3.1284 \(\int (A+B x) (d+e x)^5 (a+c x^2) \, dx\)

Optimal. Leaf size=108 \[ \frac{(d+e x)^7 \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac{(d+e x)^6 \left (a e^2+c d^2\right ) (B d-A e)}{6 e^4}-\frac{c (d+e x)^8 (3 B d-A e)}{8 e^4}+\frac{B c (d+e x)^9}{9 e^4} \]

[Out]

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

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Rubi [A]  time = 0.176489, 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} \[ \frac{(d+e x)^7 \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac{(d+e x)^6 \left (a e^2+c d^2\right ) (B d-A e)}{6 e^4}-\frac{c (d+e x)^8 (3 B d-A e)}{8 e^4}+\frac{B c (d+e x)^9}{9 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a + c*x^2),x]

[Out]

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

Mathematica [B]  time = 0.0641929, size = 233, normalized size = 2.16 \[ \frac{1}{7} e^3 x^7 \left (a B e^2+5 A c d e+10 B c d^2\right )+\frac{1}{6} e^2 x^6 \left (a A e^3+5 a B d e^2+10 A c d^2 e+10 B c d^3\right )+d e x^5 \left (a A e^3+2 a B d e^2+2 A c d^2 e+B c d^3\right )+\frac{1}{4} d^2 x^4 \left (10 a A e^3+10 a B d e^2+5 A c d^2 e+B c d^3\right )+\frac{1}{3} d^3 x^3 \left (10 a A e^2+5 a B d e+A c d^2\right )+\frac{1}{2} a d^4 x^2 (5 A e+B d)+a A d^5 x+\frac{1}{8} c e^4 x^8 (A e+5 B d)+\frac{1}{9} B c e^5 x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a + c*x^2),x]

[Out]

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

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Maple [B]  time = 0.003, size = 247, normalized size = 2.3 \begin{align*}{\frac{B{e}^{5}c{x}^{9}}{9}}+{\frac{ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) c{x}^{8}}{8}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) c+B{e}^{5}a \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) c+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) a \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) c+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) c+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{5}c+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) a{x}^{2}}{2}}+A{d}^{5}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5*(c*x^2+a),x)

[Out]

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

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Maxima [B]  time = 0.999639, size = 320, normalized size = 2.96 \begin{align*} \frac{1}{9} \, B c e^{5} x^{9} + \frac{1}{8} \,{\left (5 \, B c d e^{4} + A c e^{5}\right )} x^{8} + A a d^{5} x + \frac{1}{7} \,{\left (10 \, B c d^{2} e^{3} + 5 \, A c d e^{4} + B a e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, B c d^{3} e^{2} + 10 \, A c d^{2} e^{3} + 5 \, B a d e^{4} + A a e^{5}\right )} x^{6} +{\left (B c d^{4} e + 2 \, A c d^{3} e^{2} + 2 \, B a d^{2} e^{3} + A a d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{5} + 5 \, A c d^{4} e + 10 \, B a d^{3} e^{2} + 10 \, A a d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (A c d^{5} + 5 \, B a d^{4} e + 10 \, A a d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{5} + 5 \, A a d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.53881, size = 625, normalized size = 5.79 \begin{align*} \frac{1}{9} x^{9} e^{5} c B + \frac{5}{8} x^{8} e^{4} d c B + \frac{1}{8} x^{8} e^{5} c A + \frac{10}{7} x^{7} e^{3} d^{2} c B + \frac{1}{7} x^{7} e^{5} a B + \frac{5}{7} x^{7} e^{4} d c A + \frac{5}{3} x^{6} e^{2} d^{3} c B + \frac{5}{6} x^{6} e^{4} d a B + \frac{5}{3} x^{6} e^{3} d^{2} c A + \frac{1}{6} x^{6} e^{5} a A + x^{5} e d^{4} c B + 2 x^{5} e^{3} d^{2} a B + 2 x^{5} e^{2} d^{3} c A + x^{5} e^{4} d a A + \frac{1}{4} x^{4} d^{5} c B + \frac{5}{2} x^{4} e^{2} d^{3} a B + \frac{5}{4} x^{4} e d^{4} c A + \frac{5}{2} x^{4} e^{3} d^{2} a A + \frac{5}{3} x^{3} e d^{4} a B + \frac{1}{3} x^{3} d^{5} c A + \frac{10}{3} x^{3} e^{2} d^{3} a A + \frac{1}{2} x^{2} d^{5} a B + \frac{5}{2} x^{2} e d^{4} a A + x d^{5} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.190081, size = 287, normalized size = 2.66 \begin{align*} A a d^{5} x + \frac{B c e^{5} x^{9}}{9} + x^{8} \left (\frac{A c e^{5}}{8} + \frac{5 B c d e^{4}}{8}\right ) + x^{7} \left (\frac{5 A c d e^{4}}{7} + \frac{B a e^{5}}{7} + \frac{10 B c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{A a e^{5}}{6} + \frac{5 A c d^{2} e^{3}}{3} + \frac{5 B a d e^{4}}{6} + \frac{5 B c d^{3} e^{2}}{3}\right ) + x^{5} \left (A a d e^{4} + 2 A c d^{3} e^{2} + 2 B a d^{2} e^{3} + B c d^{4} e\right ) + x^{4} \left (\frac{5 A a d^{2} e^{3}}{2} + \frac{5 A c d^{4} e}{4} + \frac{5 B a d^{3} e^{2}}{2} + \frac{B c d^{5}}{4}\right ) + x^{3} \left (\frac{10 A a d^{3} e^{2}}{3} + \frac{A c d^{5}}{3} + \frac{5 B a d^{4} e}{3}\right ) + x^{2} \left (\frac{5 A a d^{4} e}{2} + \frac{B a d^{5}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5*(c*x**2+a),x)

[Out]

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

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Giac [B]  time = 1.15723, size = 346, normalized size = 3.2 \begin{align*} \frac{1}{9} \, B c x^{9} e^{5} + \frac{5}{8} \, B c d x^{8} e^{4} + \frac{10}{7} \, B c d^{2} x^{7} e^{3} + \frac{5}{3} \, B c d^{3} x^{6} e^{2} + B c d^{4} x^{5} e + \frac{1}{4} \, B c d^{5} x^{4} + \frac{1}{8} \, A c x^{8} e^{5} + \frac{5}{7} \, A c d x^{7} e^{4} + \frac{5}{3} \, A c d^{2} x^{6} e^{3} + 2 \, A c d^{3} x^{5} e^{2} + \frac{5}{4} \, A c d^{4} x^{4} e + \frac{1}{3} \, A c d^{5} x^{3} + \frac{1}{7} \, B a x^{7} e^{5} + \frac{5}{6} \, B a d x^{6} e^{4} + 2 \, B a d^{2} x^{5} e^{3} + \frac{5}{2} \, B a d^{3} x^{4} e^{2} + \frac{5}{3} \, B a d^{4} x^{3} e + \frac{1}{2} \, B a d^{5} x^{2} + \frac{1}{6} \, A a x^{6} e^{5} + A a d x^{5} e^{4} + \frac{5}{2} \, A a d^{2} x^{4} e^{3} + \frac{10}{3} \, A a d^{3} x^{3} e^{2} + \frac{5}{2} \, A a d^{4} x^{2} e + A a d^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a),x, algorithm="giac")

[Out]

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