3.1299 \(\int (A+B x) (d+e x)^3 (a+c x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ \frac{2 c (d+e x)^7 \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}-\frac{c (d+e x)^6 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}+\frac{(d+e x)^5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6}-\frac{c^2 (d+e x)^8 (5 B d-A e)}{8 e^6}+\frac{B c^2 (d+e x)^9}{9 e^6} \]

[Out]

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

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Rubi [A]  time = 0.216989, 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} \[ \frac{2 c (d+e x)^7 \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}-\frac{c (d+e x)^6 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}+\frac{(d+e x)^5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6}-\frac{c^2 (d+e x)^8 (5 B d-A e)}{8 e^6}+\frac{B c^2 (d+e x)^9}{9 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0672267, size = 244, normalized size = 1.18 \[ \frac{1}{5} x^5 \left (a^2 B e^3+6 a A c d e^2+6 a B c d^2 e+A c^2 d^3\right )+\frac{1}{2} a^2 d^2 x^2 (3 A e+B d)+a^2 A d^3 x+\frac{1}{7} c e x^7 \left (2 a B e^2+3 A c d e+3 B c d^2\right )+\frac{1}{6} c x^6 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac{1}{4} a x^4 \left (a A e^3+3 a B d e^2+6 A c d^2 e+2 B c d^3\right )+\frac{1}{3} a d x^3 \left (3 a A e^2+3 a B d e+2 A c d^2\right )+\frac{1}{8} c^2 e^2 x^8 (A e+3 B d)+\frac{1}{9} B c^2 e^3 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 252, normalized size = 1.2 \begin{align*}{\frac{B{c}^{2}{e}^{3}{x}^{9}}{9}}+{\frac{ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){c}^{2}{x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){c}^{2}+2\,B{e}^{3}ac \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){c}^{2}+2\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) ac \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{3}{c}^{2}+2\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) ac+{a}^{2}B{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) ac+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{3}ac+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{3}{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3*(c*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.995072, size = 351, normalized size = 1.7 \begin{align*} \frac{1}{9} \, B c^{2} e^{3} x^{9} + \frac{1}{8} \,{\left (3 \, B c^{2} d e^{2} + A c^{2} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, B c^{2} d^{2} e + 3 \, A c^{2} d e^{2} + 2 \, B a c e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac{1}{6} \,{\left (B c^{2} d^{3} + 3 \, A c^{2} d^{2} e + 6 \, B a c d e^{2} + 2 \, A a c e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (A c^{2} d^{3} + 6 \, B a c d^{2} e + 6 \, A a c d e^{2} + B a^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, B a c d^{3} + 6 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A a c d^{3} + 3 \, B a^{2} d^{2} e + 3 \, A a^{2} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{3} + 3 \, A a^{2} d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.63471, size = 655, normalized size = 3.18 \begin{align*} \frac{1}{9} x^{9} e^{3} c^{2} B + \frac{3}{8} x^{8} e^{2} d c^{2} B + \frac{1}{8} x^{8} e^{3} c^{2} A + \frac{3}{7} x^{7} e d^{2} c^{2} B + \frac{2}{7} x^{7} e^{3} c a B + \frac{3}{7} x^{7} e^{2} d c^{2} A + \frac{1}{6} x^{6} d^{3} c^{2} B + x^{6} e^{2} d c a B + \frac{1}{2} x^{6} e d^{2} c^{2} A + \frac{1}{3} x^{6} e^{3} c a A + \frac{6}{5} x^{5} e d^{2} c a B + \frac{1}{5} x^{5} e^{3} a^{2} B + \frac{1}{5} x^{5} d^{3} c^{2} A + \frac{6}{5} x^{5} e^{2} d c a A + \frac{1}{2} x^{4} d^{3} c a B + \frac{3}{4} x^{4} e^{2} d a^{2} B + \frac{3}{2} x^{4} e d^{2} c a A + \frac{1}{4} x^{4} e^{3} a^{2} A + x^{3} e d^{2} a^{2} B + \frac{2}{3} x^{3} d^{3} c a A + x^{3} e^{2} d a^{2} A + \frac{1}{2} x^{2} d^{3} a^{2} B + \frac{3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.142811, size = 303, normalized size = 1.47 \begin{align*} A a^{2} d^{3} x + \frac{B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac{A c^{2} e^{3}}{8} + \frac{3 B c^{2} d e^{2}}{8}\right ) + x^{7} \left (\frac{3 A c^{2} d e^{2}}{7} + \frac{2 B a c e^{3}}{7} + \frac{3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac{A a c e^{3}}{3} + \frac{A c^{2} d^{2} e}{2} + B a c d e^{2} + \frac{B c^{2} d^{3}}{6}\right ) + x^{5} \left (\frac{6 A a c d e^{2}}{5} + \frac{A c^{2} d^{3}}{5} + \frac{B a^{2} e^{3}}{5} + \frac{6 B a c d^{2} e}{5}\right ) + x^{4} \left (\frac{A a^{2} e^{3}}{4} + \frac{3 A a c d^{2} e}{2} + \frac{3 B a^{2} d e^{2}}{4} + \frac{B a c d^{3}}{2}\right ) + x^{3} \left (A a^{2} d e^{2} + \frac{2 A a c d^{3}}{3} + B a^{2} d^{2} e\right ) + x^{2} \left (\frac{3 A a^{2} d^{2} e}{2} + \frac{B a^{2} d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*(c*x**2+a)**2,x)

[Out]

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

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Giac [A]  time = 1.20597, size = 379, normalized size = 1.84 \begin{align*} \frac{1}{9} \, B c^{2} x^{9} e^{3} + \frac{3}{8} \, B c^{2} d x^{8} e^{2} + \frac{3}{7} \, B c^{2} d^{2} x^{7} e + \frac{1}{6} \, B c^{2} d^{3} x^{6} + \frac{1}{8} \, A c^{2} x^{8} e^{3} + \frac{3}{7} \, A c^{2} d x^{7} e^{2} + \frac{1}{2} \, A c^{2} d^{2} x^{6} e + \frac{1}{5} \, A c^{2} d^{3} x^{5} + \frac{2}{7} \, B a c x^{7} e^{3} + B a c d x^{6} e^{2} + \frac{6}{5} \, B a c d^{2} x^{5} e + \frac{1}{2} \, B a c d^{3} x^{4} + \frac{1}{3} \, A a c x^{6} e^{3} + \frac{6}{5} \, A a c d x^{5} e^{2} + \frac{3}{2} \, A a c d^{2} x^{4} e + \frac{2}{3} \, A a c d^{3} x^{3} + \frac{1}{5} \, B a^{2} x^{5} e^{3} + \frac{3}{4} \, B a^{2} d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + \frac{1}{2} \, B a^{2} d^{3} x^{2} + \frac{1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac{3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+a)^2,x, algorithm="giac")

[Out]

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