3.351 \(\int x^2 (a+b x)^n (c+d x^2) \, dx\)
Optimal. Leaf size=135 \[ \frac{a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac{2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)} \]
[Out]
(a^2*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)
) + ((b^2*c + 6*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*
x)^(5 + n))/(b^5*(5 + n))
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Rubi [A] time = 0.0815451, antiderivative size = 135, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used =
{948} \[ \frac{a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac{2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)} \]
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
(a^2*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)
) + ((b^2*c + 6*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*
x)^(5 + n))/(b^5*(5 + n))
Rule 948
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))
Rubi steps
\begin{align*} \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx &=\int \left (\frac{\left (a^2 b^2 c+a^4 d\right ) (a+b x)^n}{b^4}-\frac{2 \left (a b^2 c+2 a^3 d\right ) (a+b x)^{1+n}}{b^4}+\frac{\left (b^2 c+6 a^2 d\right ) (a+b x)^{2+n}}{b^4}-\frac{4 a d (a+b x)^{3+n}}{b^4}+\frac{d (a+b x)^{4+n}}{b^4}\right ) \, dx\\ &=\frac{a^2 \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}-\frac{2 a \left (b^2 c+2 a^2 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac{\left (b^2 c+6 a^2 d\right ) (a+b x)^{3+n}}{b^5 (3+n)}-\frac{4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac{d (a+b x)^{5+n}}{b^5 (5+n)}\\ \end{align*}
Mathematica [A] time = 0.100699, size = 114, normalized size = 0.84 \[ \frac{(a+b x)^{n+1} \left (\frac{(a+b x)^2 \left (6 a^2 d+b^2 c\right )}{n+3}-\frac{2 a (a+b x) \left (2 a^2 d+b^2 c\right )}{n+2}+\frac{a^2 b^2 c+a^4 d}{n+1}+\frac{d (a+b x)^4}{n+5}-\frac{4 a d (a+b x)^3}{n+4}\right )}{b^5} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
((a + b*x)^(1 + n)*((a^2*b^2*c + a^4*d)/(1 + n) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x))/(2 + n) + ((b^2*c + 6*a^2*
d)*(a + b*x)^2)/(3 + n) - (4*a*d*(a + b*x)^3)/(4 + n) + (d*(a + b*x)^4)/(5 + n)))/b^5
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Maple [B] time = 0.049, size = 328, normalized size = 2.4 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{4}d{n}^{4}{x}^{4}+10\,{b}^{4}d{n}^{3}{x}^{4}-4\,a{b}^{3}d{n}^{3}{x}^{3}+{b}^{4}c{n}^{4}{x}^{2}+35\,{b}^{4}d{n}^{2}{x}^{4}-24\,a{b}^{3}d{n}^{2}{x}^{3}+12\,{b}^{4}c{n}^{3}{x}^{2}+50\,{b}^{4}dn{x}^{4}+12\,{a}^{2}{b}^{2}d{n}^{2}{x}^{2}-2\,a{b}^{3}c{n}^{3}x-44\,a{b}^{3}dn{x}^{3}+49\,{b}^{4}c{n}^{2}{x}^{2}+24\,d{x}^{4}{b}^{4}+36\,{a}^{2}{b}^{2}dn{x}^{2}-20\,a{b}^{3}c{n}^{2}x-24\,ad{x}^{3}{b}^{3}+78\,{b}^{4}cn{x}^{2}-24\,{a}^{3}bdnx+2\,{a}^{2}{b}^{2}c{n}^{2}+24\,{a}^{2}{b}^{2}d{x}^{2}-58\,a{b}^{3}cnx+40\,{b}^{4}c{x}^{2}-24\,{a}^{3}bdx+18\,{a}^{2}{b}^{2}cn-40\,a{b}^{3}cx+24\,{a}^{4}d+40\,{a}^{2}{b}^{2}c \right ) }{{b}^{5} \left ({n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+120 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(b*x+a)^n*(d*x^2+c),x)
[Out]
(b*x+a)^(1+n)*(b^4*d*n^4*x^4+10*b^4*d*n^3*x^4-4*a*b^3*d*n^3*x^3+b^4*c*n^4*x^2+35*b^4*d*n^2*x^4-24*a*b^3*d*n^2*
x^3+12*b^4*c*n^3*x^2+50*b^4*d*n*x^4+12*a^2*b^2*d*n^2*x^2-2*a*b^3*c*n^3*x-44*a*b^3*d*n*x^3+49*b^4*c*n^2*x^2+24*
b^4*d*x^4+36*a^2*b^2*d*n*x^2-20*a*b^3*c*n^2*x-24*a*b^3*d*x^3+78*b^4*c*n*x^2-24*a^3*b*d*n*x+2*a^2*b^2*c*n^2+24*
a^2*b^2*d*x^2-58*a*b^3*c*n*x+40*b^4*c*x^2-24*a^3*b*d*x+18*a^2*b^2*c*n-40*a*b^3*c*x+24*a^4*d+40*a^2*b^2*c)/b^5/
(n^5+15*n^4+85*n^3+225*n^2+274*n+120)
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Maxima [A] time = 1.0251, size = 284, normalized size = 2.1 \begin{align*} \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c),x, algorithm="maxima")
[Out]
((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c/((n^3 + 6*n^2 + 11*n + 6)*
b^3) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2
+ 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*d/((n^5 + 15*n^4 + 85*n^3 +
225*n^2 + 274*n + 120)*b^5)
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Fricas [B] time = 1.93335, size = 786, normalized size = 5.82 \begin{align*} \frac{{\left (2 \, a^{3} b^{2} c n^{2} + 18 \, a^{3} b^{2} c n + 40 \, a^{3} b^{2} c + 24 \, a^{5} d +{\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} +{\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} +{\left (b^{5} c n^{4} + 40 \, b^{5} c + 4 \,{\left (3 \, b^{5} c - a^{2} b^{3} d\right )} n^{3} +{\left (49 \, b^{5} c - 12 \, a^{2} b^{3} d\right )} n^{2} + 2 \,{\left (39 \, b^{5} c - 4 \, a^{2} b^{3} d\right )} n\right )} x^{3} +{\left (a b^{4} c n^{4} + 10 \, a b^{4} c n^{3} +{\left (29 \, a b^{4} c + 12 \, a^{3} b^{2} d\right )} n^{2} + 4 \,{\left (5 \, a b^{4} c + 3 \, a^{3} b^{2} d\right )} n\right )} x^{2} - 2 \,{\left (a^{2} b^{3} c n^{3} + 9 \, a^{2} b^{3} c n^{2} + 4 \,{\left (5 \, a^{2} b^{3} c + 3 \, a^{4} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c),x, algorithm="fricas")
[Out]
(2*a^3*b^2*c*n^2 + 18*a^3*b^2*c*n + 40*a^3*b^2*c + 24*a^5*d + (b^5*d*n^4 + 10*b^5*d*n^3 + 35*b^5*d*n^2 + 50*b^
5*d*n + 24*b^5*d)*x^5 + (a*b^4*d*n^4 + 6*a*b^4*d*n^3 + 11*a*b^4*d*n^2 + 6*a*b^4*d*n)*x^4 + (b^5*c*n^4 + 40*b^5
*c + 4*(3*b^5*c - a^2*b^3*d)*n^3 + (49*b^5*c - 12*a^2*b^3*d)*n^2 + 2*(39*b^5*c - 4*a^2*b^3*d)*n)*x^3 + (a*b^4*
c*n^4 + 10*a*b^4*c*n^3 + (29*a*b^4*c + 12*a^3*b^2*d)*n^2 + 4*(5*a*b^4*c + 3*a^3*b^2*d)*n)*x^2 - 2*(a^2*b^3*c*n
^3 + 9*a^2*b^3*c*n^2 + 4*(5*a^2*b^3*c + 3*a^4*b*d)*n)*x)*(b*x + a)^n/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*
b^5*n^2 + 274*b^5*n + 120*b^5)
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Sympy [A] time = 10.4669, size = 4078, normalized size = 30.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(b*x+a)**n*(d*x**2+c),x)
[Out]
Piecewise((a**n*(c*x**3/3 + d*x**5/5), Eq(b, 0)), (12*a**6*d*log(a/b + x)/(12*a**6*b**5 + 48*a**5*b**6*x + 72*
a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 7*a**6*d/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b*
*7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 48*a**5*b*d*x*log(a/b + x)/(12*a**6*b**5 + 48*a**5*b**6*x +
72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 16*a**5*b*d*x/(12*a**6*b**5 + 48*a**5*b**6*x + 7
2*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 72*a**4*b**2*d*x**2*log(a/b + x)/(12*a**6*b**5 + 4
8*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 48*a**3*b**3*d*x**3*log(a/b + x)/
(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) - 24*a**3*b**3*d*x
**3/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 12*a**2*b**4
*d*x**4*log(a/b + x)/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**
4) - 18*a**2*b**4*d*x**4/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9
*x**4) + 4*a*b**5*c*x**3/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9
*x**4) + b**6*c*x**4/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**
4), Eq(n, -5)), (-12*a**5*d*log(a/b + x)/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) - 10
*a**5*d/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) - 36*a**4*b*d*x*log(a/b + x)/(3*a**4*
b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) - 18*a**4*b*d*x/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2
*b**7*x**2 + 3*a*b**8*x**3) - 36*a**3*b**2*d*x**2*log(a/b + x)/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2
+ 3*a*b**8*x**3) - 12*a**2*b**3*d*x**3*log(a/b + x)/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**
8*x**3) + 12*a**2*b**3*d*x**3/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) + 3*a*b**4*d*x*
*4/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) + b**5*c*x**3/(3*a**4*b**5 + 9*a**3*b**6*x
+ 9*a**2*b**7*x**2 + 3*a*b**8*x**3), Eq(n, -4)), (12*a**4*d*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x
**2) + 18*a**4*d/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 24*a**3*b*d*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6
*x + 2*b**7*x**2) + 24*a**3*b*d*x/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 2*a**2*b**2*c*log(a/b + x)/(2*a**
2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 3*a**2*b**2*c/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*a**2*b**2*d*x
**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 4*a*b**3*c*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*
x + 2*b**7*x**2) + 4*a*b**3*c*x/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 4*a*b**3*d*x**3/(2*a**2*b**5 + 4*a*
b**6*x + 2*b**7*x**2) + 2*b**4*c*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + b**4*d*x**4/(2*a
**2*b**5 + 4*a*b**6*x + 2*b**7*x**2), Eq(n, -3)), (-12*a**4*d*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 12*a**4*d/(
3*a*b**5 + 3*b**6*x) - 12*a**3*b*d*x*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 6*a**2*b**2*c*log(a/b + x)/(3*a*b**5
+ 3*b**6*x) - 6*a**2*b**2*c/(3*a*b**5 + 3*b**6*x) + 6*a**2*b**2*d*x**2/(3*a*b**5 + 3*b**6*x) - 6*a*b**3*c*x*l
og(a/b + x)/(3*a*b**5 + 3*b**6*x) - 2*a*b**3*d*x**3/(3*a*b**5 + 3*b**6*x) + 3*b**4*c*x**2/(3*a*b**5 + 3*b**6*x
) + b**4*d*x**4/(3*a*b**5 + 3*b**6*x), Eq(n, -2)), (a**4*d*log(a/b + x)/b**5 - a**3*d*x/b**4 + a**2*c*log(a/b
+ x)/b**3 + a**2*d*x**2/(2*b**3) - a*c*x/b**2 - a*d*x**3/(3*b**2) + c*x**2/(2*b) + d*x**4/(4*b), Eq(n, -1)), (
24*a**5*d*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 24*
a**4*b*d*n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) +
2*a**3*b**2*c*n**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b*
*5) + 18*a**3*b**2*c*n*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 12
0*b**5) + 40*a**3*b**2*c*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5) + 12*a**3*b**2*d*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 2
74*b**5*n + 120*b**5) + 12*a**3*b**2*d*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5
*n**2 + 274*b**5*n + 120*b**5) - 2*a**2*b**3*c*n**3*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 +
225*b**5*n**2 + 274*b**5*n + 120*b**5) - 18*a**2*b**3*c*n**2*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**
5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 40*a**2*b**3*c*n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 +
85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 4*a**2*b**3*d*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b
**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 12*a**2*b**3*d*n**2*x**3*(a + b*x)**n/(b**5
*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 8*a**2*b**3*d*n*x**3*(a + b*x)*
*n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + a*b**4*c*n**4*x**2*(a +
b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 10*a*b**4*c*n**3*
x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 29*a*b**
4*c*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) +
20*a*b**4*c*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b
**5) + a*b**4*d*n**4*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5) + 6*a*b**4*d*n**3*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*
b**5*n + 120*b**5) + 11*a*b**4*d*n**2*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n*
*2 + 274*b**5*n + 120*b**5) + 6*a*b**4*d*n*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b*
*5*n**2 + 274*b**5*n + 120*b**5) + b**5*c*n**4*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 22
5*b**5*n**2 + 274*b**5*n + 120*b**5) + 12*b**5*c*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n*
*3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 49*b**5*c*n**2*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*
b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 78*b**5*c*n*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 +
85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 40*b**5*c*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4
+ 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + b**5*d*n**4*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*
n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 10*b**5*d*n**3*x**5*(a + b*x)**n/(b**5*n**5 + 1
5*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 35*b**5*d*n**2*x**5*(a + b*x)**n/(b**5*n
**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 50*b**5*d*n*x**5*(a + b*x)**n/(b*
*5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 24*b**5*d*x**5*(a + b*x)**n/(
b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5), True))
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Giac [B] time = 1.23197, size = 842, normalized size = 6.24 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{5} d n^{4} x^{5} +{\left (b x + a\right )}^{n} a b^{4} d n^{4} x^{4} + 10 \,{\left (b x + a\right )}^{n} b^{5} d n^{3} x^{5} +{\left (b x + a\right )}^{n} b^{5} c n^{4} x^{3} + 6 \,{\left (b x + a\right )}^{n} a b^{4} d n^{3} x^{4} + 35 \,{\left (b x + a\right )}^{n} b^{5} d n^{2} x^{5} +{\left (b x + a\right )}^{n} a b^{4} c n^{4} x^{2} + 12 \,{\left (b x + a\right )}^{n} b^{5} c n^{3} x^{3} - 4 \,{\left (b x + a\right )}^{n} a^{2} b^{3} d n^{3} x^{3} + 11 \,{\left (b x + a\right )}^{n} a b^{4} d n^{2} x^{4} + 50 \,{\left (b x + a\right )}^{n} b^{5} d n x^{5} + 10 \,{\left (b x + a\right )}^{n} a b^{4} c n^{3} x^{2} + 49 \,{\left (b x + a\right )}^{n} b^{5} c n^{2} x^{3} - 12 \,{\left (b x + a\right )}^{n} a^{2} b^{3} d n^{2} x^{3} + 6 \,{\left (b x + a\right )}^{n} a b^{4} d n x^{4} + 24 \,{\left (b x + a\right )}^{n} b^{5} d x^{5} - 2 \,{\left (b x + a\right )}^{n} a^{2} b^{3} c n^{3} x + 29 \,{\left (b x + a\right )}^{n} a b^{4} c n^{2} x^{2} + 12 \,{\left (b x + a\right )}^{n} a^{3} b^{2} d n^{2} x^{2} + 78 \,{\left (b x + a\right )}^{n} b^{5} c n x^{3} - 8 \,{\left (b x + a\right )}^{n} a^{2} b^{3} d n x^{3} - 18 \,{\left (b x + a\right )}^{n} a^{2} b^{3} c n^{2} x + 20 \,{\left (b x + a\right )}^{n} a b^{4} c n x^{2} + 12 \,{\left (b x + a\right )}^{n} a^{3} b^{2} d n x^{2} + 40 \,{\left (b x + a\right )}^{n} b^{5} c x^{3} + 2 \,{\left (b x + a\right )}^{n} a^{3} b^{2} c n^{2} - 40 \,{\left (b x + a\right )}^{n} a^{2} b^{3} c n x - 24 \,{\left (b x + a\right )}^{n} a^{4} b d n x + 18 \,{\left (b x + a\right )}^{n} a^{3} b^{2} c n + 40 \,{\left (b x + a\right )}^{n} a^{3} b^{2} c + 24 \,{\left (b x + a\right )}^{n} a^{5} d}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c),x, algorithm="giac")
[Out]
((b*x + a)^n*b^5*d*n^4*x^5 + (b*x + a)^n*a*b^4*d*n^4*x^4 + 10*(b*x + a)^n*b^5*d*n^3*x^5 + (b*x + a)^n*b^5*c*n^
4*x^3 + 6*(b*x + a)^n*a*b^4*d*n^3*x^4 + 35*(b*x + a)^n*b^5*d*n^2*x^5 + (b*x + a)^n*a*b^4*c*n^4*x^2 + 12*(b*x +
a)^n*b^5*c*n^3*x^3 - 4*(b*x + a)^n*a^2*b^3*d*n^3*x^3 + 11*(b*x + a)^n*a*b^4*d*n^2*x^4 + 50*(b*x + a)^n*b^5*d*
n*x^5 + 10*(b*x + a)^n*a*b^4*c*n^3*x^2 + 49*(b*x + a)^n*b^5*c*n^2*x^3 - 12*(b*x + a)^n*a^2*b^3*d*n^2*x^3 + 6*(
b*x + a)^n*a*b^4*d*n*x^4 + 24*(b*x + a)^n*b^5*d*x^5 - 2*(b*x + a)^n*a^2*b^3*c*n^3*x + 29*(b*x + a)^n*a*b^4*c*n
^2*x^2 + 12*(b*x + a)^n*a^3*b^2*d*n^2*x^2 + 78*(b*x + a)^n*b^5*c*n*x^3 - 8*(b*x + a)^n*a^2*b^3*d*n*x^3 - 18*(b
*x + a)^n*a^2*b^3*c*n^2*x + 20*(b*x + a)^n*a*b^4*c*n*x^2 + 12*(b*x + a)^n*a^3*b^2*d*n*x^2 + 40*(b*x + a)^n*b^5
*c*x^3 + 2*(b*x + a)^n*a^3*b^2*c*n^2 - 40*(b*x + a)^n*a^2*b^3*c*n*x - 24*(b*x + a)^n*a^4*b*d*n*x + 18*(b*x + a
)^n*a^3*b^2*c*n + 40*(b*x + a)^n*a^3*b^2*c + 24*(b*x + a)^n*a^5*d)/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^
5*n^2 + 274*b^5*n + 120*b^5)