3.1300 \(\int (A+B x) (d+e x)^2 (a+c x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ \frac{c (d+e x)^6 \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}-\frac{2 c (d+e x)^5 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}+\frac{(d+e x)^4 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6}-\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{c^2 (d+e x)^7 (5 B d-A e)}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6} \]

[Out]

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

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Rubi [A]  time = 0.170831, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{c (d+e x)^6 \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}-\frac{2 c (d+e x)^5 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}+\frac{(d+e x)^4 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6}-\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{c^2 (d+e x)^7 (5 B d-A e)}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^2}{e^5}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^3}{e^5}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^4}{e^5}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^5}{e^5}+\frac{c^2 (-5 B d+A e) (d+e x)^6}{e^5}+\frac{B c^2 (d+e x)^7}{e^5}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^3}{3 e^6}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^4}{4 e^6}-\frac{2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^5}{5 e^6}+\frac{c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{3 e^6}-\frac{c^2 (5 B d-A e) (d+e x)^7}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6}\\ \end{align*}

Mathematica [A]  time = 0.0424344, size = 174, normalized size = 0.84 \[ \frac{1}{2} a^2 d x^2 (2 A e+B d)+a^2 A d^2 x+\frac{1}{6} c x^6 \left (2 a B e^2+2 A c d e+B c d^2\right )+\frac{1}{5} c x^5 \left (2 a A e^2+4 a B d e+A c d^2\right )+\frac{1}{4} a x^4 \left (a B e^2+4 A c d e+2 B c d^2\right )+\frac{1}{3} a x^3 \left (a A e^2+2 a B d e+2 A c d^2\right )+\frac{1}{7} c^2 e x^7 (A e+2 B d)+\frac{1}{8} B c^2 e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 177, normalized size = 0.9 \begin{align*}{\frac{B{c}^{2}{e}^{2}{x}^{8}}{8}}+{\frac{ \left ( A{e}^{2}+2\,Bde \right ){c}^{2}{x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{2}+2\,B{e}^{2}ac \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}{d}^{2}+2\, \left ( A{e}^{2}+2\,Bde \right ) ac \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 2\,Ade+B{d}^{2} \right ) ac+{a}^{2}B{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{2}ac+ \left ( A{e}^{2}+2\,Bde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Ade+B{d}^{2} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{2}{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0278, size = 248, normalized size = 1.2 \begin{align*} \frac{1}{8} \, B c^{2} e^{2} x^{8} + \frac{1}{7} \,{\left (2 \, B c^{2} d e + A c^{2} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{2} + 2 \, A c^{2} d e + 2 \, B a c e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac{1}{5} \,{\left (A c^{2} d^{2} + 4 \, B a c d e + 2 \, A a c e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, B a c d^{2} + 4 \, A a c d e + B a^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A a c d^{2} + 2 \, B a^{2} d e + A a^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{2} + 2 \, A a^{2} d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.61455, size = 466, normalized size = 2.26 \begin{align*} \frac{1}{8} x^{8} e^{2} c^{2} B + \frac{2}{7} x^{7} e d c^{2} B + \frac{1}{7} x^{7} e^{2} c^{2} A + \frac{1}{6} x^{6} d^{2} c^{2} B + \frac{1}{3} x^{6} e^{2} c a B + \frac{1}{3} x^{6} e d c^{2} A + \frac{4}{5} x^{5} e d c a B + \frac{1}{5} x^{5} d^{2} c^{2} A + \frac{2}{5} x^{5} e^{2} c a A + \frac{1}{2} x^{4} d^{2} c a B + \frac{1}{4} x^{4} e^{2} a^{2} B + x^{4} e d c a A + \frac{2}{3} x^{3} e d a^{2} B + \frac{2}{3} x^{3} d^{2} c a A + \frac{1}{3} x^{3} e^{2} a^{2} A + \frac{1}{2} x^{2} d^{2} a^{2} B + x^{2} e d a^{2} A + x d^{2} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.226784, size = 211, normalized size = 1.02 \begin{align*} A a^{2} d^{2} x + \frac{B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2} e^{2}}{7} + \frac{2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac{A c^{2} d e}{3} + \frac{B a c e^{2}}{3} + \frac{B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac{2 A a c e^{2}}{5} + \frac{A c^{2} d^{2}}{5} + \frac{4 B a c d e}{5}\right ) + x^{4} \left (A a c d e + \frac{B a^{2} e^{2}}{4} + \frac{B a c d^{2}}{2}\right ) + x^{3} \left (\frac{A a^{2} e^{2}}{3} + \frac{2 A a c d^{2}}{3} + \frac{2 B a^{2} d e}{3}\right ) + x^{2} \left (A a^{2} d e + \frac{B a^{2} d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32527, size = 270, normalized size = 1.31 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} e^{2} + \frac{2}{7} \, B c^{2} d x^{7} e + \frac{1}{6} \, B c^{2} d^{2} x^{6} + \frac{1}{7} \, A c^{2} x^{7} e^{2} + \frac{1}{3} \, A c^{2} d x^{6} e + \frac{1}{5} \, A c^{2} d^{2} x^{5} + \frac{1}{3} \, B a c x^{6} e^{2} + \frac{4}{5} \, B a c d x^{5} e + \frac{1}{2} \, B a c d^{2} x^{4} + \frac{2}{5} \, A a c x^{5} e^{2} + A a c d x^{4} e + \frac{2}{3} \, A a c d^{2} x^{3} + \frac{1}{4} \, B a^{2} x^{4} e^{2} + \frac{2}{3} \, B a^{2} d x^{3} e + \frac{1}{2} \, B a^{2} d^{2} x^{2} + \frac{1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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