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]

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

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Rubi [A]  time = 0.18, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [B]  time = 0.06, size = 233, normalized size = 2.16 \[ \frac {1}{7} e^3 x^7 \left (a B e^2+5 A c d e+10 B c d^2\right )+\frac {1}{3} d^3 x^3 \left (10 a A e^2+5 a B d e+A c d^2\right )+\frac {1}{6} e^2 x^6 \left (a A e^3+5 a B d e^2+10 A c d^2 e+10 B c d^3\right )+d e x^5 \left (a A e^3+2 a B d e^2+2 A c d^2 e+B c d^3\right )+\frac {1}{4} d^2 x^4 \left (10 a A e^3+10 a B d e^2+5 A c d^2 e+B c d^3\right )+\frac {1}{2} a d^4 x^2 (5 A e+B d)+a A d^5 x+\frac {1}{8} c e^4 x^8 (A e+5 B d)+\frac {1}{9} B c e^5 x^9 \]

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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fricas [B]  time = 0.50, size = 268, normalized size = 2.48 \[ \frac {1}{9} x^{9} e^{5} c B + \frac {5}{8} x^{8} e^{4} d c B + \frac {1}{8} x^{8} e^{5} c A + \frac {10}{7} x^{7} e^{3} d^{2} c B + \frac {1}{7} x^{7} e^{5} a B + \frac {5}{7} x^{7} e^{4} d c A + \frac {5}{3} x^{6} e^{2} d^{3} c B + \frac {5}{6} x^{6} e^{4} d a B + \frac {5}{3} x^{6} e^{3} d^{2} c A + \frac {1}{6} x^{6} e^{5} a A + x^{5} e d^{4} c B + 2 x^{5} e^{3} d^{2} a B + 2 x^{5} e^{2} d^{3} c A + x^{5} e^{4} d a A + \frac {1}{4} x^{4} d^{5} c B + \frac {5}{2} x^{4} e^{2} d^{3} a B + \frac {5}{4} x^{4} e d^{4} c A + \frac {5}{2} x^{4} e^{3} d^{2} a A + \frac {5}{3} x^{3} e d^{4} a B + \frac {1}{3} x^{3} d^{5} c A + \frac {10}{3} x^{3} e^{2} d^{3} a A + \frac {1}{2} x^{2} d^{5} a B + \frac {5}{2} x^{2} e d^{4} a A + x d^{5} a A \]

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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giac [B]  time = 0.16, size = 256, normalized size = 2.37 \[ \frac {1}{9} \, B c x^{9} e^{5} + \frac {5}{8} \, B c d x^{8} e^{4} + \frac {10}{7} \, B c d^{2} x^{7} e^{3} + \frac {5}{3} \, B c d^{3} x^{6} e^{2} + B c d^{4} x^{5} e + \frac {1}{4} \, B c d^{5} x^{4} + \frac {1}{8} \, A c x^{8} e^{5} + \frac {5}{7} \, A c d x^{7} e^{4} + \frac {5}{3} \, A c d^{2} x^{6} e^{3} + 2 \, A c d^{3} x^{5} e^{2} + \frac {5}{4} \, A c d^{4} x^{4} e + \frac {1}{3} \, A c d^{5} x^{3} + \frac {1}{7} \, B a x^{7} e^{5} + \frac {5}{6} \, B a d x^{6} e^{4} + 2 \, B a d^{2} x^{5} e^{3} + \frac {5}{2} \, B a d^{3} x^{4} e^{2} + \frac {5}{3} \, B a d^{4} x^{3} e + \frac {1}{2} \, B a d^{5} x^{2} + \frac {1}{6} \, A a x^{6} e^{5} + A a d x^{5} e^{4} + \frac {5}{2} \, A a d^{2} x^{4} e^{3} + \frac {10}{3} \, A a d^{3} x^{3} e^{2} + \frac {5}{2} \, A a d^{4} x^{2} e + A a d^{5} x \]

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

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maple [B]  time = 0.05, size = 247, normalized size = 2.29 \[ \frac {B c \,e^{5} x^{9}}{9}+\frac {\left (A \,e^{5}+5 B d \,e^{4}\right ) c \,x^{8}}{8}+A a \,d^{5} x +\frac {\left (B a \,e^{5}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c \right ) x^{7}}{7}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) a +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c \right ) x^{6}}{6}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a +\left (5 A \,d^{4} e +B \,d^{5}\right ) c \right ) x^{4}}{4}+\frac {\left (5 A \,d^{4} e +B \,d^{5}\right ) a \,x^{2}}{2}+\frac {\left (A c \,d^{5}+\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a \right ) x^{3}}{3} \]

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 = 0.55, size = 237, normalized size = 2.19 \[ \frac {1}{9} \, B c e^{5} x^{9} + \frac {1}{8} \, {\left (5 \, B c d e^{4} + A c e^{5}\right )} x^{8} + A a d^{5} x + \frac {1}{7} \, {\left (10 \, B c d^{2} e^{3} + 5 \, A c d e^{4} + B a e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B c d^{3} e^{2} + 10 \, A c d^{2} e^{3} + 5 \, B a d e^{4} + A a e^{5}\right )} x^{6} + {\left (B c d^{4} e + 2 \, A c d^{3} e^{2} + 2 \, B a d^{2} e^{3} + A a d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{5} + 5 \, A c d^{4} e + 10 \, B a d^{3} e^{2} + 10 \, A a d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{5} + 5 \, B a d^{4} e + 10 \, A a d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{5} + 5 \, A a d^{4} e\right )} x^{2} \]

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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mupad [B]  time = 0.12, size = 231, normalized size = 2.14 \[ x^5\,\left (B\,c\,d^4\,e+2\,A\,c\,d^3\,e^2+2\,B\,a\,d^2\,e^3+A\,a\,d\,e^4\right )+x^3\,\left (\frac {A\,c\,d^5}{3}+\frac {5\,B\,a\,d^4\,e}{3}+\frac {10\,A\,a\,d^3\,e^2}{3}\right )+x^7\,\left (\frac {10\,B\,c\,d^2\,e^3}{7}+\frac {5\,A\,c\,d\,e^4}{7}+\frac {B\,a\,e^5}{7}\right )+x^4\,\left (\frac {B\,c\,d^5}{4}+\frac {5\,A\,c\,d^4\,e}{4}+\frac {5\,B\,a\,d^3\,e^2}{2}+\frac {5\,A\,a\,d^2\,e^3}{2}\right )+x^6\,\left (\frac {5\,B\,c\,d^3\,e^2}{3}+\frac {5\,A\,c\,d^2\,e^3}{3}+\frac {5\,B\,a\,d\,e^4}{6}+\frac {A\,a\,e^5}{6}\right )+A\,a\,d^5\,x+\frac {B\,c\,e^5\,x^9}{9}+\frac {a\,d^4\,x^2\,\left (5\,A\,e+B\,d\right )}{2}+\frac {c\,e^4\,x^8\,\left (A\,e+5\,B\,d\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.11, size = 287, normalized size = 2.66 \[ A a d^{5} x + \frac {B c e^{5} x^{9}}{9} + x^{8} \left (\frac {A c e^{5}}{8} + \frac {5 B c d e^{4}}{8}\right ) + x^{7} \left (\frac {5 A c d e^{4}}{7} + \frac {B a e^{5}}{7} + \frac {10 B c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {A a e^{5}}{6} + \frac {5 A c d^{2} e^{3}}{3} + \frac {5 B a d e^{4}}{6} + \frac {5 B c d^{3} e^{2}}{3}\right ) + x^{5} \left (A a d e^{4} + 2 A c d^{3} e^{2} + 2 B a d^{2} e^{3} + B c d^{4} e\right ) + x^{4} \left (\frac {5 A a d^{2} e^{3}}{2} + \frac {5 A c d^{4} e}{4} + \frac {5 B a d^{3} e^{2}}{2} + \frac {B c d^{5}}{4}\right ) + x^{3} \left (\frac {10 A a d^{3} e^{2}}{3} + \frac {A c d^{5}}{3} + \frac {5 B a d^{4} e}{3}\right ) + x^{2} \left (\frac {5 A a d^{4} e}{2} + \frac {B a d^{5}}{2}\right ) \]

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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