4*(3*b^5*d*e^2 + 2*a*b^4*e^3)*p^
3 + 30*(2*b^5*d*e^2 + a*b^4*e^3)*p^2 + (93*b^5*d*e^2 + 37*a*b^4*e^3)*p)*x^4 + 2*(30*b^5*d^2*e + 30*a*b^4*d*e^2
+ 2*(3*b^5*d^2*e + 6*a*b^4*d*e^2 + a^2*b^3*e^3)*p^3 + 3*(11*b^5*d^2*e + 18*a*b^4*d*e^2 + a^2*b^3*e^3)*p^2 + (
57*b^5*d^2*e + 72*a*b^4*d*e^2 + a^2*b^3*e^3)*p)*x^3 + 6*(4*a^2*b^3*d^3 - a^3*b^2*d^2*e)*p^2 + (30*b^5*d^3 + 90
*a*b^4*d^2*e + 4*(b^5*d^3 + 6*a*b^4*d^2*e + 3*a^2*b^3*d*e^2)*p^3 + 6*(4*b^5*d^3 + 21*a*b^4*d^2*e + 6*a^2*b^3*d
*e^2 - a^3*b^2*e^3)*p^2 + (47*b^5*d^3 + 201*a*b^4*d^2*e + 15*a^2*b^3*d*e^2 - 3*a^3*b^2*e^3)*p)*x^2 + (47*a^2*b
^3*d^3 - 27*a^3*b^2*d^2*e + 6*a^4*b*d*e^2)*p + 2*(30*a*b^4*d^3 + 2*(2*a*b^4*d^3 + 3*a^2*b^3*d^2*e)*p^3 + 3*(8*
a*b^4*d^3 + 9*a^2*b^3*d^2*e - 2*a^3*b^2*d*e^2)*p^2 + (47*a*b^4*d^3 + 30*a^2*b^3*d^2*e - 15*a^3*b^2*d*e^2 + 3*a
^4*b*e^3)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p/(4*b^4*p^4 + 28*b^4*p^3 + 71*b^4*p^2 + 77*b^4*p + 30*b^4)

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giac [B]  time = 0.29, size = 1805, normalized size = 9.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(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^5*p^3*x^5*e^3 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d*p^3*x^4*e^2 + 12*(b^
2*x^2 + 2*a*b*x + a^2)^p*b^5*d^2*p^3*x^3*e + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d^3*p^3*x^2 + 8*(b^2*x^2 + 2*a*
b*x + a^2)^p*a*b^4*p^3*x^4*e^3 + 18*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*p^2*x^5*e^3 + 24*(b^2*x^2 + 2*a*b*x + a^2)
^p*a*b^4*d*p^3*x^3*e^2 + 60*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d*p^2*x^4*e^2 + 24*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b
^4*d^2*p^3*x^2*e + 66*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d^2*p^2*x^3*e + 8*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d^3*
p^3*x + 24*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d^3*p^2*x^2 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*p^3*x^3*e^3 + 3
0*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*p^2*x^4*e^3 + 26*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*p*x^5*e^3 + 12*(b^2*x^2 +
2*a*b*x + a^2)^p*a^2*b^3*d*p^3*x^2*e^2 + 108*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d*p^2*x^3*e^2 + 93*(b^2*x^2 +
2*a*b*x + a^2)^p*b^5*d*p*x^4*e^2 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*d^2*p^3*x*e + 126*(b^2*x^2 + 2*a*b*x
+ a^2)^p*a*b^4*d^2*p^2*x^2*e + 114*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d^2*p*x^3*e + 4*(b^2*x^2 + 2*a*b*x + a^2)^
p*a^2*b^3*d^3*p^3 + 48*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d^3*p^2*x + 47*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d^3*p*
x^2 + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*p^2*x^3*e^3 + 37*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*p*x^4*e^3 + 12*
(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*x^5*e^3 + 36*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*d*p^2*x^2*e^2 + 144*(b^2*x^2
+ 2*a*b*x + a^2)^p*a*b^4*d*p*x^3*e^2 + 45*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d*x^4*e^2 + 54*(b^2*x^2 + 2*a*b*x +
a^2)^p*a^2*b^3*d^2*p^2*x*e + 201*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d^2*p*x^2*e + 60*(b^2*x^2 + 2*a*b*x + a^2)^
p*b^5*d^2*x^3*e + 24*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*d^3*p^2 + 94*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d^3*p*
x + 30*(b^2*x^2 + 2*a*b*x + a^2)^p*b^5*d^3*x^2 - 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b^2*p^2*x^2*e^3 + 2*(b^2*x^
2 + 2*a*b*x + a^2)^p*a^2*b^3*p*x^3*e^3 + 15*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*x^4*e^3 - 12*(b^2*x^2 + 2*a*b*x
+ a^2)^p*a^3*b^2*d*p^2*x*e^2 + 15*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*d*p*x^2*e^2 + 60*(b^2*x^2 + 2*a*b*x + a^
2)^p*a*b^4*d*x^3*e^2 - 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b^2*d^2*p^2*e + 60*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^
3*d^2*p*x*e + 90*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d^2*x^2*e + 47*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^3*d^3*p +
60*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^4*d^3*x - 3*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b^2*p*x^2*e^3 - 30*(b^2*x^2 + 2
*a*b*x + a^2)^p*a^3*b^2*d*p*x*e^2 - 27*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b^2*d^2*p*e + 30*(b^2*x^2 + 2*a*b*x + a
^2)^p*a^2*b^3*d^3 + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^4*b*p*x*e^3 + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^4*b*d*p*e^2
- 30*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b^2*d^2*e + 15*(b^2*x^2 + 2*a*b*x + a^2)^p*a^4*b*d*e^2 - 3*(b^2*x^2 + 2*a
*b*x + a^2)^p*a^5*e^3)/(4*b^4*p^4 + 28*b^4*p^3 + 71*b^4*p^2 + 77*b^4*p + 30*b^4)

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maple [B]  time = 0.05, size = 407, normalized size = 2.22 \begin {gather*} -\frac {\left (-4 b^{3} e^{3} p^{3} x^{3}-12 b^{3} d \,e^{2} p^{3} x^{2}-18 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 -60 b^{3} d \,e^{2} p^{2} x^{2}-26 b^{3} e^{3} p \,x^{3}+12 a \,b^{2} d \,e^{2} p^{2} x +15 a \,b^{2} e^{3} p \,x^{2}-4 b^{3} d^{3} p^{3}-66 b^{3} d^{2} e \,p^{2} x -93 b^{3} d \,e^{2} p \,x^{2}-12 b^{3} e^{3} x^{3}-6 a^{2} b \,e^{3} p x +6 a \,b^{2} d^{2} e \,p^{2}+42 a \,b^{2} d \,e^{2} p x +9 a \,b^{2} e^{3} x^{2}-24 b^{3} d^{3} p^{2}-114 b^{3} d^{2} e p x -45 b^{3} d \,e^{2} x^{2}-6 a^{2} b d \,e^{2} p -6 a^{2} b \,e^{3} x +27 a \,b^{2} d^{2} e p +30 a \,b^{2} d \,e^{2} x -47 b^{3} d^{3} p -60 b^{3} d^{2} e x +3 a^{3} e^{3}-15 a^{2} b d \,e^{2}+30 a \,b^{2} d^{2} e -30 b^{3} d^{3}\right ) \left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{2 \left (4 p^{4}+28 p^{3}+71 p^{2}+77 p +30\right ) b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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^2-18*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-60*b^3*d*e^2*p^2*x^2-26*b^3*e^3*p*x^3+12*a*b^2*d*e^2*p^2*x+15*a*b^2*e^3*p*x^2-4*b^3*d^3*p^3-
66*b^3*d^2*e*p^2*x-93*b^3*d*e^2*p*x^2-12*b^3*e^3*x^3-6*a^2*b*e^3*p*x+6*a*b^2*d^2*e*p^2+42*a*b^2*d*e^2*p*x+9*a*
b^2*e^3*x^2-24*b^3*d^3*p^2-114*b^3*d^2*e*p*x-45*b^3*d*e^2*x^2-6*a^2*b*d*e^2*p-6*a^2*b*e^3*x+27*a*b^2*d^2*e*p+3
0*a*b^2*d*e^2*x-47*b^3*d^3*p-60*b^3*d^2*e*x+3*a^3*e^3-15*a^2*b*d*e^2+30*a*b^2*d^2*e-30*b^3*d^3)*(b*x+a)^2/b^4/
(4*p^4+28*p^3+71*p^2+77*p+30)

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maxima [B]  time = 0.75, size = 679, normalized size = 3.71 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} a d^{3}}{b {\left (2 \, p + 1\right )}} + \frac {{\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^{3}}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b} + \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} a 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^{2} e}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\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} a d e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac {3 \, {\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} d e^{2}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\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} a e^{3}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} + \frac {{\left ({\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{5} x^{5} + {\left (4 \, p^{4} + 12 \, p^{3} + 11 \, p^{2} + 3 \, p\right )} a b^{4} x^{4} - 4 \, {\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a^{2} b^{3} x^{3} + 6 \, {\left (2 \, p^{2} + p\right )} a^{3} b^{2} x^{2} - 12 \, a^{4} b p x + 6 \, a^{5}\right )} {\left (b x + a\right )}^{2 \, p} e^{3}}{{\left (8 \, p^{5} + 60 \, p^{4} + 170 \, p^{3} + 225 \, p^{2} + 137 \, p + 30\right )} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(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)*a*d^3/(b*(2*p + 1)) + 1/2*(b^2*(2*p + 1)*x^2 + 2*a*b*p*x - a^2)*(b*x + a)^(2*p)*d^3/
((2*p^2 + 3*p + 1)*b) + 3/2*(b^2*(2*p + 1)*x^2 + 2*a*b*p*x - a^2)*(b*x + a)^(2*p)*a*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^2*e/((4*p^3
+ 12*p^2 + 11*p + 3)*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)*a*d*e^2/((4*p^3 + 12*p^2 + 11*p + 3)*b^3) + 3/2*((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)*d*e^2/((4*p^4 + 20*p^3 + 3
5*p^2 + 25*p + 6)*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)*a*e^3/((4*p^4 + 20*p^3 + 35*p^2 + 25*p + 6)*b^4) + ((
4*p^4 + 20*p^3 + 35*p^2 + 25*p + 6)*b^5*x^5 + (4*p^4 + 12*p^3 + 11*p^2 + 3*p)*a*b^4*x^4 - 4*(2*p^3 + 3*p^2 + p
)*a^2*b^3*x^3 + 6*(2*p^2 + p)*a^3*b^2*x^2 - 12*a^4*b*p*x + 6*a^5)*(b*x + a)^(2*p)*e^3/((8*p^5 + 60*p^4 + 170*p
^3 + 225*p^2 + 137*p + 30)*b^4)

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mupad [B]  time = 2.49, size = 683, normalized size = 3.73 \begin {gather*} {\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {x^2\,\left (-6\,a^3\,b^2\,e^3\,p^2-3\,a^3\,b^2\,e^3\,p+12\,a^2\,b^3\,d\,e^2\,p^3+36\,a^2\,b^3\,d\,e^2\,p^2+15\,a^2\,b^3\,d\,e^2\,p+24\,a\,b^4\,d^2\,e\,p^3+126\,a\,b^4\,d^2\,e\,p^2+201\,a\,b^4\,d^2\,e\,p+90\,a\,b^4\,d^2\,e+4\,b^5\,d^3\,p^3+24\,b^5\,d^3\,p^2+47\,b^5\,d^3\,p+30\,b^5\,d^3\right )}{2\,b^4\,\left (4\,p^4+28\,p^3+71\,p^2+77\,p+30\right )}+\frac {a^2\,\left (-3\,a^3\,e^3+6\,a^2\,b\,d\,e^2\,p+15\,a^2\,b\,d\,e^2-6\,a\,b^2\,d^2\,e\,p^2-27\,a\,b^2\,d^2\,e\,p-30\,a\,b^2\,d^2\,e+4\,b^3\,d^3\,p^3+24\,b^3\,d^3\,p^2+47\,b^3\,d^3\,p+30\,b^3\,d^3\right )}{2\,b^4\,\left (4\,p^4+28\,p^3+71\,p^2+77\,p+30\right )}+\frac {b\,e^3\,x^5\,\left (2\,p^3+9\,p^2+13\,p+6\right )}{4\,p^4+28\,p^3+71\,p^2+77\,p+30}+\frac {a\,x\,\left (3\,a^3\,e^3\,p-6\,a^2\,b\,d\,e^2\,p^2-15\,a^2\,b\,d\,e^2\,p+6\,a\,b^2\,d^2\,e\,p^3+27\,a\,b^2\,d^2\,e\,p^2+30\,a\,b^2\,d^2\,e\,p+4\,b^3\,d^3\,p^3+24\,b^3\,d^3\,p^2+47\,b^3\,d^3\,p+30\,b^3\,d^3\right )}{b^3\,\left (4\,p^4+28\,p^3+71\,p^2+77\,p+30\right )}+\frac {e^2\,x^4\,\left (2\,p^2+5\,p+3\right )\,\left (5\,a\,e+15\,b\,d+4\,a\,e\,p+6\,b\,d\,p\right )}{2\,\left (4\,p^4+28\,p^3+71\,p^2+77\,p+30\right )}+\frac {e\,x^3\,\left (p+1\right )\,\left (2\,a^2\,e^2\,p^2+a^2\,e^2\,p+12\,a\,b\,d\,e\,p^2+42\,a\,b\,d\,e\,p+30\,a\,b\,d\,e+6\,b^2\,d^2\,p^2+27\,b^2\,d^2\,p+30\,b^2\,d^2\right )}{b\,\left (4\,p^4+28\,p^3+71\,p^2+77\,p+30\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(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*((x^2*(30*b^5*d^3 + 47*b^5*d^3*p + 24*b^5*d^3*p^2 + 4*b^5*d^3*p^3 - 3*a^3*b^2*e^3*
p - 6*a^3*b^2*e^3*p^2 + 90*a*b^4*d^2*e + 201*a*b^4*d^2*e*p + 15*a^2*b^3*d*e^2*p + 126*a*b^4*d^2*e*p^2 + 24*a*b
^4*d^2*e*p^3 + 36*a^2*b^3*d*e^2*p^2 + 12*a^2*b^3*d*e^2*p^3))/(2*b^4*(77*p + 71*p^2 + 28*p^3 + 4*p^4 + 30)) + (
a^2*(30*b^3*d^3 - 3*a^3*e^3 + 47*b^3*d^3*p + 24*b^3*d^3*p^2 + 4*b^3*d^3*p^3 - 30*a*b^2*d^2*e + 15*a^2*b*d*e^2
- 27*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*(77*p + 71*p^2 + 28*p^3 + 4*p^4 + 30)) + (b*
e^3*x^5*(13*p + 9*p^2 + 2*p^3 + 6))/(77*p + 71*p^2 + 28*p^3 + 4*p^4 + 30) + (a*x*(30*b^3*d^3 + 3*a^3*e^3*p + 4
7*b^3*d^3*p + 24*b^3*d^3*p^2 + 4*b^3*d^3*p^3 + 30*a*b^2*d^2*e*p - 15*a^2*b*d*e^2*p + 27*a*b^2*d^2*e*p^2 - 6*a^
2*b*d*e^2*p^2 + 6*a*b^2*d^2*e*p^3))/(b^3*(77*p + 71*p^2 + 28*p^3 + 4*p^4 + 30)) + (e^2*x^4*(5*p + 2*p^2 + 3)*(
5*a*e + 15*b*d + 4*a*e*p + 6*b*d*p))/(2*(77*p + 71*p^2 + 28*p^3 + 4*p^4 + 30)) + (e*x^3*(p + 1)*(30*b^2*d^2 +
a^2*e^2*p + 27*b^2*d^2*p + 2*a^2*e^2*p^2 + 6*b^2*d^2*p^2 + 30*a*b*d*e + 42*a*b*d*e*p + 12*a*b*d*e*p^2))/(b*(77
*p + 71*p^2 + 28*p^3 + 4*p^4 + 30)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*(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)), (Integral((a + b*x)*
(d + e*x)**3/((a + b*x)**2)**(5/2), x), Eq(p, -5/2)), (-6*a**3*e**3*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2
*b**6*x**2) - 11*a**3*e**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6*a**2*b*d*e**2*log(a/b + x)/(2*a**2*b**
4 + 4*a*b**5*x + 2*b**6*x**2) + 9*a**2*b*d*e**2/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*e**3*x*lo
g(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 16*a**2*b*e**3*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2
) - 3*a*b**2*d**2*e/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 12*a*b**2*d*e**2*x*log(a/b + x)/(2*a**2*b**4 +
4*a*b**5*x + 2*b**6*x**2) + 12*a*b**2*d*e**2*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*e**3*x**2*l
og(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*a*b**2*e**3*x**2/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x
**2) - b**3*d**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*b**3*d**2*e*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6
*x**2) + 6*b**3*d*e**2*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*e**3*x**3/(2*a**2*b
**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(p, -2)), (Integral((a + b*x)*(d + e*x)**3/((a + b*x)**2)**(3/2), x), Eq(p,
-3/2)), (-a**3*e**3*log(a/b + x)/b**4 + 3*a**2*d*e**2*log(a/b + x)/b**3 + a**2*e**3*x/b**3 - 3*a*d**2*e*log(a
/b + x)/b**2 - 3*a*d*e**2*x/b**2 - a*e**3*x**2/(2*b**2) + d**3*log(a/b + x)/b + 3*d**2*e*x/b + 3*d*e**2*x**2/(
2*b) + e**3*x**3/(3*b), Eq(p, -1)), (-3*a**5*e**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3
+ 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 6*a**4*b*d*e**2*p*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*
b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 15*a**4*b*d*e**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p
**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 6*a**4*b*e**3*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p
/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) - 6*a**3*b**2*d**2*e*p**2*(a**2 + 2*a*b*x
+ b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) - 27*a**3*b**2*d**2*e*p*(
a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) - 30*a**3*b
**2*d**2*e*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4)
- 12*a**3*b**2*d*e**2*p**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 15
4*b**4*p + 60*b**4) - 30*a**3*b**2*d*e**2*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 14
2*b**4*p**2 + 154*b**4*p + 60*b**4) - 6*a**3*b**2*e**3*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4
+ 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) - 3*a**3*b**2*e**3*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)*
*p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 4*a**2*b**3*d**3*p**3*(a**2 + 2*a*b*x
+ b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 24*a**2*b**3*d**3*p**2*
(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 47*a**2*
b**3*d**3*p*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4
) + 30*a**2*b**3*d**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p
+ 60*b**4) + 12*a**2*b**3*d**2*e*p**3*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**
4*p**2 + 154*b**4*p + 60*b**4) + 54*a**2*b**3*d**2*e*p**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*
b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 60*a**2*b**3*d**2*e*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8
*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 12*a**2*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 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 36*a**2*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 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4)
+ 15*a**2*b**3*d*e**2*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154
*b**4*p + 60*b**4) + 4*a**2*b**3*e**3*p**3*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 +
142*b**4*p**2 + 154*b**4*p + 60*b**4) + 6*a**2*b**3*e**3*p**2*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**
4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 2*a**2*b**3*e**3*p*x**3*(a**2 + 2*a*b*x + b**2*x**2
)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 8*a*b**4*d**3*p**3*x*(a**2 + 2*a*b*
x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 48*a*b**4*d**3*p**2*x*
(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 94*a*b**
4*d**3*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4)
+ 60*a*b**4*d**3*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p +
60*b**4) + 24*a*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 + 56*b**4*p**3 + 142*b**4*
p**2 + 154*b**4*p + 60*b**4) + 126*a*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 + 56*b
**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 201*a*b**4*d**2*e*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8
*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 90*a*b**4*d**2*e*x**2*(a**2 + 2*a*b*x + b*
*2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 24*a*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 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 108*a*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 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p +
60*b**4) + 144*a*b**4*d*e**2*p*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**
2 + 154*b**4*p + 60*b**4) + 60*a*b**4*d*e**2*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3
+ 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 8*a*b**4*e**3*p**3*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4
+ 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 30*a*b**4*e**3*p**2*x**4*(a**2 + 2*a*b*x + b**2*x**2
)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 37*a*b**4*e**3*p*x**4*(a**2 + 2*a*b
*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 15*a*b**4*e**3*x**4*(
a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 4*b**5*d*
*3*p**3*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**
4) + 24*b**5*d**3*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*
b**4*p + 60*b**4) + 47*b**5*d**3*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4
*p**2 + 154*b**4*p + 60*b**4) + 30*b**5*d**3*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3
+ 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 12*b**5*d**2*e*p**3*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**
4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 66*b**5*d**2*e*p**2*x**3*(a**2 + 2*a*b*x + b**2*x**
2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 114*b**5*d**2*e*p*x**3*(a**2 + 2*a
*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 60*b**5*d**2*e*x**3
*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 12*b**5
*d*e**2*p**3*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 6
0*b**4) + 60*b**5*d*e**2*p**2*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2
+ 154*b**4*p + 60*b**4) + 93*b**5*d*e**2*p*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 +
142*b**4*p**2 + 154*b**4*p + 60*b**4) + 45*b**5*d*e**2*x**4*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56
*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 4*b**5*e**3*p**3*x**5*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*
b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 18*b**5*e**3*p**2*x**5*(a**2 + 2*a*b*x + b*
*2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 26*b**5*e**3*p*x**5*(a**2 +
2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4) + 12*b**5*e**3*x**
5*(a**2 + 2*a*b*x + b**2*x**2)**p/(8*b**4*p**4 + 56*b**4*p**3 + 142*b**4*p**2 + 154*b**4*p + 60*b**4), True))

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