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]

1/6*(a*e^2+c*d^2)^4*(e*x+d)^6/e^9-8/7*c*d*(a*e^2+c*d^2)^3*(e*x+d)^7/e^9+1/2*c*(a*e^2+c*d^2)^2*(a*e^2+7*c*d^2)*
(e*x+d)^8/e^9-8/9*c^2*d*(a*e^2+c*d^2)*(3*a*e^2+7*c*d^2)*(e*x+d)^9/e^9+1/5*c^2*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2
*d^4)*(e*x+d)^10/e^9-8/11*c^3*d*(3*a*e^2+7*c*d^2)*(e*x+d)^11/e^9+1/3*c^3*(a*e^2+7*c*d^2)*(e*x+d)^12/e^9-8/13*c
^4*d*(e*x+d)^13/e^9+1/14*c^4*(e*x+d)^14/e^9

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Rubi [A]  time = 0.41, antiderivative size = 278, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.09, size = 307, normalized size = 1.10 \[ \frac {x \left (15015 a^4 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+2145 a^3 c x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+429 a^2 c^2 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+65 a c^3 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )+5 c^4 x^8 \left (2002 d^5+9009 d^4 e x+16380 d^3 e^2 x^2+15015 d^2 e^3 x^3+6930 d e^4 x^4+1287 e^5 x^5\right )\right )}{90090} \]

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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fricas [A]  time = 1.11, size = 397, normalized size = 1.43 \[ \frac {1}{14} x^{14} e^{5} c^{4} + \frac {5}{13} x^{13} e^{4} d c^{4} + \frac {5}{6} x^{12} e^{3} d^{2} c^{4} + \frac {1}{3} x^{12} e^{5} c^{3} a + \frac {10}{11} x^{11} e^{2} d^{3} c^{4} + \frac {20}{11} x^{11} e^{4} d c^{3} a + \frac {1}{2} x^{10} e d^{4} c^{4} + 4 x^{10} e^{3} d^{2} c^{3} a + \frac {3}{5} x^{10} e^{5} c^{2} a^{2} + \frac {1}{9} x^{9} d^{5} c^{4} + \frac {40}{9} x^{9} e^{2} d^{3} c^{3} a + \frac {10}{3} x^{9} e^{4} d c^{2} a^{2} + \frac {5}{2} x^{8} e d^{4} c^{3} a + \frac {15}{2} x^{8} e^{3} d^{2} c^{2} a^{2} + \frac {1}{2} x^{8} e^{5} c a^{3} + \frac {4}{7} x^{7} d^{5} c^{3} a + \frac {60}{7} x^{7} e^{2} d^{3} c^{2} a^{2} + \frac {20}{7} x^{7} e^{4} d c a^{3} + 5 x^{6} e d^{4} c^{2} a^{2} + \frac {20}{3} x^{6} e^{3} d^{2} c a^{3} + \frac {1}{6} x^{6} e^{5} a^{4} + \frac {6}{5} x^{5} d^{5} c^{2} a^{2} + 8 x^{5} e^{2} d^{3} c a^{3} + x^{5} e^{4} d a^{4} + 5 x^{4} e d^{4} c a^{3} + \frac {5}{2} x^{4} e^{3} d^{2} a^{4} + \frac {4}{3} x^{3} d^{5} c a^{3} + \frac {10}{3} x^{3} e^{2} d^{3} a^{4} + \frac {5}{2} x^{2} e d^{4} a^{4} + x d^{5} a^{4} \]

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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giac [A]  time = 0.16, size = 382, normalized size = 1.37 \[ \frac {1}{14} \, c^{4} x^{14} e^{5} + \frac {5}{13} \, c^{4} d x^{13} e^{4} + \frac {5}{6} \, c^{4} d^{2} x^{12} e^{3} + \frac {10}{11} \, c^{4} d^{3} x^{11} e^{2} + \frac {1}{2} \, c^{4} d^{4} x^{10} e + \frac {1}{9} \, c^{4} d^{5} x^{9} + \frac {1}{3} \, a c^{3} x^{12} e^{5} + \frac {20}{11} \, a c^{3} d x^{11} e^{4} + 4 \, a c^{3} d^{2} x^{10} e^{3} + \frac {40}{9} \, a c^{3} d^{3} x^{9} e^{2} + \frac {5}{2} \, a c^{3} d^{4} x^{8} e + \frac {4}{7} \, a c^{3} d^{5} x^{7} + \frac {3}{5} \, a^{2} c^{2} x^{10} e^{5} + \frac {10}{3} \, a^{2} c^{2} d x^{9} e^{4} + \frac {15}{2} \, a^{2} c^{2} d^{2} x^{8} e^{3} + \frac {60}{7} \, a^{2} c^{2} d^{3} x^{7} e^{2} + 5 \, a^{2} c^{2} d^{4} x^{6} e + \frac {6}{5} \, a^{2} c^{2} d^{5} x^{5} + \frac {1}{2} \, a^{3} c x^{8} e^{5} + \frac {20}{7} \, a^{3} c d x^{7} e^{4} + \frac {20}{3} \, a^{3} c d^{2} x^{6} e^{3} + 8 \, a^{3} c d^{3} x^{5} e^{2} + 5 \, a^{3} c d^{4} x^{4} e + \frac {4}{3} \, a^{3} c d^{5} x^{3} + \frac {1}{6} \, a^{4} x^{6} e^{5} + a^{4} d x^{5} e^{4} + \frac {5}{2} \, a^{4} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{4} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{4} d^{4} x^{2} e + a^{4} d^{5} x \]

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

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maple [A]  time = 0.04, size = 379, normalized size = 1.36 \[ \frac {c^{4} e^{5} x^{14}}{14}+\frac {5 c^{4} d \,e^{4} x^{13}}{13}+\frac {\left (4 e^{5} a \,c^{3}+10 d^{2} e^{3} c^{4}\right ) x^{12}}{12}+\frac {5 a^{4} d^{4} e \,x^{2}}{2}+\frac {\left (20 d \,e^{4} a \,c^{3}+10 d^{3} e^{2} c^{4}\right ) x^{11}}{11}+a^{4} d^{5} x +\frac {\left (6 e^{5} a^{2} c^{2}+40 d^{2} e^{3} a \,c^{3}+5 d^{4} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (30 d \,e^{4} a^{2} c^{2}+40 d^{3} e^{2} a \,c^{3}+c^{4} d^{5}\right ) x^{9}}{9}+\frac {\left (4 e^{5} a^{3} c +60 d^{2} e^{3} a^{2} c^{2}+20 d^{4} e a \,c^{3}\right ) x^{8}}{8}+\frac {\left (20 d \,e^{4} a^{3} c +60 d^{3} e^{2} a^{2} c^{2}+4 d^{5} a \,c^{3}\right ) x^{7}}{7}+\frac {\left (e^{5} a^{4}+40 d^{2} e^{3} a^{3} c +30 d^{4} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{4}+40 d^{3} e^{2} a^{3} c +6 d^{5} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{4}+20 d^{4} e \,a^{3} c \right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{4}+4 d^{5} a^{3} c \right ) x^{3}}{3} \]

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 = 1.35, size = 374, normalized size = 1.35 \[ \frac {1}{14} \, c^{4} e^{5} x^{14} + \frac {5}{13} \, c^{4} d e^{4} x^{13} + \frac {1}{6} \, {\left (5 \, c^{4} d^{2} e^{3} + 2 \, a c^{3} e^{5}\right )} x^{12} + \frac {10}{11} \, {\left (c^{4} d^{3} e^{2} + 2 \, a c^{3} d e^{4}\right )} x^{11} + \frac {5}{2} \, a^{4} d^{4} e x^{2} + \frac {1}{10} \, {\left (5 \, c^{4} d^{4} e + 40 \, a c^{3} d^{2} e^{3} + 6 \, a^{2} c^{2} e^{5}\right )} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (c^{4} d^{5} + 40 \, a c^{3} d^{3} e^{2} + 30 \, a^{2} c^{2} d e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, a c^{3} d^{4} e + 15 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{8} + \frac {4}{7} \, {\left (a c^{3} d^{5} + 15 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (30 \, a^{2} c^{2} d^{4} e + 40 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{5} + 40 \, a^{3} c d^{3} e^{2} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} c d^{4} e + a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} c d^{5} + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} \]

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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mupad [B]  time = 0.17, size = 357, normalized size = 1.28 \[ x^3\,\left (\frac {10\,a^4\,d^3\,e^2}{3}+\frac {4\,c\,a^3\,d^5}{3}\right )+x^{12}\,\left (\frac {5\,c^4\,d^2\,e^3}{6}+\frac {a\,c^3\,e^5}{3}\right )+x^5\,\left (a^4\,d\,e^4+8\,a^3\,c\,d^3\,e^2+\frac {6\,a^2\,c^2\,d^5}{5}\right )+x^6\,\left (\frac {a^4\,e^5}{6}+\frac {20\,a^3\,c\,d^2\,e^3}{3}+5\,a^2\,c^2\,d^4\,e\right )+x^9\,\left (\frac {10\,a^2\,c^2\,d\,e^4}{3}+\frac {40\,a\,c^3\,d^3\,e^2}{9}+\frac {c^4\,d^5}{9}\right )+x^{10}\,\left (\frac {3\,a^2\,c^2\,e^5}{5}+4\,a\,c^3\,d^2\,e^3+\frac {c^4\,d^4\,e}{2}\right )+a^4\,d^5\,x+\frac {c^4\,e^5\,x^{14}}{14}+\frac {5\,a^4\,d^4\,e\,x^2}{2}+\frac {5\,c^4\,d\,e^4\,x^{13}}{13}+\frac {4\,a\,c\,d\,x^7\,\left (5\,a^2\,e^4+15\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{7}+\frac {a\,c\,e\,x^8\,\left (a^2\,e^4+15\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{2}+\frac {5\,a^3\,d^2\,e\,x^4\,\left (2\,c\,d^2+a\,e^2\right )}{2}+\frac {10\,c^3\,d\,e^2\,x^{11}\,\left (c\,d^2+2\,a\,e^2\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^4*(d + e*x)^5,x)

[Out]

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

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sympy [A]  time = 0.14, size = 418, normalized size = 1.50 \[ a^{4} d^{5} x + \frac {5 a^{4} d^{4} e x^{2}}{2} + \frac {5 c^{4} d e^{4} x^{13}}{13} + \frac {c^{4} e^{5} x^{14}}{14} + x^{12} \left (\frac {a c^{3} e^{5}}{3} + \frac {5 c^{4} d^{2} e^{3}}{6}\right ) + x^{11} \left (\frac {20 a c^{3} d e^{4}}{11} + \frac {10 c^{4} d^{3} e^{2}}{11}\right ) + x^{10} \left (\frac {3 a^{2} c^{2} e^{5}}{5} + 4 a c^{3} d^{2} e^{3} + \frac {c^{4} d^{4} e}{2}\right ) + x^{9} \left (\frac {10 a^{2} c^{2} d e^{4}}{3} + \frac {40 a c^{3} d^{3} e^{2}}{9} + \frac {c^{4} d^{5}}{9}\right ) + x^{8} \left (\frac {a^{3} c e^{5}}{2} + \frac {15 a^{2} c^{2} d^{2} e^{3}}{2} + \frac {5 a c^{3} d^{4} e}{2}\right ) + x^{7} \left (\frac {20 a^{3} c d e^{4}}{7} + \frac {60 a^{2} c^{2} d^{3} e^{2}}{7} + \frac {4 a c^{3} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{4} e^{5}}{6} + \frac {20 a^{3} c d^{2} e^{3}}{3} + 5 a^{2} c^{2} d^{4} e\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} c d^{3} e^{2} + \frac {6 a^{2} c^{2} d^{5}}{5}\right ) + x^{4} \left (\frac {5 a^{4} d^{2} e^{3}}{2} + 5 a^{3} c d^{4} e\right ) + x^{3} \left (\frac {10 a^{4} d^{3} e^{2}}{3} + \frac {4 a^{3} c d^{5}}{3}\right ) \]

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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