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

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

[Out]

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

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Rubi [A]  time = 0.338185, 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 \[ \frac{3 c^2 (d+e x)^7 \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{2 c^2 d (d+e x)^6 \left (3 a e^2+5 c d^2\right )}{3 e^7}+\frac{3 c (d+e x)^5 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{3 c d (d+e x)^4 \left (a e^2+c d^2\right )^2}{2 e^7}+\frac{(d+e x)^3 \left (a e^2+c d^2\right )^3}{3 e^7}+\frac{c^3 (d+e x)^9}{9 e^7}-\frac{3 c^3 d (d+e x)^8}{4 e^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0500313, size = 116, normalized size = 0.61 \[ a^3 \left (d^2 x+d e x^2+\frac{e^2 x^3}{3}\right )+\frac{1}{10} a^2 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+\frac{1}{35} a c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac{1}{252} c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/10 + (a*c^2*x^5*(21*d^2 + 35*d*e*x +
15*e^2*x^2))/35 + (c^3*x^7*(36*d^2 + 63*d*e*x + 28*e^2*x^2))/252 + a^3*(d^2*x +
d*e*x^2 + (e^2*x^3)/3)

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Maple [A]  time = 0.001, size = 129, normalized size = 0.7 \[{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{de{c}^{3}{x}^{8}}{4}}+{\frac{ \left ( 3\,{e}^{2}a{c}^{2}+{d}^{2}{c}^{3} \right ){x}^{7}}{7}}+a{c}^{2}de{x}^{6}+{\frac{ \left ( 3\,{e}^{2}{a}^{2}c+3\,{d}^{2}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{3\,de{a}^{2}c{x}^{4}}{2}}+{\frac{ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ){x}^{3}}{3}}+de{a}^{3}{x}^{2}+{a}^{3}{d}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.697987, size = 170, normalized size = 0.89 \[ \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{4} \, c^{3} d e x^{8} + a c^{2} d e x^{6} + \frac{3}{2} \, a^{2} c d e x^{4} + \frac{1}{7} \,{\left (c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{7} + a^{3} d e x^{2} + a^{3} d^{2} x + \frac{3}{5} \,{\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a^{2} c d^{2} + a^{3} e^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.182362, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{3}{7} x^{7} e^{2} c^{2} a + x^{6} e d c^{2} a + \frac{3}{5} x^{5} d^{2} c^{2} a + \frac{3}{5} x^{5} e^{2} c a^{2} + \frac{3}{2} x^{4} e d c a^{2} + x^{3} d^{2} c a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + x^{2} e d a^{3} + x d^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.171685, size = 139, normalized size = 0.73 \[ a^{3} d^{2} x + a^{3} d e x^{2} + \frac{3 a^{2} c d e x^{4}}{2} + a c^{2} d e x^{6} + \frac{c^{3} d e x^{8}}{4} + \frac{c^{3} e^{2} x^{9}}{9} + x^{7} \left (\frac{3 a c^{2} e^{2}}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{5} \left (\frac{3 a^{2} c e^{2}}{5} + \frac{3 a c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + a^{2} c d^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209536, size = 174, normalized size = 0.92 \[ \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{7} \, a c^{2} x^{7} e^{2} + a c^{2} d x^{6} e + \frac{3}{5} \, a c^{2} d^{2} x^{5} + \frac{3}{5} \, a^{2} c x^{5} e^{2} + \frac{3}{2} \, a^{2} c d x^{4} e + a^{2} c d^{2} x^{3} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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