\(\int (A+B x) (d+e x)^4 (a+c x^2) \, dx\) [1285]

Optimal result

Integrand size = 20, antiderivative size = 108 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{6 e^4}-\frac {c (3 B d-A e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {786} \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac {(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac {c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \]

[In]

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

[Out]

-1/5*((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^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 786

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*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=a A d^4 x+\frac {1}{2} a d^3 (B d+4 A e) x^2+\frac {1}{3} d^2 \left (A c d^2+4 a B d e+6 a A e^2\right ) x^3+\frac {1}{4} d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right ) x^4+\frac {1}{5} e \left (4 B c d^3+6 A c d^2 e+4 a B d e^2+a A e^3\right ) x^5+\frac {1}{6} e^2 \left (6 B c d^2+4 A c d e+a B e^2\right ) x^6+\frac {1}{7} c e^3 (4 B d+A e) x^7+\frac {1}{8} B c e^4 x^8 \]

[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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.78

method	result	size
norman	\(\frac {B \,e^{4} c \,x^{8}}{8}+\left (\frac {1}{7} A c \,e^{4}+\frac {4}{7} B c d \,e^{3}\right ) x^{7}+\left (\frac {2}{3} A c d \,e^{3}+\frac {1}{6} B \,e^{4} a +B c \,d^{2} e^{2}\right ) x^{6}+\left (\frac {1}{5} A a \,e^{4}+\frac {6}{5} A c \,d^{2} e^{2}+\frac {4}{5} B a d \,e^{3}+\frac {4}{5} B c \,d^{3} e \right ) x^{5}+\left (A a d \,e^{3}+A c \,d^{3} e +\frac {3}{2} B a \,d^{2} e^{2}+\frac {1}{4} B c \,d^{4}\right ) x^{4}+\left (2 A a \,d^{2} e^{2}+\frac {1}{3} A c \,d^{4}+\frac {4}{3} B a \,d^{3} e \right ) x^{3}+\left (2 A a \,d^{3} e +\frac {1}{2} B a \,d^{4}\right ) x^{2}+d^{4} A a x\)	\(192\)
default	\(\frac {B \,e^{4} c \,x^{8}}{8}+\frac {\left (e^{4} A +4 e^{3} d B \right ) c \,x^{7}}{7}+\frac {\left (\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) c +B \,e^{4} a \right ) x^{6}}{6}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c +\left (e^{4} A +4 e^{3} d B \right ) a \right ) x^{5}}{5}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c +\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A c \,d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a \right ) x^{3}}{3}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a \,x^{2}}{2}+d^{4} A a x\)	\(199\)
gosper	\(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {1}{6} x^{6} B \,e^{4} a +x^{6} B c \,d^{2} e^{2}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {4}{5} x^{5} B c \,d^{3} e +x^{4} A a d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {1}{4} x^{4} B c \,d^{4}+2 x^{3} A a \,d^{2} e^{2}+\frac {1}{3} x^{3} A c \,d^{4}+\frac {4}{3} x^{3} B a \,d^{3} e +2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} B a \,d^{4}+d^{4} A a x\)	\(216\)
risch	\(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {1}{6} x^{6} B \,e^{4} a +x^{6} B c \,d^{2} e^{2}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {4}{5} x^{5} B c \,d^{3} e +x^{4} A a d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {1}{4} x^{4} B c \,d^{4}+2 x^{3} A a \,d^{2} e^{2}+\frac {1}{3} x^{3} A c \,d^{4}+\frac {4}{3} x^{3} B a \,d^{3} e +2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} B a \,d^{4}+d^{4} A a x\)	\(216\)
parallelrisch	\(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {1}{6} x^{6} B \,e^{4} a +x^{6} B c \,d^{2} e^{2}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {4}{5} x^{5} B c \,d^{3} e +x^{4} A a d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {1}{4} x^{4} B c \,d^{4}+2 x^{3} A a \,d^{2} e^{2}+\frac {1}{3} x^{3} A c \,d^{4}+\frac {4}{3} x^{3} B a \,d^{3} e +2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} B a \,d^{4}+d^{4} A a x\)	\(216\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{4} + 4 \, A a d^{3} e\right )} x^{2} \]

[In]

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

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (102) = 204\).

Time = 0.03 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.09 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=A a d^{4} x + \frac {B c e^{4} x^{8}}{8} + x^{7} \left (\frac {A c e^{4}}{7} + \frac {4 B c d e^{3}}{7}\right ) + x^{6} \cdot \left (\frac {2 A c d e^{3}}{3} + \frac {B a e^{4}}{6} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac {A a e^{4}}{5} + \frac {6 A c d^{2} e^{2}}{5} + \frac {4 B a d e^{3}}{5} + \frac {4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + A c d^{3} e + \frac {3 B a d^{2} e^{2}}{2} + \frac {B c d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a d^{2} e^{2} + \frac {A c d^{4}}{3} + \frac {4 B a d^{3} e}{3}\right ) + x^{2} \cdot \left (2 A a d^{3} e + \frac {B a d^{4}}{2}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{4} + 4 \, A a d^{3} e\right )} x^{2} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (100) = 200\).

Time = 0.26 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.99 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {4}{7} \, B c d e^{3} x^{7} + \frac {1}{7} \, A c e^{4} x^{7} + B c d^{2} e^{2} x^{6} + \frac {2}{3} \, A c d e^{3} x^{6} + \frac {1}{6} \, B a e^{4} x^{6} + \frac {4}{5} \, B c d^{3} e x^{5} + \frac {6}{5} \, A c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a d e^{3} x^{5} + \frac {1}{5} \, A a e^{4} x^{5} + \frac {1}{4} \, B c d^{4} x^{4} + A c d^{3} e x^{4} + \frac {3}{2} \, B a d^{2} e^{2} x^{4} + A a d e^{3} x^{4} + \frac {1}{3} \, A c d^{4} x^{3} + \frac {4}{3} \, B a d^{3} e x^{3} + 2 \, A a d^{2} e^{2} x^{3} + \frac {1}{2} \, B a d^{4} x^{2} + 2 \, A a d^{3} e x^{2} + A a d^{4} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.71 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=x^3\,\left (\frac {A\,c\,d^4}{3}+\frac {4\,B\,a\,d^3\,e}{3}+2\,A\,a\,d^2\,e^2\right )+x^6\,\left (B\,c\,d^2\,e^2+\frac {2\,A\,c\,d\,e^3}{3}+\frac {B\,a\,e^4}{6}\right )+x^4\,\left (\frac {B\,c\,d^4}{4}+A\,c\,d^3\,e+\frac {3\,B\,a\,d^2\,e^2}{2}+A\,a\,d\,e^3\right )+x^5\,\left (\frac {4\,B\,c\,d^3\,e}{5}+\frac {6\,A\,c\,d^2\,e^2}{5}+\frac {4\,B\,a\,d\,e^3}{5}+\frac {A\,a\,e^4}{5}\right )+A\,a\,d^4\,x+\frac {B\,c\,e^4\,x^8}{8}+\frac {a\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c\,e^3\,x^7\,\left (A\,e+4\,B\,d\right )}{7} \]

[In]

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

[Out]

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