3.14.3 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=124 \[ \frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {c (d+e x)^6 (2 c d-b e)}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4} \]

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Rubi [A]  time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {c (d+e x)^6 (2 c d-b e)}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 175, normalized size = 1.41 \begin {gather*} \frac {1}{5} e x^5 \left (c e (2 a e+9 b d)+b^2 e^2+6 c^2 d^2\right )+d x^3 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac {1}{2} d^2 x^2 \left (3 a b e+2 a c d+b^2 d\right )+\frac {1}{4} x^4 \left (3 c d e (2 a e+3 b d)+b e^2 (a e+3 b d)+2 c^2 d^3\right )+a b d^3 x+\frac {1}{2} c e^2 x^6 (b e+2 c d)+\frac {2}{7} c^2 e^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*b*d^3*x + (d^2*(b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + d*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^3 + ((2*c^
2*d^3 + b*e^2*(3*b*d + a*e) + 3*c*d*e*(3*b*d + 2*a*e))*x^4)/4 + (e*(6*c^2*d^2 + b^2*e^2 + c*e*(9*b*d + 2*a*e))
*x^5)/5 + (c*e^2*(2*c*d + b*e)*x^6)/2 + (2*c^2*e^3*x^7)/7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2), x]

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fricas [A]  time = 0.37, size = 211, normalized size = 1.70 \begin {gather*} \frac {2}{7} x^{7} e^{3} c^{2} + x^{6} e^{2} d c^{2} + \frac {1}{2} x^{6} e^{3} c b + \frac {6}{5} x^{5} e d^{2} c^{2} + \frac {9}{5} x^{5} e^{2} d c b + \frac {1}{5} x^{5} e^{3} b^{2} + \frac {2}{5} x^{5} e^{3} c a + \frac {1}{2} x^{4} d^{3} c^{2} + \frac {9}{4} x^{4} e d^{2} c b + \frac {3}{4} x^{4} e^{2} d b^{2} + \frac {3}{2} x^{4} e^{2} d c a + \frac {1}{4} x^{4} e^{3} b a + x^{3} d^{3} c b + x^{3} e d^{2} b^{2} + 2 x^{3} e d^{2} c a + x^{3} e^{2} d b a + \frac {1}{2} x^{2} d^{3} b^{2} + x^{2} d^{3} c a + \frac {3}{2} x^{2} e d^{2} b a + x d^{3} b a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*x^7*e^3*c^2 + x^6*e^2*d*c^2 + 1/2*x^6*e^3*c*b + 6/5*x^5*e*d^2*c^2 + 9/5*x^5*e^2*d*c*b + 1/5*x^5*e^3*b^2 +
2/5*x^5*e^3*c*a + 1/2*x^4*d^3*c^2 + 9/4*x^4*e*d^2*c*b + 3/4*x^4*e^2*d*b^2 + 3/2*x^4*e^2*d*c*a + 1/4*x^4*e^3*b*
a + x^3*d^3*c*b + x^3*e*d^2*b^2 + 2*x^3*e*d^2*c*a + x^3*e^2*d*b*a + 1/2*x^2*d^3*b^2 + x^2*d^3*c*a + 3/2*x^2*e*
d^2*b*a + x*d^3*b*a

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giac [A]  time = 0.16, size = 206, normalized size = 1.66 \begin {gather*} \frac {2}{7} \, c^{2} x^{7} e^{3} + c^{2} d x^{6} e^{2} + \frac {6}{5} \, c^{2} d^{2} x^{5} e + \frac {1}{2} \, c^{2} d^{3} x^{4} + \frac {1}{2} \, b c x^{6} e^{3} + \frac {9}{5} \, b c d x^{5} e^{2} + \frac {9}{4} \, b c d^{2} x^{4} e + b c d^{3} x^{3} + \frac {1}{5} \, b^{2} x^{5} e^{3} + \frac {2}{5} \, a c x^{5} e^{3} + \frac {3}{4} \, b^{2} d x^{4} e^{2} + \frac {3}{2} \, a c d x^{4} e^{2} + b^{2} d^{2} x^{3} e + 2 \, a c d^{2} x^{3} e + \frac {1}{2} \, b^{2} d^{3} x^{2} + a c d^{3} x^{2} + \frac {1}{4} \, a b x^{4} e^{3} + a b d x^{3} e^{2} + \frac {3}{2} \, a b d^{2} x^{2} e + a b d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c^2*x^7*e^3 + c^2*d*x^6*e^2 + 6/5*c^2*d^2*x^5*e + 1/2*c^2*d^3*x^4 + 1/2*b*c*x^6*e^3 + 9/5*b*c*d*x^5*e^2 +
9/4*b*c*d^2*x^4*e + b*c*d^3*x^3 + 1/5*b^2*x^5*e^3 + 2/5*a*c*x^5*e^3 + 3/4*b^2*d*x^4*e^2 + 3/2*a*c*d*x^4*e^2 +
b^2*d^2*x^3*e + 2*a*c*d^2*x^3*e + 1/2*b^2*d^3*x^2 + a*c*d^3*x^2 + 1/4*a*b*x^4*e^3 + a*b*d*x^3*e^2 + 3/2*a*b*d^
2*x^2*e + a*b*d^3*x

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maple [A]  time = 0.05, size = 221, normalized size = 1.78 \begin {gather*} \frac {2 c^{2} e^{3} x^{7}}{7}+a b \,d^{3} x +\frac {\left (2 b c \,e^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) c \right ) x^{6}}{6}+\frac {\left (2 a c \,e^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) b +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (b \,e^{3}+6 d \,e^{2} c \right ) a +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) c \right ) x^{4}}{4}+\frac {\left (b c \,d^{3}+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) b \right ) x^{3}}{3}+\frac {\left (b^{2} d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) a \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 174, normalized size = 1.40 \begin {gather*} \frac {2}{7} \, c^{2} e^{3} x^{7} + \frac {1}{2} \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{6} + a b d^{3} x + \frac {1}{5} \, {\left (6 \, c^{2} d^{2} e + 9 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, c^{2} d^{3} + 9 \, b c d^{2} e + a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{4} + {\left (b c d^{3} + a b d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a b d^{2} e + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c^2*e^3*x^7 + 1/2*(2*c^2*d*e^2 + b*c*e^3)*x^6 + a*b*d^3*x + 1/5*(6*c^2*d^2*e + 9*b*c*d*e^2 + (b^2 + 2*a*c)
*e^3)*x^5 + 1/4*(2*c^2*d^3 + 9*b*c*d^2*e + a*b*e^3 + 3*(b^2 + 2*a*c)*d*e^2)*x^4 + (b*c*d^3 + a*b*d*e^2 + (b^2
+ 2*a*c)*d^2*e)*x^3 + 1/2*(3*a*b*d^2*e + (b^2 + 2*a*c)*d^3)*x^2

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mupad [B]  time = 0.07, size = 179, normalized size = 1.44 \begin {gather*} x^4\,\left (\frac {3\,b^2\,d\,e^2}{4}+\frac {9\,b\,c\,d^2\,e}{4}+\frac {a\,b\,e^3}{4}+\frac {c^2\,d^3}{2}+\frac {3\,a\,c\,d\,e^2}{2}\right )+x^3\,\left (b^2\,d^2\,e+c\,b\,d^3+a\,b\,d\,e^2+2\,a\,c\,d^2\,e\right )+x^2\,\left (\frac {b^2\,d^3}{2}+\frac {3\,a\,e\,b\,d^2}{2}+a\,c\,d^3\right )+x^5\,\left (\frac {b^2\,e^3}{5}+\frac {9\,b\,c\,d\,e^2}{5}+\frac {6\,c^2\,d^2\,e}{5}+\frac {2\,a\,c\,e^3}{5}\right )+\frac {2\,c^2\,e^3\,x^7}{7}+\frac {c\,e^2\,x^6\,\left (b\,e+2\,c\,d\right )}{2}+a\,b\,d^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((c^2*d^3)/2 + (3*b^2*d*e^2)/4 + (a*b*e^3)/4 + (3*a*c*d*e^2)/2 + (9*b*c*d^2*e)/4) + x^3*(b^2*d^2*e + b*c*d
^3 + a*b*d*e^2 + 2*a*c*d^2*e) + x^2*((b^2*d^3)/2 + a*c*d^3 + (3*a*b*d^2*e)/2) + x^5*((b^2*e^3)/5 + (6*c^2*d^2*
e)/5 + (2*a*c*e^3)/5 + (9*b*c*d*e^2)/5) + (2*c^2*e^3*x^7)/7 + (c*e^2*x^6*(b*e + 2*c*d))/2 + a*b*d^3*x

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sympy [A]  time = 0.10, size = 211, normalized size = 1.70 \begin {gather*} a b d^{3} x + \frac {2 c^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac {b c e^{3}}{2} + c^{2} d e^{2}\right ) + x^{5} \left (\frac {2 a c e^{3}}{5} + \frac {b^{2} e^{3}}{5} + \frac {9 b c d e^{2}}{5} + \frac {6 c^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac {a b e^{3}}{4} + \frac {3 a c d e^{2}}{2} + \frac {3 b^{2} d e^{2}}{4} + \frac {9 b c d^{2} e}{4} + \frac {c^{2} d^{3}}{2}\right ) + x^{3} \left (a b d e^{2} + 2 a c d^{2} e + b^{2} d^{2} e + b c d^{3}\right ) + x^{2} \left (\frac {3 a b d^{2} e}{2} + a c d^{3} + \frac {b^{2} d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b*d**3*x + 2*c**2*e**3*x**7/7 + x**6*(b*c*e**3/2 + c**2*d*e**2) + x**5*(2*a*c*e**3/5 + b**2*e**3/5 + 9*b*c*d
*e**2/5 + 6*c**2*d**2*e/5) + x**4*(a*b*e**3/4 + 3*a*c*d*e**2/2 + 3*b**2*d*e**2/4 + 9*b*c*d**2*e/4 + c**2*d**3/
2) + x**3*(a*b*d*e**2 + 2*a*c*d**2*e + b**2*d**2*e + b*c*d**3) + x**2*(3*a*b*d**2*e/2 + a*c*d**3 + b**2*d**3/2
)

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