### 3.490 $$\int (d+e x)^5 (a+c x^2)^4 \, dx$$

Optimal. Leaf size=278 $\frac{c^2 (d+e x)^{10} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{5 e^9}+\frac{c^3 (d+e x)^{12} \left (a e^2+7 c d^2\right )}{3 e^9}-\frac{8 c^3 d (d+e x)^{11} \left (3 a e^2+7 c d^2\right )}{11 e^9}-\frac{8 c^2 d (d+e x)^9 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{9 e^9}+\frac{c (d+e x)^8 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{2 e^9}-\frac{8 c d (d+e x)^7 \left (a e^2+c d^2\right )^3}{7 e^9}+\frac{(d+e x)^6 \left (a e^2+c d^2\right )^4}{6 e^9}+\frac{c^4 (d+e x)^{14}}{14 e^9}-\frac{8 c^4 d (d+e x)^{13}}{13 e^9}$

[Out]

((c*d^2 + a*e^2)^4*(d + e*x)^6)/(6*e^9) - (8*c*d*(c*d^2 + a*e^2)^3*(d + e*x)^7)/(7*e^9) + (c*(c*d^2 + a*e^2)^2
*(7*c*d^2 + a*e^2)*(d + e*x)^8)/(2*e^9) - (8*c^2*d*(c*d^2 + a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^9) +
(c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^10)/(5*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^
11)/(11*e^9) + (c^3*(7*c*d^2 + a*e^2)*(d + e*x)^12)/(3*e^9) - (8*c^4*d*(d + e*x)^13)/(13*e^9) + (c^4*(d + e*x)
^14)/(14*e^9)

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Rubi [A]  time = 0.406954, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $\frac{c^2 (d+e x)^{10} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{5 e^9}+\frac{c^3 (d+e x)^{12} \left (a e^2+7 c d^2\right )}{3 e^9}-\frac{8 c^3 d (d+e x)^{11} \left (3 a e^2+7 c d^2\right )}{11 e^9}-\frac{8 c^2 d (d+e x)^9 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{9 e^9}+\frac{c (d+e x)^8 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{2 e^9}-\frac{8 c d (d+e x)^7 \left (a e^2+c d^2\right )^3}{7 e^9}+\frac{(d+e x)^6 \left (a e^2+c d^2\right )^4}{6 e^9}+\frac{c^4 (d+e x)^{14}}{14 e^9}-\frac{8 c^4 d (d+e x)^{13}}{13 e^9}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5*(a + c*x^2)^4,x]

[Out]

((c*d^2 + a*e^2)^4*(d + e*x)^6)/(6*e^9) - (8*c*d*(c*d^2 + a*e^2)^3*(d + e*x)^7)/(7*e^9) + (c*(c*d^2 + a*e^2)^2
*(7*c*d^2 + a*e^2)*(d + e*x)^8)/(2*e^9) - (8*c^2*d*(c*d^2 + a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^9) +
(c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^10)/(5*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^
11)/(11*e^9) + (c^3*(7*c*d^2 + a*e^2)*(d + e*x)^12)/(3*e^9) - (8*c^4*d*(d + e*x)^13)/(13*e^9) + (c^4*(d + e*x)
^14)/(14*e^9)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^4 (d+e x)^5}{e^8}-\frac{8 c d \left (c d^2+a e^2\right )^3 (d+e x)^6}{e^8}+\frac{4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^7}{e^8}+\frac{8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^8}{e^8}+\frac{2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^9}{e^8}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{10}}{e^8}+\frac{4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{11}}{e^8}-\frac{8 c^4 d (d+e x)^{12}}{e^8}+\frac{c^4 (d+e x)^{13}}{e^8}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^4 (d+e x)^6}{6 e^9}-\frac{8 c d \left (c d^2+a e^2\right )^3 (d+e x)^7}{7 e^9}+\frac{c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^8}{2 e^9}-\frac{8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^9}{9 e^9}+\frac{c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{10}}{5 e^9}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{11}}{11 e^9}+\frac{c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{12}}{3 e^9}-\frac{8 c^4 d (d+e x)^{13}}{13 e^9}+\frac{c^4 (d+e x)^{14}}{14 e^9}\\ \end{align*}

Mathematica [A]  time = 0.0963532, size = 307, normalized size = 1.1 $\frac{x \left (429 a^2 c^2 x^4 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )+2145 a^3 c x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+15015 a^4 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+65 a c^3 x^6 \left (6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+3465 d^4 e x+792 d^5+2520 d e^4 x^4+462 e^5 x^5\right )+5 c^4 x^8 \left (16380 d^3 e^2 x^2+15015 d^2 e^3 x^3+9009 d^4 e x+2002 d^5+6930 d e^4 x^4+1287 e^5 x^5\right )\right )}{90090}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5*(a + c*x^2)^4,x]

[Out]

(x*(15015*a^4*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 2145*a^3*c*x^2*
(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 429*a^2*c^2*x^4*(252
*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 65*a*c^3*x^6*(792*d
^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + 5*c^4*x^8*(2002*d^5
+ 9009*d^4*e*x + 16380*d^3*e^2*x^2 + 15015*d^2*e^3*x^3 + 6930*d*e^4*x^4 + 1287*e^5*x^5)))/90090

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Maple [A]  time = 0.05, size = 379, normalized size = 1.4 \begin{align*}{\frac{{e}^{5}{c}^{4}{x}^{14}}{14}}+{\frac{5\,d{e}^{4}{c}^{4}{x}^{13}}{13}}+{\frac{ \left ( 4\,{e}^{5}a{c}^{3}+10\,{d}^{2}{e}^{3}{c}^{4} \right ){x}^{12}}{12}}+{\frac{ \left ( 20\,d{e}^{4}a{c}^{3}+10\,{d}^{3}{e}^{2}{c}^{4} \right ){x}^{11}}{11}}+{\frac{ \left ( 6\,{e}^{5}{a}^{2}{c}^{2}+40\,{d}^{2}{e}^{3}a{c}^{3}+5\,{d}^{4}e{c}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 30\,d{e}^{4}{a}^{2}{c}^{2}+40\,{d}^{3}{e}^{2}a{c}^{3}+{c}^{4}{d}^{5} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,{e}^{5}{a}^{3}c+60\,{d}^{2}{e}^{3}{a}^{2}{c}^{2}+20\,{d}^{4}ea{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 20\,d{e}^{4}{a}^{3}c+60\,{d}^{3}{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{5}a{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{5}{a}^{4}+40\,{d}^{2}{e}^{3}{a}^{3}c+30\,{d}^{4}e{a}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,d{e}^{4}{a}^{4}+40\,{d}^{3}{e}^{2}{a}^{3}c+6\,{d}^{5}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}{a}^{4}+20\,{d}^{4}e{a}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}{a}^{4}+4\,{d}^{5}{a}^{3}c \right ){x}^{3}}{3}}+{\frac{5\,{d}^{4}e{a}^{4}{x}^{2}}{2}}+{d}^{5}{a}^{4}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(c*x^2+a)^4,x)

[Out]

1/14*e^5*c^4*x^14+5/13*d*e^4*c^4*x^13+1/12*(4*a*c^3*e^5+10*c^4*d^2*e^3)*x^12+1/11*(20*a*c^3*d*e^4+10*c^4*d^3*e
^2)*x^11+1/10*(6*a^2*c^2*e^5+40*a*c^3*d^2*e^3+5*c^4*d^4*e)*x^10+1/9*(30*a^2*c^2*d*e^4+40*a*c^3*d^3*e^2+c^4*d^5
)*x^9+1/8*(4*a^3*c*e^5+60*a^2*c^2*d^2*e^3+20*a*c^3*d^4*e)*x^8+1/7*(20*a^3*c*d*e^4+60*a^2*c^2*d^3*e^2+4*a*c^3*d
^5)*x^7+1/6*(a^4*e^5+40*a^3*c*d^2*e^3+30*a^2*c^2*d^4*e)*x^6+1/5*(5*a^4*d*e^4+40*a^3*c*d^3*e^2+6*a^2*c^2*d^5)*x
^5+1/4*(10*a^4*d^2*e^3+20*a^3*c*d^4*e)*x^4+1/3*(10*a^4*d^3*e^2+4*a^3*c*d^5)*x^3+5/2*d^4*e*a^4*x^2+d^5*a^4*x

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Maxima [A]  time = 1.20822, size = 505, normalized size = 1.82 \begin{align*} \frac{1}{14} \, c^{4} e^{5} x^{14} + \frac{5}{13} \, c^{4} d e^{4} x^{13} + \frac{1}{6} \,{\left (5 \, c^{4} d^{2} e^{3} + 2 \, a c^{3} e^{5}\right )} x^{12} + \frac{10}{11} \,{\left (c^{4} d^{3} e^{2} + 2 \, a c^{3} d e^{4}\right )} x^{11} + \frac{5}{2} \, a^{4} d^{4} e x^{2} + \frac{1}{10} \,{\left (5 \, c^{4} d^{4} e + 40 \, a c^{3} d^{2} e^{3} + 6 \, a^{2} c^{2} e^{5}\right )} x^{10} + a^{4} d^{5} x + \frac{1}{9} \,{\left (c^{4} d^{5} + 40 \, a c^{3} d^{3} e^{2} + 30 \, a^{2} c^{2} d e^{4}\right )} x^{9} + \frac{1}{2} \,{\left (5 \, a c^{3} d^{4} e + 15 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{8} + \frac{4}{7} \,{\left (a c^{3} d^{5} + 15 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (30 \, a^{2} c^{2} d^{4} e + 40 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, a^{2} c^{2} d^{5} + 40 \, a^{3} c d^{3} e^{2} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (2 \, a^{3} c d^{4} e + a^{4} d^{2} e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, a^{3} c d^{5} + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

1/14*c^4*e^5*x^14 + 5/13*c^4*d*e^4*x^13 + 1/6*(5*c^4*d^2*e^3 + 2*a*c^3*e^5)*x^12 + 10/11*(c^4*d^3*e^2 + 2*a*c^
3*d*e^4)*x^11 + 5/2*a^4*d^4*e*x^2 + 1/10*(5*c^4*d^4*e + 40*a*c^3*d^2*e^3 + 6*a^2*c^2*e^5)*x^10 + a^4*d^5*x + 1
/9*(c^4*d^5 + 40*a*c^3*d^3*e^2 + 30*a^2*c^2*d*e^4)*x^9 + 1/2*(5*a*c^3*d^4*e + 15*a^2*c^2*d^2*e^3 + a^3*c*e^5)*
x^8 + 4/7*(a*c^3*d^5 + 15*a^2*c^2*d^3*e^2 + 5*a^3*c*d*e^4)*x^7 + 1/6*(30*a^2*c^2*d^4*e + 40*a^3*c*d^2*e^3 + a^
4*e^5)*x^6 + 1/5*(6*a^2*c^2*d^5 + 40*a^3*c*d^3*e^2 + 5*a^4*d*e^4)*x^5 + 5/2*(2*a^3*c*d^4*e + a^4*d^2*e^3)*x^4
+ 2/3*(2*a^3*c*d^5 + 5*a^4*d^3*e^2)*x^3

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Fricas [A]  time = 1.75772, size = 879, normalized size = 3.16 \begin{align*} \frac{1}{14} x^{14} e^{5} c^{4} + \frac{5}{13} x^{13} e^{4} d c^{4} + \frac{5}{6} x^{12} e^{3} d^{2} c^{4} + \frac{1}{3} x^{12} e^{5} c^{3} a + \frac{10}{11} x^{11} e^{2} d^{3} c^{4} + \frac{20}{11} x^{11} e^{4} d c^{3} a + \frac{1}{2} x^{10} e d^{4} c^{4} + 4 x^{10} e^{3} d^{2} c^{3} a + \frac{3}{5} x^{10} e^{5} c^{2} a^{2} + \frac{1}{9} x^{9} d^{5} c^{4} + \frac{40}{9} x^{9} e^{2} d^{3} c^{3} a + \frac{10}{3} x^{9} e^{4} d c^{2} a^{2} + \frac{5}{2} x^{8} e d^{4} c^{3} a + \frac{15}{2} x^{8} e^{3} d^{2} c^{2} a^{2} + \frac{1}{2} x^{8} e^{5} c a^{3} + \frac{4}{7} x^{7} d^{5} c^{3} a + \frac{60}{7} x^{7} e^{2} d^{3} c^{2} a^{2} + \frac{20}{7} x^{7} e^{4} d c a^{3} + 5 x^{6} e d^{4} c^{2} a^{2} + \frac{20}{3} x^{6} e^{3} d^{2} c a^{3} + \frac{1}{6} x^{6} e^{5} a^{4} + \frac{6}{5} x^{5} d^{5} c^{2} a^{2} + 8 x^{5} e^{2} d^{3} c a^{3} + x^{5} e^{4} d a^{4} + 5 x^{4} e d^{4} c a^{3} + \frac{5}{2} x^{4} e^{3} d^{2} a^{4} + \frac{4}{3} x^{3} d^{5} c a^{3} + \frac{10}{3} x^{3} e^{2} d^{3} a^{4} + \frac{5}{2} x^{2} e d^{4} a^{4} + x d^{5} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/14*x^14*e^5*c^4 + 5/13*x^13*e^4*d*c^4 + 5/6*x^12*e^3*d^2*c^4 + 1/3*x^12*e^5*c^3*a + 10/11*x^11*e^2*d^3*c^4 +
20/11*x^11*e^4*d*c^3*a + 1/2*x^10*e*d^4*c^4 + 4*x^10*e^3*d^2*c^3*a + 3/5*x^10*e^5*c^2*a^2 + 1/9*x^9*d^5*c^4 +
40/9*x^9*e^2*d^3*c^3*a + 10/3*x^9*e^4*d*c^2*a^2 + 5/2*x^8*e*d^4*c^3*a + 15/2*x^8*e^3*d^2*c^2*a^2 + 1/2*x^8*e^
5*c*a^3 + 4/7*x^7*d^5*c^3*a + 60/7*x^7*e^2*d^3*c^2*a^2 + 20/7*x^7*e^4*d*c*a^3 + 5*x^6*e*d^4*c^2*a^2 + 20/3*x^6
*e^3*d^2*c*a^3 + 1/6*x^6*e^5*a^4 + 6/5*x^5*d^5*c^2*a^2 + 8*x^5*e^2*d^3*c*a^3 + x^5*e^4*d*a^4 + 5*x^4*e*d^4*c*a
^3 + 5/2*x^4*e^3*d^2*a^4 + 4/3*x^3*d^5*c*a^3 + 10/3*x^3*e^2*d^3*a^4 + 5/2*x^2*e*d^4*a^4 + x*d^5*a^4

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Sympy [A]  time = 0.124665, size = 418, normalized size = 1.5 \begin{align*} a^{4} d^{5} x + \frac{5 a^{4} d^{4} e x^{2}}{2} + \frac{5 c^{4} d e^{4} x^{13}}{13} + \frac{c^{4} e^{5} x^{14}}{14} + x^{12} \left (\frac{a c^{3} e^{5}}{3} + \frac{5 c^{4} d^{2} e^{3}}{6}\right ) + x^{11} \left (\frac{20 a c^{3} d e^{4}}{11} + \frac{10 c^{4} d^{3} e^{2}}{11}\right ) + x^{10} \left (\frac{3 a^{2} c^{2} e^{5}}{5} + 4 a c^{3} d^{2} e^{3} + \frac{c^{4} d^{4} e}{2}\right ) + x^{9} \left (\frac{10 a^{2} c^{2} d e^{4}}{3} + \frac{40 a c^{3} d^{3} e^{2}}{9} + \frac{c^{4} d^{5}}{9}\right ) + x^{8} \left (\frac{a^{3} c e^{5}}{2} + \frac{15 a^{2} c^{2} d^{2} e^{3}}{2} + \frac{5 a c^{3} d^{4} e}{2}\right ) + x^{7} \left (\frac{20 a^{3} c d e^{4}}{7} + \frac{60 a^{2} c^{2} d^{3} e^{2}}{7} + \frac{4 a c^{3} d^{5}}{7}\right ) + x^{6} \left (\frac{a^{4} e^{5}}{6} + \frac{20 a^{3} c d^{2} e^{3}}{3} + 5 a^{2} c^{2} d^{4} e\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} c d^{3} e^{2} + \frac{6 a^{2} c^{2} d^{5}}{5}\right ) + x^{4} \left (\frac{5 a^{4} d^{2} e^{3}}{2} + 5 a^{3} c d^{4} e\right ) + x^{3} \left (\frac{10 a^{4} d^{3} e^{2}}{3} + \frac{4 a^{3} c d^{5}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(c*x**2+a)**4,x)

[Out]

a**4*d**5*x + 5*a**4*d**4*e*x**2/2 + 5*c**4*d*e**4*x**13/13 + c**4*e**5*x**14/14 + x**12*(a*c**3*e**5/3 + 5*c*
*4*d**2*e**3/6) + x**11*(20*a*c**3*d*e**4/11 + 10*c**4*d**3*e**2/11) + x**10*(3*a**2*c**2*e**5/5 + 4*a*c**3*d*
*2*e**3 + c**4*d**4*e/2) + x**9*(10*a**2*c**2*d*e**4/3 + 40*a*c**3*d**3*e**2/9 + c**4*d**5/9) + x**8*(a**3*c*e
**5/2 + 15*a**2*c**2*d**2*e**3/2 + 5*a*c**3*d**4*e/2) + x**7*(20*a**3*c*d*e**4/7 + 60*a**2*c**2*d**3*e**2/7 +
4*a*c**3*d**5/7) + x**6*(a**4*e**5/6 + 20*a**3*c*d**2*e**3/3 + 5*a**2*c**2*d**4*e) + x**5*(a**4*d*e**4 + 8*a**
3*c*d**3*e**2 + 6*a**2*c**2*d**5/5) + x**4*(5*a**4*d**2*e**3/2 + 5*a**3*c*d**4*e) + x**3*(10*a**4*d**3*e**2/3
+ 4*a**3*c*d**5/3)

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Giac [A]  time = 1.2696, size = 516, normalized size = 1.86 \begin{align*} \frac{1}{14} \, c^{4} x^{14} e^{5} + \frac{5}{13} \, c^{4} d x^{13} e^{4} + \frac{5}{6} \, c^{4} d^{2} x^{12} e^{3} + \frac{10}{11} \, c^{4} d^{3} x^{11} e^{2} + \frac{1}{2} \, c^{4} d^{4} x^{10} e + \frac{1}{9} \, c^{4} d^{5} x^{9} + \frac{1}{3} \, a c^{3} x^{12} e^{5} + \frac{20}{11} \, a c^{3} d x^{11} e^{4} + 4 \, a c^{3} d^{2} x^{10} e^{3} + \frac{40}{9} \, a c^{3} d^{3} x^{9} e^{2} + \frac{5}{2} \, a c^{3} d^{4} x^{8} e + \frac{4}{7} \, a c^{3} d^{5} x^{7} + \frac{3}{5} \, a^{2} c^{2} x^{10} e^{5} + \frac{10}{3} \, a^{2} c^{2} d x^{9} e^{4} + \frac{15}{2} \, a^{2} c^{2} d^{2} x^{8} e^{3} + \frac{60}{7} \, a^{2} c^{2} d^{3} x^{7} e^{2} + 5 \, a^{2} c^{2} d^{4} x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{5} x^{5} + \frac{1}{2} \, a^{3} c x^{8} e^{5} + \frac{20}{7} \, a^{3} c d x^{7} e^{4} + \frac{20}{3} \, a^{3} c d^{2} x^{6} e^{3} + 8 \, a^{3} c d^{3} x^{5} e^{2} + 5 \, a^{3} c d^{4} x^{4} e + \frac{4}{3} \, a^{3} c d^{5} x^{3} + \frac{1}{6} \, a^{4} x^{6} e^{5} + a^{4} d x^{5} e^{4} + \frac{5}{2} \, a^{4} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{4} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{4} d^{4} x^{2} e + a^{4} d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(c*x^2+a)^4,x, algorithm="giac")

[Out]

1/14*c^4*x^14*e^5 + 5/13*c^4*d*x^13*e^4 + 5/6*c^4*d^2*x^12*e^3 + 10/11*c^4*d^3*x^11*e^2 + 1/2*c^4*d^4*x^10*e +
1/9*c^4*d^5*x^9 + 1/3*a*c^3*x^12*e^5 + 20/11*a*c^3*d*x^11*e^4 + 4*a*c^3*d^2*x^10*e^3 + 40/9*a*c^3*d^3*x^9*e^2
+ 5/2*a*c^3*d^4*x^8*e + 4/7*a*c^3*d^5*x^7 + 3/5*a^2*c^2*x^10*e^5 + 10/3*a^2*c^2*d*x^9*e^4 + 15/2*a^2*c^2*d^2*
x^8*e^3 + 60/7*a^2*c^2*d^3*x^7*e^2 + 5*a^2*c^2*d^4*x^6*e + 6/5*a^2*c^2*d^5*x^5 + 1/2*a^3*c*x^8*e^5 + 20/7*a^3*
c*d*x^7*e^4 + 20/3*a^3*c*d^2*x^6*e^3 + 8*a^3*c*d^3*x^5*e^2 + 5*a^3*c*d^4*x^4*e + 4/3*a^3*c*d^5*x^3 + 1/6*a^4*x
^6*e^5 + a^4*d*x^5*e^4 + 5/2*a^4*d^2*x^4*e^3 + 10/3*a^4*d^3*x^3*e^2 + 5/2*a^4*d^4*x^2*e + a^4*d^5*x