3.473 \(\int (d+e x)^3 \left (a+c x^2\right )^3 \, dx\)

Optimal. Leaf size=190 \[ \frac{3 c^2 (d+e x)^8 \left (a e^2+5 c d^2\right )}{8 e^7}-\frac{4 c^2 d (d+e x)^7 \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac{c (d+e x)^6 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7}-\frac{6 c d (d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^7}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^3}{4 e^7}+\frac{c^3 (d+e x)^{10}}{10 e^7}-\frac{2 c^3 d (d+e x)^9}{3 e^7} \]

[Out]

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

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Rubi [A]  time = 0.426314, antiderivative size = 190, 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 \[ \frac{3 c^2 (d+e x)^8 \left (a e^2+5 c d^2\right )}{8 e^7}-\frac{4 c^2 d (d+e x)^7 \left (3 a e^2+5 c d^2\right )}{7 e^7}+\frac{c (d+e x)^6 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7}-\frac{6 c d (d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^7}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^3}{4 e^7}+\frac{c^3 (d+e x)^{10}}{10 e^7}-\frac{2 c^3 d (d+e x)^9}{3 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(a + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 a^{3} d^{2} e \int x\, dx + a^{2} d x^{3} \left (a e^{2} + c d^{2}\right ) + \frac{a^{2} e x^{4} \left (a e^{2} + 9 c d^{2}\right )}{4} + \frac{3 a c d x^{5} \left (3 a e^{2} + c d^{2}\right )}{5} + \frac{a c e x^{6} \left (a e^{2} + 3 c d^{2}\right )}{2} + \frac{c^{3} d e^{2} x^{9}}{3} + \frac{c^{3} e^{3} x^{10}}{10} + \frac{c^{2} d x^{7} \left (9 a e^{2} + c d^{2}\right )}{7} + \frac{3 c^{2} e x^{8} \left (a e^{2} + c d^{2}\right )}{8} + d^{3} \int a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**2+a)**3,x)

[Out]

3*a**3*d**2*e*Integral(x, x) + a**2*d*x**3*(a*e**2 + c*d**2) + a**2*e*x**4*(a*e*
*2 + 9*c*d**2)/4 + 3*a*c*d*x**5*(3*a*e**2 + c*d**2)/5 + a*c*e*x**6*(a*e**2 + 3*c
*d**2)/2 + c**3*d*e**2*x**9/3 + c**3*e**3*x**10/10 + c**2*d*x**7*(9*a*e**2 + c*d
**2)/7 + 3*c**2*e*x**8*(a*e**2 + c*d**2)/8 + d**3*Integral(a**3, x)

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Mathematica [A]  time = 0.0958845, size = 155, normalized size = 0.82 \[ \frac{1}{840} x \left (210 a^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+42 a^2 c x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+9 a c^2 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+c^3 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a + c*x^2)^3,x]

[Out]

(x*(210*a^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 42*a^2*c*x^2*(20*d^3 +
45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 9*a*c^2*x^4*(56*d^3 + 140*d^2*e*x + 1
20*d*e^2*x^2 + 35*e^3*x^3) + c^3*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84
*e^3*x^3)))/840

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Maple [A]  time = 0.001, size = 189, normalized size = 1. \[{\frac{{c}^{3}{e}^{3}{x}^{10}}{10}}+{\frac{d{e}^{2}{c}^{3}{x}^{9}}{3}}+{\frac{ \left ( 3\,{e}^{3}a{c}^{2}+3\,{d}^{2}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 9\,d{e}^{2}a{c}^{2}+{c}^{3}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}c{e}^{3}+9\,{d}^{2}ea{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 9\,d{e}^{2}{a}^{2}c+3\,{d}^{3}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{3}+9\,{d}^{2}e{a}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{3}+3\,{d}^{3}{a}^{2}c \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}e{a}^{3}{x}^{2}}{2}}+{a}^{3}{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.698914, size = 244, normalized size = 1.28 \[ \frac{1}{10} \, c^{3} e^{3} x^{10} + \frac{1}{3} \, c^{3} d e^{2} x^{9} + \frac{3}{8} \,{\left (c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{8} + \frac{3}{2} \, a^{3} d^{2} e x^{2} + \frac{1}{7} \,{\left (c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (3 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (9 \, a^{2} c d^{2} e + a^{3} e^{3}\right )} x^{4} +{\left (a^{2} c d^{3} + a^{3} d e^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(e*x + d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.186199, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} e^{3} c^{3} + \frac{1}{3} x^{9} e^{2} d c^{3} + \frac{3}{8} x^{8} e d^{2} c^{3} + \frac{3}{8} x^{8} e^{3} c^{2} a + \frac{1}{7} x^{7} d^{3} c^{3} + \frac{9}{7} x^{7} e^{2} d c^{2} a + \frac{3}{2} x^{6} e d^{2} c^{2} a + \frac{1}{2} x^{6} e^{3} c a^{2} + \frac{3}{5} x^{5} d^{3} c^{2} a + \frac{9}{5} x^{5} e^{2} d c a^{2} + \frac{9}{4} x^{4} e d^{2} c a^{2} + \frac{1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} c a^{2} + x^{3} e^{2} d a^{3} + \frac{3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(e*x + d)^3,x, algorithm="fricas")

[Out]

1/10*x^10*e^3*c^3 + 1/3*x^9*e^2*d*c^3 + 3/8*x^8*e*d^2*c^3 + 3/8*x^8*e^3*c^2*a +
1/7*x^7*d^3*c^3 + 9/7*x^7*e^2*d*c^2*a + 3/2*x^6*e*d^2*c^2*a + 1/2*x^6*e^3*c*a^2
+ 3/5*x^5*d^3*c^2*a + 9/5*x^5*e^2*d*c*a^2 + 9/4*x^4*e*d^2*c*a^2 + 1/4*x^4*e^3*a^
3 + x^3*d^3*c*a^2 + x^3*e^2*d*a^3 + 3/2*x^2*e*d^2*a^3 + x*d^3*a^3

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Sympy [A]  time = 0.192814, size = 202, normalized size = 1.06 \[ a^{3} d^{3} x + \frac{3 a^{3} d^{2} e x^{2}}{2} + \frac{c^{3} d e^{2} x^{9}}{3} + \frac{c^{3} e^{3} x^{10}}{10} + x^{8} \left (\frac{3 a c^{2} e^{3}}{8} + \frac{3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac{9 a c^{2} d e^{2}}{7} + \frac{c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c e^{3}}{2} + \frac{3 a c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{9 a^{2} c d e^{2}}{5} + \frac{3 a c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{3} e^{3}}{4} + \frac{9 a^{2} c d^{2} e}{4}\right ) + x^{3} \left (a^{3} d e^{2} + a^{2} c d^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**2+a)**3,x)

[Out]

a**3*d**3*x + 3*a**3*d**2*e*x**2/2 + c**3*d*e**2*x**9/3 + c**3*e**3*x**10/10 + x
**8*(3*a*c**2*e**3/8 + 3*c**3*d**2*e/8) + x**7*(9*a*c**2*d*e**2/7 + c**3*d**3/7)
+ x**6*(a**2*c*e**3/2 + 3*a*c**2*d**2*e/2) + x**5*(9*a**2*c*d*e**2/5 + 3*a*c**2
*d**3/5) + x**4*(a**3*e**3/4 + 9*a**2*c*d**2*e/4) + x**3*(a**3*d*e**2 + a**2*c*d
**3)

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GIAC/XCAS [A]  time = 0.210279, size = 248, normalized size = 1.31 \[ \frac{1}{10} \, c^{3} x^{10} e^{3} + \frac{1}{3} \, c^{3} d x^{9} e^{2} + \frac{3}{8} \, c^{3} d^{2} x^{8} e + \frac{1}{7} \, c^{3} d^{3} x^{7} + \frac{3}{8} \, a c^{2} x^{8} e^{3} + \frac{9}{7} \, a c^{2} d x^{7} e^{2} + \frac{3}{2} \, a c^{2} d^{2} x^{6} e + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{1}{2} \, a^{2} c x^{6} e^{3} + \frac{9}{5} \, a^{2} c d x^{5} e^{2} + \frac{9}{4} \, a^{2} c d^{2} x^{4} e + a^{2} c d^{3} x^{3} + \frac{1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac{3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(e*x + d)^3,x, algorithm="giac")

[Out]

1/10*c^3*x^10*e^3 + 1/3*c^3*d*x^9*e^2 + 3/8*c^3*d^2*x^8*e + 1/7*c^3*d^3*x^7 + 3/
8*a*c^2*x^8*e^3 + 9/7*a*c^2*d*x^7*e^2 + 3/2*a*c^2*d^2*x^6*e + 3/5*a*c^2*d^3*x^5
+ 1/2*a^2*c*x^6*e^3 + 9/5*a^2*c*d*x^5*e^2 + 9/4*a^2*c*d^2*x^4*e + a^2*c*d^3*x^3
+ 1/4*a^3*x^4*e^3 + a^3*d*x^3*e^2 + 3/2*a^3*d^2*x^2*e + a^3*d^3*x