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

Optimal. Leaf size=115 \[ \frac{2 c (d+e x)^5 \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{c d (d+e x)^4 \left (a e^2+c d^2\right )}{e^5}+\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}-\frac{2 c^2 d (d+e x)^6}{3 e^5} \]

[Out]

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

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Rubi [A]  time = 0.195079, antiderivative size = 115, 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 \[ \frac{2 c (d+e x)^5 \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{c d (d+e x)^4 \left (a e^2+c d^2\right )}{e^5}+\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}-\frac{2 c^2 d (d+e x)^6}{3 e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0259199, size = 91, normalized size = 0.79 \[ a^2 d^2 x+a^2 d e x^2+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+a c d e x^4+\frac{1}{3} c^2 d e x^6+\frac{1}{7} c^2 e^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 88, normalized size = 0.8 \[{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{de{c}^{2}{x}^{6}}{3}}+{\frac{ \left ( 2\,{e}^{2}ac+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+acde{x}^{4}+{\frac{ \left ({a}^{2}{e}^{2}+2\,{d}^{2}ac \right ){x}^{3}}{3}}+de{a}^{2}{x}^{2}+{a}^{2}{d}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.693889, size = 117, normalized size = 1.02 \[ \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac{1}{5} \,{\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{3} \,{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.180352, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{2}{5} x^{5} e^{2} c a + x^{4} e d c a + \frac{2}{3} x^{3} d^{2} c a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} e d a^{2} + x d^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.148985, size = 95, normalized size = 0.83 \[ a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac{c^{2} d e x^{6}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{2 a c d^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210736, size = 120, normalized size = 1.04 \[ \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{2}{5} \, a c x^{5} e^{2} + a c d x^{4} e + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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