3.1104 \(\int (A+B x) (d+e x)^3 (b x+c x^2) \, dx\)

Optimal. Leaf size=118 \[ -\frac {(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac {(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 118, normalized size of antiderivative = 1.00, 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 = {771} \[ -\frac {(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac {(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 135, normalized size = 1.14 \[ \frac {1}{3} d^2 x^3 (3 A b e+A c d+b B d)+\frac {1}{6} e^2 x^6 (A c e+b B e+3 B c d)+\frac {1}{5} e x^5 (A e (b e+3 c d)+3 B d (b e+c d))+\frac {1}{4} d x^4 (3 A e (b e+c d)+B d (3 b e+c d))+\frac {1}{2} A b d^3 x^2+\frac {1}{7} B c e^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.80, size = 168, normalized size = 1.42 \[ \frac {1}{7} x^{7} e^{3} c B + \frac {1}{2} x^{6} e^{2} d c B + \frac {1}{6} x^{6} e^{3} b B + \frac {1}{6} x^{6} e^{3} c A + \frac {3}{5} x^{5} e d^{2} c B + \frac {3}{5} x^{5} e^{2} d b B + \frac {3}{5} x^{5} e^{2} d c A + \frac {1}{5} x^{5} e^{3} b A + \frac {1}{4} x^{4} d^{3} c B + \frac {3}{4} x^{4} e d^{2} b B + \frac {3}{4} x^{4} e d^{2} c A + \frac {3}{4} x^{4} e^{2} d b A + \frac {1}{3} x^{3} d^{3} b B + \frac {1}{3} x^{3} d^{3} c A + x^{3} e d^{2} b A + \frac {1}{2} x^{2} d^{3} b A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 164, normalized size = 1.39 \[ \frac {1}{7} \, B c x^{7} e^{3} + \frac {1}{2} \, B c d x^{6} e^{2} + \frac {3}{5} \, B c d^{2} x^{5} e + \frac {1}{4} \, B c d^{3} x^{4} + \frac {1}{6} \, B b x^{6} e^{3} + \frac {1}{6} \, A c x^{6} e^{3} + \frac {3}{5} \, B b d x^{5} e^{2} + \frac {3}{5} \, A c d x^{5} e^{2} + \frac {3}{4} \, B b d^{2} x^{4} e + \frac {3}{4} \, A c d^{2} x^{4} e + \frac {1}{3} \, B b d^{3} x^{3} + \frac {1}{3} \, A c d^{3} x^{3} + \frac {1}{5} \, A b x^{5} e^{3} + \frac {3}{4} \, A b d x^{4} e^{2} + A b d^{2} x^{3} e + \frac {1}{2} \, A b d^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 152, normalized size = 1.29 \[ \frac {B c \,e^{3} x^{7}}{7}+\frac {A b \,d^{3} x^{2}}{2}+\frac {\left (B b \,e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) c \right ) x^{6}}{6}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) b +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b +\left (3 A \,d^{2} e +B \,d^{3}\right ) c \right ) x^{4}}{4}+\frac {\left (A c \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 137, normalized size = 1.16 \[ \frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, A b d^{3} x^{2} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B c d^{2} e + A b e^{3} + 3 \, {\left (B b + A c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A b d e^{2} + 3 \, {\left (B b + A c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A b d^{2} e + {\left (B b + A c\right )} d^{3}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.37, size = 146, normalized size = 1.24 \[ x^3\,\left (\frac {A\,c\,d^3}{3}+\frac {B\,b\,d^3}{3}+A\,b\,d^2\,e\right )+x^6\,\left (\frac {A\,c\,e^3}{6}+\frac {B\,b\,e^3}{6}+\frac {B\,c\,d\,e^2}{2}\right )+x^4\,\left (\frac {B\,c\,d^3}{4}+\frac {3\,A\,b\,d\,e^2}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,B\,b\,d^2\,e}{4}\right )+x^5\,\left (\frac {A\,b\,e^3}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {3\,B\,b\,d\,e^2}{5}+\frac {3\,B\,c\,d^2\,e}{5}\right )+\frac {A\,b\,d^3\,x^2}{2}+\frac {B\,c\,e^3\,x^7}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 177, normalized size = 1.50 \[ \frac {A b d^{3} x^{2}}{2} + \frac {B c e^{3} x^{7}}{7} + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B b e^{3}}{6} + \frac {B c d e^{2}}{2}\right ) + x^{5} \left (\frac {A b e^{3}}{5} + \frac {3 A c d e^{2}}{5} + \frac {3 B b d e^{2}}{5} + \frac {3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac {3 A b d e^{2}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B b d^{2} e}{4} + \frac {B c d^{3}}{4}\right ) + x^{3} \left (A b d^{2} e + \frac {A c d^{3}}{3} + \frac {B b d^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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