3.22.20 \(\int (d+e x)^2 (a+b x+c x^2)^2 \, dx\) [2120]

Optimal. Leaf size=156 \[ \frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{2 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^5}-\frac {c (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 156, 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 = {712} \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^5}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac {c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 712

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 153, normalized size = 0.98 \begin {gather*} a^2 d^2 x+a d (b d+a e) x^2+\frac {1}{3} \left (b^2 d^2+2 a c d^2+4 a b d e+a^2 e^2\right ) x^3+\frac {1}{2} \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2+2 a c e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.73, size = 155, normalized size = 0.99

method	result	size
default	\(\frac {c^{2} e^{2} x^{7}}{7}+\frac {\left (2 b c \,e^{2}+2 c^{2} d e \right ) x^{6}}{6}+\frac {\left (c^{2} d^{2}+4 b c d e +e^{2} \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 b c \,d^{2}+2 d e \left (2 a c +b^{2}\right )+2 a b \,e^{2}\right ) x^{4}}{4}+\frac {\left (d^{2} \left (2 a c +b^{2}\right )+4 a b d e +a^{2} e^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{2}+2 a b \,d^{2}\right ) x^{2}}{2}+a^{2} d^{2} x\)	\(155\)
norman	\(\frac {c^{2} e^{2} x^{7}}{7}+\left (\frac {1}{3} b c \,e^{2}+\frac {1}{3} c^{2} d e \right ) x^{6}+\left (\frac {2}{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2}+\frac {4}{5} b c d e +\frac {1}{5} c^{2} d^{2}\right ) x^{5}+\left (\frac {1}{2} a b \,e^{2}+a c d e +\frac {1}{2} b^{2} d e +\frac {1}{2} b c \,d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} e^{2}+\frac {4}{3} a b d e +\frac {2}{3} a c \,d^{2}+\frac {1}{3} b^{2} d^{2}\right ) x^{3}+\left (d e \,a^{2}+a b \,d^{2}\right ) x^{2}+a^{2} d^{2} x\)	\(156\)
gosper	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} c^{2} d e \,x^{6}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2} x^{5}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} x^{5} c^{2} d^{2}+\frac {1}{2} x^{4} a b \,e^{2}+a c d e \,x^{4}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} d^{2} a c \,x^{3}+\frac {1}{3} b^{2} d^{2} x^{3}+d e \,a^{2} x^{2}+a b \,d^{2} x^{2}+a^{2} d^{2} x\)	\(179\)
risch	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} c^{2} d e \,x^{6}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2} x^{5}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} x^{5} c^{2} d^{2}+\frac {1}{2} x^{4} a b \,e^{2}+a c d e \,x^{4}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} d^{2} a c \,x^{3}+\frac {1}{3} b^{2} d^{2} x^{3}+d e \,a^{2} x^{2}+a b \,d^{2} x^{2}+a^{2} d^{2} x\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 150, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} e^{2} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b c d^{2} + a b e^{2} + {\left (b^{2} e + 2 \, a c e\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a b d e + {\left (b^{2} + 2 \, a c\right )} d^{2} + a^{2} e^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.48, size = 156, normalized size = 1.00 \begin {gather*} \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {1}{2} \, b c d^{2} x^{4} + a b d^{2} x^{2} + \frac {1}{3} \, {\left (b^{2} + 2 \, a c\right )} d^{2} x^{3} + a^{2} d^{2} x + \frac {1}{210} \, {\left (30 \, c^{2} x^{7} + 70 \, b c x^{6} + 105 \, a b x^{4} + 42 \, {\left (b^{2} + 2 \, a c\right )} x^{5} + 70 \, a^{2} x^{3}\right )} e^{2} + \frac {1}{30} \, {\left (10 \, c^{2} d x^{6} + 24 \, b c d x^{5} + 40 \, a b d x^{3} + 15 \, {\left (b^{2} + 2 \, a c\right )} d x^{4} + 30 \, a^{2} d x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*d^2*x^5 + 1/2*b*c*d^2*x^4 + a*b*d^2*x^2 + 1/3*(b^2 + 2*a*c)*d^2*x^3 + a^2*d^2*x + 1/210*(30*c^2*x^7 +
70*b*c*x^6 + 105*a*b*x^4 + 42*(b^2 + 2*a*c)*x^5 + 70*a^2*x^3)*e^2 + 1/30*(10*c^2*d*x^6 + 24*b*c*d*x^5 + 40*a*b
*d*x^3 + 15*(b^2 + 2*a*c)*d*x^4 + 30*a^2*d*x^2)*e

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Sympy [A]
time = 0.02, size = 173, normalized size = 1.11 \begin {gather*} a^{2} d^{2} x + \frac {c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac {b c e^{2}}{3} + \frac {c^{2} d e}{3}\right ) + x^{5} \cdot \left (\frac {2 a c e^{2}}{5} + \frac {b^{2} e^{2}}{5} + \frac {4 b c d e}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {a b e^{2}}{2} + a c d e + \frac {b^{2} d e}{2} + \frac {b c d^{2}}{2}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {4 a b d e}{3} + \frac {2 a c d^{2}}{3} + \frac {b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.11, size = 178, normalized size = 1.14 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} e^{2} + \frac {1}{3} \, c^{2} d x^{6} e + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {1}{3} \, b c x^{6} e^{2} + \frac {4}{5} \, b c d x^{5} e + \frac {1}{2} \, b c d^{2} x^{4} + \frac {1}{5} \, b^{2} x^{5} e^{2} + \frac {2}{5} \, a c x^{5} e^{2} + \frac {1}{2} \, b^{2} d x^{4} e + a c d x^{4} e + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {2}{3} \, a c d^{2} x^{3} + \frac {1}{2} \, a b x^{4} e^{2} + \frac {4}{3} \, a b d x^{3} e + a b d^{2} x^{2} + \frac {1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.06, size = 146, normalized size = 0.94 \begin {gather*} x^3\,\left (\frac {a^2\,e^2}{3}+\frac {4\,a\,b\,d\,e}{3}+\frac {2\,c\,a\,d^2}{3}+\frac {b^2\,d^2}{3}\right )+x^5\,\left (\frac {b^2\,e^2}{5}+\frac {4\,b\,c\,d\,e}{5}+\frac {c^2\,d^2}{5}+\frac {2\,a\,c\,e^2}{5}\right )+x^4\,\left (\frac {b^2\,d\,e}{2}+\frac {c\,b\,d^2}{2}+\frac {a\,b\,e^2}{2}+a\,c\,d\,e\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^7}{7}+a\,d\,x^2\,\left (a\,e+b\,d\right )+\frac {c\,e\,x^6\,\left (b\,e+c\,d\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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