3.1285 \(\int (A+B x) (d+e x)^4 (a+c x^2) \, dx\)

Optimal. Leaf size=108 \[ \frac{(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac{(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac{c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]

[Out]

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

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Rubi [A]  time = 0.137743, antiderivative size = 108, 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, Rules used = {772} \[ \frac{(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac{(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac{c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(a + c*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0532217, size = 194, normalized size = 1.8 \[ \frac{1}{6} e^2 x^6 \left (a B e^2+4 A c d e+6 B c d^2\right )+\frac{1}{5} e x^5 \left (a A e^3+4 a B d e^2+6 A c d^2 e+4 B c d^3\right )+\frac{1}{4} d x^4 \left (4 a A e^3+6 a B d e^2+4 A c d^2 e+B c d^3\right )+\frac{1}{3} d^2 x^3 \left (6 a A e^2+4 a B d e+A c d^2\right )+\frac{1}{2} a d^3 x^2 (4 A e+B d)+a A d^4 x+\frac{1}{7} c e^3 x^7 (A e+4 B d)+\frac{1}{8} B c e^4 x^8 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a + c*x^2),x]

[Out]

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

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Maple [A]  time = 0.001, size = 199, normalized size = 1.8 \begin{align*}{\frac{B{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) c{x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) c+B{e}^{4}a \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) a{x}^{2}}{2}}+A{d}^{4}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(c*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.03321, size = 262, normalized size = 2.43 \begin{align*} \frac{1}{8} \, B c e^{4} x^{8} + \frac{1}{7} \,{\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac{1}{6} \,{\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{4} + 4 \, A a d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.60382, size = 498, normalized size = 4.61 \begin{align*} \frac{1}{8} x^{8} e^{4} c B + \frac{4}{7} x^{7} e^{3} d c B + \frac{1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac{1}{6} x^{6} e^{4} a B + \frac{2}{3} x^{6} e^{3} d c A + \frac{4}{5} x^{5} e d^{3} c B + \frac{4}{5} x^{5} e^{3} d a B + \frac{6}{5} x^{5} e^{2} d^{2} c A + \frac{1}{5} x^{5} e^{4} a A + \frac{1}{4} x^{4} d^{4} c B + \frac{3}{2} x^{4} e^{2} d^{2} a B + x^{4} e d^{3} c A + x^{4} e^{3} d a A + \frac{4}{3} x^{3} e d^{3} a B + \frac{1}{3} x^{3} d^{4} c A + 2 x^{3} e^{2} d^{2} a A + \frac{1}{2} x^{2} d^{4} a B + 2 x^{2} e d^{3} a A + x d^{4} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.169384, size = 226, normalized size = 2.09 \begin{align*} A a d^{4} x + \frac{B c e^{4} x^{8}}{8} + x^{7} \left (\frac{A c e^{4}}{7} + \frac{4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac{2 A c d e^{3}}{3} + \frac{B a e^{4}}{6} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac{A a e^{4}}{5} + \frac{6 A c d^{2} e^{2}}{5} + \frac{4 B a d e^{3}}{5} + \frac{4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + A c d^{3} e + \frac{3 B a d^{2} e^{2}}{2} + \frac{B c d^{4}}{4}\right ) + x^{3} \left (2 A a d^{2} e^{2} + \frac{A c d^{4}}{3} + \frac{4 B a d^{3} e}{3}\right ) + x^{2} \left (2 A a d^{3} e + \frac{B a d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(c*x**2+a),x)

[Out]

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

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Giac [B]  time = 1.14459, size = 279, normalized size = 2.58 \begin{align*} \frac{1}{8} \, B c x^{8} e^{4} + \frac{4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac{4}{5} \, B c d^{3} x^{5} e + \frac{1}{4} \, B c d^{4} x^{4} + \frac{1}{7} \, A c x^{7} e^{4} + \frac{2}{3} \, A c d x^{6} e^{3} + \frac{6}{5} \, A c d^{2} x^{5} e^{2} + A c d^{3} x^{4} e + \frac{1}{3} \, A c d^{4} x^{3} + \frac{1}{6} \, B a x^{6} e^{4} + \frac{4}{5} \, B a d x^{5} e^{3} + \frac{3}{2} \, B a d^{2} x^{4} e^{2} + \frac{4}{3} \, B a d^{3} x^{3} e + \frac{1}{2} \, B a d^{4} x^{2} + \frac{1}{5} \, A a x^{5} e^{4} + A a d x^{4} e^{3} + 2 \, A a d^{2} x^{3} e^{2} + 2 \, A a d^{3} x^{2} e + A a d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a),x, algorithm="giac")

[Out]

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