3.1906 \(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=146 \[ \frac{e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac{10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac{5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac{5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac{(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac{e^5 (a+b x)^{11}}{11 b^6} \]

[Out]

((b*d - a*e)^5*(a + b*x)^6)/(6*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^7)/(7*b^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)
^8)/(4*b^6) + (10*e^3*(b*d - a*e)^2*(a + b*x)^9)/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^10)/(2*b^6) + (e^5*(a +
b*x)^11)/(11*b^6)

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Rubi [A]  time = 0.250853, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac{10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac{5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac{5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac{(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac{e^5 (a+b x)^{11}}{11 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

((b*d - a*e)^5*(a + b*x)^6)/(6*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^7)/(7*b^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)
^8)/(4*b^6) + (10*e^3*(b*d - a*e)^2*(a + b*x)^9)/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^10)/(2*b^6) + (e^5*(a +
b*x)^11)/(11*b^6)

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 (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^5 \, dx\\ &=\int \left (\frac{(b d-a e)^5 (a+b x)^5}{b^5}+\frac{5 e (b d-a e)^4 (a+b x)^6}{b^5}+\frac{10 e^2 (b d-a e)^3 (a+b x)^7}{b^5}+\frac{10 e^3 (b d-a e)^2 (a+b x)^8}{b^5}+\frac{5 e^4 (b d-a e) (a+b x)^9}{b^5}+\frac{e^5 (a+b x)^{10}}{b^5}\right ) \, dx\\ &=\frac{(b d-a e)^5 (a+b x)^6}{6 b^6}+\frac{5 e (b d-a e)^4 (a+b x)^7}{7 b^6}+\frac{5 e^2 (b d-a e)^3 (a+b x)^8}{4 b^6}+\frac{10 e^3 (b d-a e)^2 (a+b x)^9}{9 b^6}+\frac{e^4 (b d-a e) (a+b x)^{10}}{2 b^6}+\frac{e^5 (a+b x)^{11}}{11 b^6}\\ \end{align*}

Mathematica [B]  time = 0.0504335, size = 413, normalized size = 2.83 \[ \frac{5}{9} b^3 e^3 x^9 \left (2 a^2 e^2+5 a b d e+2 b^2 d^2\right )+\frac{5}{4} b^2 e^2 x^8 \left (5 a^2 b d e^2+a^3 e^3+5 a b^2 d^2 e+b^3 d^3\right )+\frac{5}{7} b e x^7 \left (20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4+10 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{6} x^6 \left (100 a^2 b^3 d^3 e^2+100 a^3 b^2 d^2 e^3+25 a^4 b d e^4+a^5 e^5+25 a b^4 d^4 e+b^5 d^5\right )+a d x^5 \left (20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4+10 a b^3 d^3 e+b^4 d^4\right )+\frac{5}{2} a^2 d^2 x^4 \left (5 a^2 b d e^2+a^3 e^3+5 a b^2 d^2 e+b^3 d^3\right )+\frac{5}{3} a^3 d^3 x^3 \left (2 a^2 e^2+5 a b d e+2 b^2 d^2\right )+\frac{5}{2} a^4 d^4 x^2 (a e+b d)+a^5 d^5 x+\frac{1}{2} b^4 e^4 x^{10} (a e+b d)+\frac{1}{11} b^5 e^5 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^5*d^5*x + (5*a^4*d^4*(b*d + a*e)*x^2)/2 + (5*a^3*d^3*(2*b^2*d^2 + 5*a*b*d*e + 2*a^2*e^2)*x^3)/3 + (5*a^2*d^2
*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3)*x^4)/2 + a*d*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e
^2 + 10*a^3*b*d*e^3 + a^4*e^4)*x^5 + ((b^5*d^5 + 25*a*b^4*d^4*e + 100*a^2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 +
25*a^4*b*d*e^4 + a^5*e^5)*x^6)/6 + (5*b*e*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^
4*e^4)*x^7)/7 + (5*b^2*e^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3)*x^8)/4 + (5*b^3*e^3*(2*b^2*d^2
+ 5*a*b*d*e + 2*a^2*e^2)*x^9)/9 + (b^4*e^4*(b*d + a*e)*x^10)/2 + (b^5*e^5*x^11)/11

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Maple [B]  time = 0.001, size = 688, normalized size = 4.7 \begin{align*}{\frac{{b}^{5}{e}^{5}{x}^{11}}{11}}+{\frac{ \left ( \left ( a{e}^{5}+5\,bd{e}^{4} \right ){b}^{4}+4\,{b}^{4}{e}^{5}a \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){b}^{4}+4\, \left ( a{e}^{5}+5\,bd{e}^{4} \right ) a{b}^{3}+6\,{b}^{3}{e}^{5}{a}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){b}^{4}+4\, \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ) a{b}^{3}+6\, \left ( a{e}^{5}+5\,bd{e}^{4} \right ){a}^{2}{b}^{2}+4\,{b}^{2}{e}^{5}{a}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){b}^{4}+4\, \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ) a{b}^{3}+6\, \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){a}^{2}{b}^{2}+4\, \left ( a{e}^{5}+5\,bd{e}^{4} \right ){a}^{3}b+b{e}^{5}{a}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){b}^{4}+4\, \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ) a{b}^{3}+6\, \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){a}^{3}b+ \left ( a{e}^{5}+5\,bd{e}^{4} \right ){a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( a{d}^{5}{b}^{4}+4\, \left ( 5\,a{d}^{4}e+b{d}^{5} \right ) a{b}^{3}+6\, \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){a}^{2}{b}^{2}+4\, \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){a}^{3}b+ \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}{d}^{5}{b}^{3}+6\, \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){a}^{2}{b}^{2}+4\, \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){a}^{3}b+ \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}{d}^{5}{b}^{2}+4\, \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){a}^{3}b+ \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{4}{d}^{5}b+ \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){a}^{4} \right ){x}^{2}}{2}}+{a}^{5}{d}^{5}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/11*b^5*e^5*x^11+1/10*((a*e^5+5*b*d*e^4)*b^4+4*b^4*e^5*a)*x^10+1/9*((5*a*d*e^4+10*b*d^2*e^3)*b^4+4*(a*e^5+5*b
*d*e^4)*a*b^3+6*b^3*e^5*a^2)*x^9+1/8*((10*a*d^2*e^3+10*b*d^3*e^2)*b^4+4*(5*a*d*e^4+10*b*d^2*e^3)*a*b^3+6*(a*e^
5+5*b*d*e^4)*a^2*b^2+4*b^2*e^5*a^3)*x^8+1/7*((10*a*d^3*e^2+5*b*d^4*e)*b^4+4*(10*a*d^2*e^3+10*b*d^3*e^2)*a*b^3+
6*(5*a*d*e^4+10*b*d^2*e^3)*a^2*b^2+4*(a*e^5+5*b*d*e^4)*a^3*b+b*e^5*a^4)*x^7+1/6*((5*a*d^4*e+b*d^5)*b^4+4*(10*a
*d^3*e^2+5*b*d^4*e)*a*b^3+6*(10*a*d^2*e^3+10*b*d^3*e^2)*a^2*b^2+4*(5*a*d*e^4+10*b*d^2*e^3)*a^3*b+(a*e^5+5*b*d*
e^4)*a^4)*x^6+1/5*(a*d^5*b^4+4*(5*a*d^4*e+b*d^5)*a*b^3+6*(10*a*d^3*e^2+5*b*d^4*e)*a^2*b^2+4*(10*a*d^2*e^3+10*b
*d^3*e^2)*a^3*b+(5*a*d*e^4+10*b*d^2*e^3)*a^4)*x^5+1/4*(4*a^2*d^5*b^3+6*(5*a*d^4*e+b*d^5)*a^2*b^2+4*(10*a*d^3*e
^2+5*b*d^4*e)*a^3*b+(10*a*d^2*e^3+10*b*d^3*e^2)*a^4)*x^4+1/3*(6*a^3*d^5*b^2+4*(5*a*d^4*e+b*d^5)*a^3*b+(10*a*d^
3*e^2+5*b*d^4*e)*a^4)*x^3+1/2*(4*a^4*d^5*b+(5*a*d^4*e+b*d^5)*a^4)*x^2+a^5*d^5*x

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Maxima [B]  time = 1.08735, size = 576, normalized size = 3.95 \begin{align*} \frac{1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac{1}{2} \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac{5}{4} \,{\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac{5}{7} \,{\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} +{\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac{5}{2} \,{\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2*b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*
b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a*b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a
*b^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6*(b^5*d^5 + 25*a*b^4*d^4*e + 100*a^
2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b
^2*d^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2*d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^
2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b*d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*d^4*e)*x^2

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Fricas [B]  time = 1.31352, size = 1075, normalized size = 7.36 \begin{align*} \frac{1}{11} x^{11} e^{5} b^{5} + \frac{1}{2} x^{10} e^{4} d b^{5} + \frac{1}{2} x^{10} e^{5} b^{4} a + \frac{10}{9} x^{9} e^{3} d^{2} b^{5} + \frac{25}{9} x^{9} e^{4} d b^{4} a + \frac{10}{9} x^{9} e^{5} b^{3} a^{2} + \frac{5}{4} x^{8} e^{2} d^{3} b^{5} + \frac{25}{4} x^{8} e^{3} d^{2} b^{4} a + \frac{25}{4} x^{8} e^{4} d b^{3} a^{2} + \frac{5}{4} x^{8} e^{5} b^{2} a^{3} + \frac{5}{7} x^{7} e d^{4} b^{5} + \frac{50}{7} x^{7} e^{2} d^{3} b^{4} a + \frac{100}{7} x^{7} e^{3} d^{2} b^{3} a^{2} + \frac{50}{7} x^{7} e^{4} d b^{2} a^{3} + \frac{5}{7} x^{7} e^{5} b a^{4} + \frac{1}{6} x^{6} d^{5} b^{5} + \frac{25}{6} x^{6} e d^{4} b^{4} a + \frac{50}{3} x^{6} e^{2} d^{3} b^{3} a^{2} + \frac{50}{3} x^{6} e^{3} d^{2} b^{2} a^{3} + \frac{25}{6} x^{6} e^{4} d b a^{4} + \frac{1}{6} x^{6} e^{5} a^{5} + x^{5} d^{5} b^{4} a + 10 x^{5} e d^{4} b^{3} a^{2} + 20 x^{5} e^{2} d^{3} b^{2} a^{3} + 10 x^{5} e^{3} d^{2} b a^{4} + x^{5} e^{4} d a^{5} + \frac{5}{2} x^{4} d^{5} b^{3} a^{2} + \frac{25}{2} x^{4} e d^{4} b^{2} a^{3} + \frac{25}{2} x^{4} e^{2} d^{3} b a^{4} + \frac{5}{2} x^{4} e^{3} d^{2} a^{5} + \frac{10}{3} x^{3} d^{5} b^{2} a^{3} + \frac{25}{3} x^{3} e d^{4} b a^{4} + \frac{10}{3} x^{3} e^{2} d^{3} a^{5} + \frac{5}{2} x^{2} d^{5} b a^{4} + \frac{5}{2} x^{2} e d^{4} a^{5} + x d^{5} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/11*x^11*e^5*b^5 + 1/2*x^10*e^4*d*b^5 + 1/2*x^10*e^5*b^4*a + 10/9*x^9*e^3*d^2*b^5 + 25/9*x^9*e^4*d*b^4*a + 10
/9*x^9*e^5*b^3*a^2 + 5/4*x^8*e^2*d^3*b^5 + 25/4*x^8*e^3*d^2*b^4*a + 25/4*x^8*e^4*d*b^3*a^2 + 5/4*x^8*e^5*b^2*a
^3 + 5/7*x^7*e*d^4*b^5 + 50/7*x^7*e^2*d^3*b^4*a + 100/7*x^7*e^3*d^2*b^3*a^2 + 50/7*x^7*e^4*d*b^2*a^3 + 5/7*x^7
*e^5*b*a^4 + 1/6*x^6*d^5*b^5 + 25/6*x^6*e*d^4*b^4*a + 50/3*x^6*e^2*d^3*b^3*a^2 + 50/3*x^6*e^3*d^2*b^2*a^3 + 25
/6*x^6*e^4*d*b*a^4 + 1/6*x^6*e^5*a^5 + x^5*d^5*b^4*a + 10*x^5*e*d^4*b^3*a^2 + 20*x^5*e^2*d^3*b^2*a^3 + 10*x^5*
e^3*d^2*b*a^4 + x^5*e^4*d*a^5 + 5/2*x^4*d^5*b^3*a^2 + 25/2*x^4*e*d^4*b^2*a^3 + 25/2*x^4*e^2*d^3*b*a^4 + 5/2*x^
4*e^3*d^2*a^5 + 10/3*x^3*d^5*b^2*a^3 + 25/3*x^3*e*d^4*b*a^4 + 10/3*x^3*e^2*d^3*a^5 + 5/2*x^2*d^5*b*a^4 + 5/2*x
^2*e*d^4*a^5 + x*d^5*a^5

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Sympy [B]  time = 0.1293, size = 500, normalized size = 3.42 \begin{align*} a^{5} d^{5} x + \frac{b^{5} e^{5} x^{11}}{11} + x^{10} \left (\frac{a b^{4} e^{5}}{2} + \frac{b^{5} d e^{4}}{2}\right ) + x^{9} \left (\frac{10 a^{2} b^{3} e^{5}}{9} + \frac{25 a b^{4} d e^{4}}{9} + \frac{10 b^{5} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac{5 a^{3} b^{2} e^{5}}{4} + \frac{25 a^{2} b^{3} d e^{4}}{4} + \frac{25 a b^{4} d^{2} e^{3}}{4} + \frac{5 b^{5} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac{5 a^{4} b e^{5}}{7} + \frac{50 a^{3} b^{2} d e^{4}}{7} + \frac{100 a^{2} b^{3} d^{2} e^{3}}{7} + \frac{50 a b^{4} d^{3} e^{2}}{7} + \frac{5 b^{5} d^{4} e}{7}\right ) + x^{6} \left (\frac{a^{5} e^{5}}{6} + \frac{25 a^{4} b d e^{4}}{6} + \frac{50 a^{3} b^{2} d^{2} e^{3}}{3} + \frac{50 a^{2} b^{3} d^{3} e^{2}}{3} + \frac{25 a b^{4} d^{4} e}{6} + \frac{b^{5} d^{5}}{6}\right ) + x^{5} \left (a^{5} d e^{4} + 10 a^{4} b d^{2} e^{3} + 20 a^{3} b^{2} d^{3} e^{2} + 10 a^{2} b^{3} d^{4} e + a b^{4} d^{5}\right ) + x^{4} \left (\frac{5 a^{5} d^{2} e^{3}}{2} + \frac{25 a^{4} b d^{3} e^{2}}{2} + \frac{25 a^{3} b^{2} d^{4} e}{2} + \frac{5 a^{2} b^{3} d^{5}}{2}\right ) + x^{3} \left (\frac{10 a^{5} d^{3} e^{2}}{3} + \frac{25 a^{4} b d^{4} e}{3} + \frac{10 a^{3} b^{2} d^{5}}{3}\right ) + x^{2} \left (\frac{5 a^{5} d^{4} e}{2} + \frac{5 a^{4} b d^{5}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**5*d**5*x + b**5*e**5*x**11/11 + x**10*(a*b**4*e**5/2 + b**5*d*e**4/2) + x**9*(10*a**2*b**3*e**5/9 + 25*a*b*
*4*d*e**4/9 + 10*b**5*d**2*e**3/9) + x**8*(5*a**3*b**2*e**5/4 + 25*a**2*b**3*d*e**4/4 + 25*a*b**4*d**2*e**3/4
+ 5*b**5*d**3*e**2/4) + x**7*(5*a**4*b*e**5/7 + 50*a**3*b**2*d*e**4/7 + 100*a**2*b**3*d**2*e**3/7 + 50*a*b**4*
d**3*e**2/7 + 5*b**5*d**4*e/7) + x**6*(a**5*e**5/6 + 25*a**4*b*d*e**4/6 + 50*a**3*b**2*d**2*e**3/3 + 50*a**2*b
**3*d**3*e**2/3 + 25*a*b**4*d**4*e/6 + b**5*d**5/6) + x**5*(a**5*d*e**4 + 10*a**4*b*d**2*e**3 + 20*a**3*b**2*d
**3*e**2 + 10*a**2*b**3*d**4*e + a*b**4*d**5) + x**4*(5*a**5*d**2*e**3/2 + 25*a**4*b*d**3*e**2/2 + 25*a**3*b**
2*d**4*e/2 + 5*a**2*b**3*d**5/2) + x**3*(10*a**5*d**3*e**2/3 + 25*a**4*b*d**4*e/3 + 10*a**3*b**2*d**5/3) + x**
2*(5*a**5*d**4*e/2 + 5*a**4*b*d**5/2)

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Giac [B]  time = 1.13859, size = 635, normalized size = 4.35 \begin{align*} \frac{1}{11} \, b^{5} x^{11} e^{5} + \frac{1}{2} \, b^{5} d x^{10} e^{4} + \frac{10}{9} \, b^{5} d^{2} x^{9} e^{3} + \frac{5}{4} \, b^{5} d^{3} x^{8} e^{2} + \frac{5}{7} \, b^{5} d^{4} x^{7} e + \frac{1}{6} \, b^{5} d^{5} x^{6} + \frac{1}{2} \, a b^{4} x^{10} e^{5} + \frac{25}{9} \, a b^{4} d x^{9} e^{4} + \frac{25}{4} \, a b^{4} d^{2} x^{8} e^{3} + \frac{50}{7} \, a b^{4} d^{3} x^{7} e^{2} + \frac{25}{6} \, a b^{4} d^{4} x^{6} e + a b^{4} d^{5} x^{5} + \frac{10}{9} \, a^{2} b^{3} x^{9} e^{5} + \frac{25}{4} \, a^{2} b^{3} d x^{8} e^{4} + \frac{100}{7} \, a^{2} b^{3} d^{2} x^{7} e^{3} + \frac{50}{3} \, a^{2} b^{3} d^{3} x^{6} e^{2} + 10 \, a^{2} b^{3} d^{4} x^{5} e + \frac{5}{2} \, a^{2} b^{3} d^{5} x^{4} + \frac{5}{4} \, a^{3} b^{2} x^{8} e^{5} + \frac{50}{7} \, a^{3} b^{2} d x^{7} e^{4} + \frac{50}{3} \, a^{3} b^{2} d^{2} x^{6} e^{3} + 20 \, a^{3} b^{2} d^{3} x^{5} e^{2} + \frac{25}{2} \, a^{3} b^{2} d^{4} x^{4} e + \frac{10}{3} \, a^{3} b^{2} d^{5} x^{3} + \frac{5}{7} \, a^{4} b x^{7} e^{5} + \frac{25}{6} \, a^{4} b d x^{6} e^{4} + 10 \, a^{4} b d^{2} x^{5} e^{3} + \frac{25}{2} \, a^{4} b d^{3} x^{4} e^{2} + \frac{25}{3} \, a^{4} b d^{4} x^{3} e + \frac{5}{2} \, a^{4} b d^{5} x^{2} + \frac{1}{6} \, a^{5} x^{6} e^{5} + a^{5} d x^{5} e^{4} + \frac{5}{2} \, a^{5} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{5} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{5} d^{4} x^{2} e + a^{5} d^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/11*b^5*x^11*e^5 + 1/2*b^5*d*x^10*e^4 + 10/9*b^5*d^2*x^9*e^3 + 5/4*b^5*d^3*x^8*e^2 + 5/7*b^5*d^4*x^7*e + 1/6*
b^5*d^5*x^6 + 1/2*a*b^4*x^10*e^5 + 25/9*a*b^4*d*x^9*e^4 + 25/4*a*b^4*d^2*x^8*e^3 + 50/7*a*b^4*d^3*x^7*e^2 + 25
/6*a*b^4*d^4*x^6*e + a*b^4*d^5*x^5 + 10/9*a^2*b^3*x^9*e^5 + 25/4*a^2*b^3*d*x^8*e^4 + 100/7*a^2*b^3*d^2*x^7*e^3
+ 50/3*a^2*b^3*d^3*x^6*e^2 + 10*a^2*b^3*d^4*x^5*e + 5/2*a^2*b^3*d^5*x^4 + 5/4*a^3*b^2*x^8*e^5 + 50/7*a^3*b^2*
d*x^7*e^4 + 50/3*a^3*b^2*d^2*x^6*e^3 + 20*a^3*b^2*d^3*x^5*e^2 + 25/2*a^3*b^2*d^4*x^4*e + 10/3*a^3*b^2*d^5*x^3
+ 5/7*a^4*b*x^7*e^5 + 25/6*a^4*b*d*x^6*e^4 + 10*a^4*b*d^2*x^5*e^3 + 25/2*a^4*b*d^3*x^4*e^2 + 25/3*a^4*b*d^4*x^
3*e + 5/2*a^4*b*d^5*x^2 + 1/6*a^5*x^6*e^5 + a^5*d*x^5*e^4 + 5/2*a^5*d^2*x^4*e^3 + 10/3*a^5*d^3*x^3*e^2 + 5/2*a
^5*d^4*x^2*e + a^5*d^5*x