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

Optimal. Leaf size=134 \[ \frac {2 e (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)}+\frac {(a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+1)}+\frac {e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+2)} \]

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Rubi [A]  time = 0.09, antiderivative size = 134, 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 {2 e (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)}+\frac {(a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+1)}+\frac {e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.07, size = 109, normalized size = 0.81 \begin {gather*} \frac {\left ((a+b x)^2\right )^{p+1} \left (a^2 e^2-2 a b e (d (p+2)+e (p+1) x)+b^2 \left (d^2 \left (2 p^2+7 p+6\right )+4 d e \left (p^2+3 p+2\right ) x+e^2 \left (2 p^2+5 p+3\right ) x^2\right )\right )}{2 b^3 (p+1) (p+2) (2 p+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((a + b*x)^2)^(1 + p)*(a^2*e^2 - 2*a*b*e*(d*(2 + p) + e*(1 + p)*x) + b^2*(d^2*(6 + 7*p + 2*p^2) + 4*d*e*(2 +
3*p + p^2)*x + e^2*(3 + 5*p + 2*p^2)*x^2)))/(2*b^3*(1 + p)*(2 + p)*(3 + 2*p))

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IntegrateAlgebraic [F]  time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^2 \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)^2*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

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

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fricas [B]  time = 0.45, size = 353, normalized size = 2.63 \begin {gather*} \frac {{\left (2 \, a^{2} b^{2} d^{2} p^{2} + 6 \, a^{2} b^{2} d^{2} - 4 \, a^{3} b d e + a^{4} e^{2} + {\left (2 \, b^{4} e^{2} p^{2} + 5 \, b^{4} e^{2} p + 3 \, b^{4} e^{2}\right )} x^{4} + 4 \, {\left (2 \, b^{4} d e + a b^{3} e^{2} + {\left (b^{4} d e + a b^{3} e^{2}\right )} p^{2} + {\left (3 \, b^{4} d e + 2 \, a b^{3} e^{2}\right )} p\right )} x^{3} + {\left (6 \, b^{4} d^{2} + 12 \, a b^{3} d e + 2 \, {\left (b^{4} d^{2} + 4 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} p^{2} + {\left (7 \, b^{4} d^{2} + 22 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} p\right )} x^{2} + {\left (7 \, a^{2} b^{2} d^{2} - 2 \, a^{3} b d e\right )} p + 2 \, {\left (6 \, a b^{3} d^{2} + 2 \, {\left (a b^{3} d^{2} + a^{2} b^{2} d e\right )} p^{2} + {\left (7 \, a b^{3} d^{2} + 4 \, a^{2} b^{2} d e - a^{3} b e^{2}\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, {\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^2*b^2*d^2*p^2 + 6*a^2*b^2*d^2 - 4*a^3*b*d*e + a^4*e^2 + (2*b^4*e^2*p^2 + 5*b^4*e^2*p + 3*b^4*e^2)*x^4
+ 4*(2*b^4*d*e + a*b^3*e^2 + (b^4*d*e + a*b^3*e^2)*p^2 + (3*b^4*d*e + 2*a*b^3*e^2)*p)*x^3 + (6*b^4*d^2 + 12*a
*b^3*d*e + 2*(b^4*d^2 + 4*a*b^3*d*e + a^2*b^2*e^2)*p^2 + (7*b^4*d^2 + 22*a*b^3*d*e + a^2*b^2*e^2)*p)*x^2 + (7*
a^2*b^2*d^2 - 2*a^3*b*d*e)*p + 2*(6*a*b^3*d^2 + 2*(a*b^3*d^2 + a^2*b^2*d*e)*p^2 + (7*a*b^3*d^2 + 4*a^2*b^2*d*e
- a^3*b*e^2)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p/(2*b^3*p^3 + 9*b^3*p^2 + 13*b^3*p + 6*b^3)

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giac [B]  time = 0.32, size = 903, normalized size = 6.74 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} p^{2} x^{4} e^{2} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d p^{2} x^{3} e + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} p^{2} x^{2} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} p^{2} x^{3} e^{2} + 5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} p x^{4} e^{2} + 8 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d p^{2} x^{2} e + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d p x^{3} e + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{2} p^{2} x + 7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} p x^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} p^{2} x^{2} e^{2} + 8 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} p x^{3} e^{2} + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} x^{4} e^{2} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d p^{2} x e + 22 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d p x^{2} e + 8 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d x^{3} e + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d^{2} p^{2} + 14 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{2} p x + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} x^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} p x^{2} e^{2} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} x^{3} e^{2} + 8 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d p x e + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d x^{2} e + 7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d^{2} p + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{2} x - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} b p x e^{2} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} b d p e + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d^{2} - 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} b d e + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{4} e^{2}}{2 \, {\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*p^2*x^4*e^2 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*p^2*x^3*e + 2*(b^2*x^
2 + 2*a*b*x + a^2)^p*b^4*d^2*p^2*x^2 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*p^2*x^3*e^2 + 5*(b^2*x^2 + 2*a*b*x
+ a^2)^p*b^4*p*x^4*e^2 + 8*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*p^2*x^2*e + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*
d*p*x^3*e + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*p^2*x + 7*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^2*p*x^2 + 2*(b
^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*p^2*x^2*e^2 + 8*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*p*x^3*e^2 + 3*(b^2*x^2 + 2
*a*b*x + a^2)^p*b^4*x^4*e^2 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d*p^2*x*e + 22*(b^2*x^2 + 2*a*b*x + a^2)^p
*a*b^3*d*p*x^2*e + 8*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*x^3*e + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d^2*p^2 +
14*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*p*x + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^2*x^2 + (b^2*x^2 + 2*a*b*x
+ a^2)^p*a^2*b^2*p*x^2*e^2 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*x^3*e^2 + 8*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*
b^2*d*p*x*e + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*x^2*e + 7*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d^2*p + 12*
(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*x - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b*p*x*e^2 - 2*(b^2*x^2 + 2*a*b*x +
a^2)^p*a^3*b*d*p*e + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d^2 - 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a^3*b*d*e + (b
^2*x^2 + 2*a*b*x + a^2)^p*a^4*e^2)/(2*b^3*p^3 + 9*b^3*p^2 + 13*b^3*p + 6*b^3)

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maple [A]  time = 0.05, size = 179, normalized size = 1.34 \begin {gather*} \frac {\left (b x +a \right )^{2} \left (2 b^{2} e^{2} p^{2} x^{2}+4 b^{2} d e \,p^{2} x +5 b^{2} e^{2} p \,x^{2}-2 a b \,e^{2} p x +2 b^{2} d^{2} p^{2}+12 b^{2} d e p x +3 e^{2} x^{2} b^{2}-2 a b d e p -2 a b \,e^{2} x +7 b^{2} d^{2} p +8 b^{2} d e x +a^{2} e^{2}-4 a b d e +6 b^{2} d^{2}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{2 \left (2 p^{3}+9 p^{2}+13 p +6\right ) b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x+a)^2*(2*b^2*e^2*p^2*x^2+4*b^2*d*e*p^2*x+5*b^2*e^2*p*x^2-2*a*b*e^2*p*x+2*b^2*d^2*p^2+12*b^2*d*e*p*x+3*
b^2*e^2*x^2-2*a*b*d*e*p-2*a*b*e^2*x+7*b^2*d^2*p+8*b^2*d*e*x+a^2*e^2-4*a*b*d*e+6*b^2*d^2)*(b^2*x^2+2*a*b*x+a^2)
^p/b^3/(2*p^3+9*p^2+13*p+6)

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maxima [B]  time = 0.64, size = 404, normalized size = 3.01 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} a d^{2}}{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^{2}}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b} + \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} a d e}{{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac {2 \, {\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}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{2}} + \frac {{\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 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^{2}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 2.27, size = 355, normalized size = 2.65 \begin {gather*} {\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {a^2\,\left (a^2\,e^2-2\,a\,b\,d\,e\,p-4\,a\,b\,d\,e+2\,b^2\,d^2\,p^2+7\,b^2\,d^2\,p+6\,b^2\,d^2\right )}{2\,b^3\,\left (2\,p^3+9\,p^2+13\,p+6\right )}+\frac {x^2\,\left (2\,a^2\,b^2\,e^2\,p^2+a^2\,b^2\,e^2\,p+8\,a\,b^3\,d\,e\,p^2+22\,a\,b^3\,d\,e\,p+12\,a\,b^3\,d\,e+2\,b^4\,d^2\,p^2+7\,b^4\,d^2\,p+6\,b^4\,d^2\right )}{2\,b^3\,\left (2\,p^3+9\,p^2+13\,p+6\right )}+\frac {2\,e\,x^3\,\left (p+1\right )\,\left (a\,e+2\,b\,d+a\,e\,p+b\,d\,p\right )}{2\,p^3+9\,p^2+13\,p+6}+\frac {a\,x\,\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 )}{b^2\,\left (2\,p^3+9\,p^2+13\,p+6\right )}+\frac {b\,e^2\,x^4\,\left (2\,p^2+5\,p+3\right )}{2\,\left (2\,p^3+9\,p^2+13\,p+6\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^2*(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^2*(a^2*e^2 + 6*b^2*d^2 + 7*b^2*d^2*p + 2*b^2*d^2*p^2 - 4*a*b*d*e - 2*a*b*d*e*p
))/(2*b^3*(13*p + 9*p^2 + 2*p^3 + 6)) + (x^2*(6*b^4*d^2 + 7*b^4*d^2*p + 2*b^4*d^2*p^2 + a^2*b^2*e^2*p + 12*a*b
^3*d*e + 2*a^2*b^2*e^2*p^2 + 8*a*b^3*d*e*p^2 + 22*a*b^3*d*e*p))/(2*b^3*(13*p + 9*p^2 + 2*p^3 + 6)) + (2*e*x^3*
(p + 1)*(a*e + 2*b*d + a*e*p + b*d*p))/(13*p + 9*p^2 + 2*p^3 + 6) + (a*x*(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))/(b^2*(13*p + 9*p^2 + 2*p^3 + 6)) + (b*e^2*x^4*(5*p + 2*p^2 + 3
))/(2*(13*p + 9*p^2 + 2*p^3 + 6)))

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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)**2*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Piecewise((a*(d**2*x + d*e*x**2 + e**2*x**3/3)*(a**2)**p, Eq(b, 0)), (2*a**2*e**2*log(a/b + x)/(2*a**2*b**3 +
4*a*b**4*x + 2*b**5*x**2) + 3*a**2*e**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 2*a*b*d*e/(2*a**2*b**3 + 4*
a*b**4*x + 2*b**5*x**2) + 4*a*b*e**2*x*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*e**2*x/(2
*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - b**2*d**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 4*b**2*d*e*x/(2*
a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 2*b**2*e**2*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2)
, Eq(p, -2)), (Integral((a + b*x)*(d + e*x)**2/((a + b*x)**2)**(3/2), x), Eq(p, -3/2)), (a**2*e**2*log(a/b + x
)/b**3 - 2*a*d*e*log(a/b + x)/b**2 - a*e**2*x/b**2 + d**2*log(a/b + x)/b + 2*d*e*x/b + e**2*x**2/(2*b), Eq(p,
-1)), (a**4*e**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) - 2*a**3*b
*d*e*p*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) - 4*a**3*b*d*e*(a**2
+ 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) - 2*a**3*b*e**2*p*x*(a**2 + 2*a*
b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 2*a**2*b**2*d**2*p**2*(a**2 + 2*a*b*x
+ b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 7*a**2*b**2*d**2*p*(a**2 + 2*a*b*x + b**
2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 6*a**2*b**2*d**2*(a**2 + 2*a*b*x + b**2*x**2)*
*p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 4*a**2*b**2*d*e*p**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p
/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 8*a**2*b**2*d*e*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b
**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 2*a**2*b**2*e**2*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4
*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + a**2*b**2*e**2*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b*
*3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 4*a*b**3*d**2*p**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p
**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 14*a*b**3*d**2*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 +
18*b**3*p**2 + 26*b**3*p + 12*b**3) + 12*a*b**3*d**2*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*
p**2 + 26*b**3*p + 12*b**3) + 8*a*b**3*d*e*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p*
*2 + 26*b**3*p + 12*b**3) + 22*a*b**3*d*e*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 +
26*b**3*p + 12*b**3) + 12*a*b**3*d*e*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b*
*3*p + 12*b**3) + 4*a*b**3*e**2*p**2*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**
3*p + 12*b**3) + 8*a*b**3*e**2*p*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p
+ 12*b**3) + 4*a*b**3*e**2*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b
**3) + 2*b**4*d**2*p**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3
) + 7*b**4*d**2*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 6*
b**4*d**2*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 4*b**4*d*e
*p**2*x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 12*b**4*d*e*p*
x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 8*b**4*d*e*x**3*(a**
2 + 2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 2*b**4*e**2*p**2*x**4*(a**2 +
2*a*b*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 5*b**4*e**2*p*x**4*(a**2 + 2*a*b
*x + b**2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3) + 3*b**4*e**2*x**4*(a**2 + 2*a*b*x + b**
2*x**2)**p/(4*b**3*p**3 + 18*b**3*p**2 + 26*b**3*p + 12*b**3), True))

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