3.1301 \(\int (A+B x) (d+e x) \left (a+c x^2\right )^2 \, dx\)

Optimal. Leaf size=106 \[ \frac{1}{2} a^2 x^2 (A e+B d)+a^2 A d x+\frac{1}{5} c x^5 (2 a B e+A c d)+\frac{1}{2} a c x^4 (A e+B d)+\frac{1}{3} a x^3 (a B e+2 A c d)+\frac{1}{6} c^2 x^6 (A e+B d)+\frac{1}{7} B c^2 e x^7 \]

[Out]

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

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Rubi [A]  time = 0.266123, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1}{2} a^2 x^2 (A e+B d)+a^2 A d x+\frac{1}{5} c x^5 (2 a B e+A c d)+\frac{1}{2} a c x^4 (A e+B d)+\frac{1}{3} a x^3 (a B e+2 A c d)+\frac{1}{6} c^2 x^6 (A e+B d)+\frac{1}{7} B c^2 e x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0845411, size = 95, normalized size = 0.9 \[ \frac{1}{210} x \left (35 a^2 (3 A (2 d+e x)+B x (3 d+2 e x))+7 a c x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+c^2 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(35*a^2*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)) + 7*a*c*x^2*(5*A*(4*d + 3*e*x)
+ 3*B*x*(5*d + 4*e*x)) + c^2*x^4*(7*A*(6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x))))/210

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Maple [A]  time = 0.001, size = 99, normalized size = 0.9 \[{\frac{B{c}^{2}e{x}^{7}}{7}}+{\frac{{c}^{2} \left ( Ae+Bd \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}d+2\,Beac \right ){x}^{5}}{5}}+{\frac{ac \left ( Ae+Bd \right ){x}^{4}}{2}}+{\frac{ \left ( 2\,Adac+Be{a}^{2} \right ){x}^{3}}{3}}+{\frac{{a}^{2} \left ( Ae+Bd \right ){x}^{2}}{2}}+{a}^{2}Adx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)*(c*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.696954, size = 143, normalized size = 1.35 \[ \frac{1}{7} \, B c^{2} e x^{7} + \frac{1}{6} \,{\left (B c^{2} d + A c^{2} e\right )} x^{6} + \frac{1}{5} \,{\left (A c^{2} d + 2 \, B a c e\right )} x^{5} + A a^{2} d x + \frac{1}{2} \,{\left (B a c d + A a c e\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A a c d + B a^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d + A a^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244264, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e c^{2} B + \frac{1}{6} x^{6} d c^{2} B + \frac{1}{6} x^{6} e c^{2} A + \frac{2}{5} x^{5} e c a B + \frac{1}{5} x^{5} d c^{2} A + \frac{1}{2} x^{4} d c a B + \frac{1}{2} x^{4} e c a A + \frac{1}{3} x^{3} e a^{2} B + \frac{2}{3} x^{3} d c a A + \frac{1}{2} x^{2} d a^{2} B + \frac{1}{2} x^{2} e a^{2} A + x d a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.157717, size = 124, normalized size = 1.17 \[ A a^{2} d x + \frac{B c^{2} e x^{7}}{7} + x^{6} \left (\frac{A c^{2} e}{6} + \frac{B c^{2} d}{6}\right ) + x^{5} \left (\frac{A c^{2} d}{5} + \frac{2 B a c e}{5}\right ) + x^{4} \left (\frac{A a c e}{2} + \frac{B a c d}{2}\right ) + x^{3} \left (\frac{2 A a c d}{3} + \frac{B a^{2} e}{3}\right ) + x^{2} \left (\frac{A a^{2} e}{2} + \frac{B a^{2} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.281462, size = 162, normalized size = 1.53 \[ \frac{1}{7} \, B c^{2} x^{7} e + \frac{1}{6} \, B c^{2} d x^{6} + \frac{1}{6} \, A c^{2} x^{6} e + \frac{1}{5} \, A c^{2} d x^{5} + \frac{2}{5} \, B a c x^{5} e + \frac{1}{2} \, B a c d x^{4} + \frac{1}{2} \, A a c x^{4} e + \frac{2}{3} \, A a c d x^{3} + \frac{1}{3} \, B a^{2} x^{3} e + \frac{1}{2} \, B a^{2} d x^{2} + \frac{1}{2} \, A a^{2} x^{2} e + A a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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