3.1316 \(\int (A+B x) (d+e x)^2 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=334 \[ -\frac{c (d+e x)^6 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8}+\frac{3 c^2 (d+e x)^8 \left (a B e^2-2 A c d e+7 B c d^2\right )}{8 e^8}-\frac{c^2 (d+e x)^7 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}-\frac{3 c (d+e x)^5 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8}-\frac{(d+e x)^3 \left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8}-\frac{c^3 (d+e x)^9 (7 B d-A e)}{9 e^8}+\frac{B c^3 (d+e x)^{10}}{10 e^8} \]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^3)/(3*e^8) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d
+ e*x)^4)/(4*e^8) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^5)/(5*e^
8) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^6)/(6*e^8) - (
c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^7)/(7*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*
e + a*B*e^2)*(d + e*x)^8)/(8*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^9)/(9*e^8) + (B*c^3*(d + e*x)^10)/(10*e^8)

________________________________________________________________________________________

Rubi [A]  time = 0.307445, antiderivative size = 334, 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} \[ -\frac{c (d+e x)^6 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8}+\frac{3 c^2 (d+e x)^8 \left (a B e^2-2 A c d e+7 B c d^2\right )}{8 e^8}-\frac{c^2 (d+e x)^7 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}-\frac{3 c (d+e x)^5 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8}-\frac{(d+e x)^3 \left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8}-\frac{c^3 (d+e x)^9 (7 B d-A e)}{9 e^8}+\frac{B c^3 (d+e x)^{10}}{10 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^3)/(3*e^8) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d
+ e*x)^4)/(4*e^8) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^5)/(5*e^
8) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^6)/(6*e^8) - (
c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^7)/(7*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*
e + a*B*e^2)*(d + e*x)^8)/(8*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^9)/(9*e^8) + (B*c^3*(d + e*x)^10)/(10*e^8)

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 )^3 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3 (d+e x)^2}{e^7}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^3}{e^7}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^4}{e^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^5}{e^7}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^6}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^7}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^8}{e^7}+\frac{B c^3 (d+e x)^9}{e^7}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^3}{3 e^8}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^4}{4 e^8}-\frac{3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^5}{5 e^8}-\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^6}{6 e^8}-\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^7}{7 e^8}+\frac{3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^8}{8 e^8}-\frac{c^3 (7 B d-A e) (d+e x)^9}{9 e^8}+\frac{B c^3 (d+e x)^{10}}{10 e^8}\\ \end{align*}

Mathematica [A]  time = 0.0629274, size = 238, normalized size = 0.71 \[ \frac{1}{4} a^2 x^4 \left (a B e^2+6 A c d e+3 B c d^2\right )+\frac{1}{3} a^2 x^3 \left (a A e^2+2 a B d e+3 A c d^2\right )+\frac{1}{2} a^3 d x^2 (2 A e+B d)+a^3 A d^2 x+\frac{1}{8} c^2 x^8 \left (3 a B e^2+2 A c d e+B c d^2\right )+\frac{1}{7} c^2 x^7 \left (3 a A e^2+6 a B d e+A c d^2\right )+\frac{1}{2} a c x^6 \left (a B e^2+2 A c d e+B c d^2\right )+\frac{3}{5} a c x^5 \left (a A e^2+2 a B d e+A c d^2\right )+\frac{1}{9} c^3 e x^9 (A e+2 B d)+\frac{1}{10} B c^3 e^2 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 251, normalized size = 0.8 \begin{align*}{\frac{B{c}^{3}{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( A{e}^{2}+2\,Bde \right ){c}^{3}{x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{3}+3\,B{e}^{2}a{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}{d}^{2}+3\, \left ( A{e}^{2}+2\,Bde \right ) a{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\, \left ( 2\,Ade+B{d}^{2} \right ) a{c}^{2}+3\,B{e}^{2}{a}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{2}a{c}^{2}+3\, \left ( A{e}^{2}+2\,Bde \right ){a}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\, \left ( 2\,Ade+B{d}^{2} \right ){a}^{2}c+B{e}^{2}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{d}^{2}{a}^{2}c+ \left ( A{e}^{2}+2\,Bde \right ){a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Ade+B{d}^{2} \right ){a}^{3}{x}^{2}}{2}}+A{d}^{2}{a}^{3}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)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02587, size = 354, normalized size = 1.06 \begin{align*} \frac{1}{10} \, B c^{3} e^{2} x^{10} + \frac{1}{9} \,{\left (2 \, B c^{3} d e + A c^{3} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{2} + 2 \, A c^{3} d e + 3 \, B a c^{2} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (A c^{3} d^{2} + 6 \, B a c^{2} d e + 3 \, A a c^{2} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac{1}{2} \,{\left (B a c^{2} d^{2} + 2 \, A a c^{2} d e + B a^{2} c e^{2}\right )} x^{6} + \frac{3}{5} \,{\left (A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d^{2} + 6 \, A a^{2} c d e + B a^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{2} c d^{2} + 2 \, B a^{3} d e + A a^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d^{2} + 2 \, A a^{3} 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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.61463, size = 657, normalized size = 1.97 \begin{align*} \frac{1}{10} x^{10} e^{2} c^{3} B + \frac{2}{9} x^{9} e d c^{3} B + \frac{1}{9} x^{9} e^{2} c^{3} A + \frac{1}{8} x^{8} d^{2} c^{3} B + \frac{3}{8} x^{8} e^{2} c^{2} a B + \frac{1}{4} x^{8} e d c^{3} A + \frac{6}{7} x^{7} e d c^{2} a B + \frac{1}{7} x^{7} d^{2} c^{3} A + \frac{3}{7} x^{7} e^{2} c^{2} a A + \frac{1}{2} x^{6} d^{2} c^{2} a B + \frac{1}{2} x^{6} e^{2} c a^{2} B + x^{6} e d c^{2} a A + \frac{6}{5} x^{5} e d c a^{2} B + \frac{3}{5} x^{5} d^{2} c^{2} a A + \frac{3}{5} x^{5} e^{2} c a^{2} A + \frac{3}{4} x^{4} d^{2} c a^{2} B + \frac{1}{4} x^{4} e^{2} a^{3} B + \frac{3}{2} x^{4} e d c a^{2} A + \frac{2}{3} x^{3} e d a^{3} B + x^{3} d^{2} c a^{2} A + \frac{1}{3} x^{3} e^{2} a^{3} A + \frac{1}{2} x^{2} d^{2} a^{3} B + x^{2} e d a^{3} A + x d^{2} a^{3} 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)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.157006, size = 306, normalized size = 0.92 \begin{align*} A a^{3} d^{2} x + \frac{B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac{A c^{3} e^{2}}{9} + \frac{2 B c^{3} d e}{9}\right ) + x^{8} \left (\frac{A c^{3} d e}{4} + \frac{3 B a c^{2} e^{2}}{8} + \frac{B c^{3} d^{2}}{8}\right ) + x^{7} \left (\frac{3 A a c^{2} e^{2}}{7} + \frac{A c^{3} d^{2}}{7} + \frac{6 B a c^{2} d e}{7}\right ) + x^{6} \left (A a c^{2} d e + \frac{B a^{2} c e^{2}}{2} + \frac{B a c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{3 A a^{2} c e^{2}}{5} + \frac{3 A a c^{2} d^{2}}{5} + \frac{6 B a^{2} c d e}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c d e}{2} + \frac{B a^{3} e^{2}}{4} + \frac{3 B a^{2} c d^{2}}{4}\right ) + x^{3} \left (\frac{A a^{3} e^{2}}{3} + A a^{2} c d^{2} + \frac{2 B a^{3} d e}{3}\right ) + x^{2} \left (A a^{3} d e + \frac{B a^{3} 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)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21846, size = 387, normalized size = 1.16 \begin{align*} \frac{1}{10} \, B c^{3} x^{10} e^{2} + \frac{2}{9} \, B c^{3} d x^{9} e + \frac{1}{8} \, B c^{3} d^{2} x^{8} + \frac{1}{9} \, A c^{3} x^{9} e^{2} + \frac{1}{4} \, A c^{3} d x^{8} e + \frac{1}{7} \, A c^{3} d^{2} x^{7} + \frac{3}{8} \, B a c^{2} x^{8} e^{2} + \frac{6}{7} \, B a c^{2} d x^{7} e + \frac{1}{2} \, B a c^{2} d^{2} x^{6} + \frac{3}{7} \, A a c^{2} x^{7} e^{2} + A a c^{2} d x^{6} e + \frac{3}{5} \, A a c^{2} d^{2} x^{5} + \frac{1}{2} \, B a^{2} c x^{6} e^{2} + \frac{6}{5} \, B a^{2} c d x^{5} e + \frac{3}{4} \, B a^{2} c d^{2} x^{4} + \frac{3}{5} \, A a^{2} c x^{5} e^{2} + \frac{3}{2} \, A a^{2} c d x^{4} e + A a^{2} c d^{2} x^{3} + \frac{1}{4} \, B a^{3} x^{4} e^{2} + \frac{2}{3} \, B a^{3} d x^{3} e + \frac{1}{2} \, B a^{3} d^{2} x^{2} + \frac{1}{3} \, A a^{3} x^{3} e^{2} + A a^{3} d x^{2} e + A a^{3} 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)^3,x, algorithm="giac")

[Out]

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