3.3.9 \(\int (d+e x)^4 (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=137 \[ \frac {(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac {d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]

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Rubi [A]  time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} \frac {(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac {d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(b*x + c*x^2)^2,x]

[Out]

(d^2*(c*d - b*e)^2*(d + e*x)^5)/(5*e^5) - (d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^6)/(3*e^5) + ((6*c^2*d^2 - 6*
b*c*d*e + b^2*e^2)*(d + e*x)^7)/(7*e^5) - (c*(2*c*d - b*e)*(d + e*x)^8)/(4*e^5) + (c^2*(d + e*x)^9)/(9*e^5)

Rule 698

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)^4 \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 (d+e x)^4}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^5}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^6}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^7}{e^4}+\frac {c^2 (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac {d^2 (c d-b e)^2 (d+e x)^5}{5 e^5}-\frac {d (c d-b e) (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 159, normalized size = 1.16 \begin {gather*} \frac {1}{7} e^2 x^7 \left (b^2 e^2+8 b c d e+6 c^2 d^2\right )+\frac {2}{3} d e x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac {1}{5} d^2 x^5 \left (6 b^2 e^2+8 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^4 x^3+\frac {1}{2} b d^3 x^4 (2 b e+c d)+\frac {1}{4} c e^3 x^8 (b e+2 c d)+\frac {1}{9} c^2 e^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(b*x + c*x^2)^2,x]

[Out]

(b^2*d^4*x^3)/3 + (b*d^3*(c*d + 2*b*e)*x^4)/2 + (d^2*(c^2*d^2 + 8*b*c*d*e + 6*b^2*e^2)*x^5)/5 + (2*d*e*(c^2*d^
2 + 3*b*c*d*e + b^2*e^2)*x^6)/3 + (e^2*(6*c^2*d^2 + 8*b*c*d*e + b^2*e^2)*x^7)/7 + (c*e^3*(2*c*d + b*e)*x^8)/4
+ (c^2*e^4*x^9)/9

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^4*(b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(d + e*x)^4*(b*x + c*x^2)^2, x]

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fricas [A]  time = 0.36, size = 175, normalized size = 1.28 \begin {gather*} \frac {1}{9} x^{9} e^{4} c^{2} + \frac {1}{2} x^{8} e^{3} d c^{2} + \frac {1}{4} x^{8} e^{4} c b + \frac {6}{7} x^{7} e^{2} d^{2} c^{2} + \frac {8}{7} x^{7} e^{3} d c b + \frac {1}{7} x^{7} e^{4} b^{2} + \frac {2}{3} x^{6} e d^{3} c^{2} + 2 x^{6} e^{2} d^{2} c b + \frac {2}{3} x^{6} e^{3} d b^{2} + \frac {1}{5} x^{5} d^{4} c^{2} + \frac {8}{5} x^{5} e d^{3} c b + \frac {6}{5} x^{5} e^{2} d^{2} b^{2} + \frac {1}{2} x^{4} d^{4} c b + x^{4} e d^{3} b^{2} + \frac {1}{3} x^{3} d^{4} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^4*c^2 + 1/2*x^8*e^3*d*c^2 + 1/4*x^8*e^4*c*b + 6/7*x^7*e^2*d^2*c^2 + 8/7*x^7*e^3*d*c*b + 1/7*x^7*e^4*
b^2 + 2/3*x^6*e*d^3*c^2 + 2*x^6*e^2*d^2*c*b + 2/3*x^6*e^3*d*b^2 + 1/5*x^5*d^4*c^2 + 8/5*x^5*e*d^3*c*b + 6/5*x^
5*e^2*d^2*b^2 + 1/2*x^4*d^4*c*b + x^4*e*d^3*b^2 + 1/3*x^3*d^4*b^2

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giac [A]  time = 0.15, size = 169, normalized size = 1.23 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} e^{4} + \frac {1}{2} \, c^{2} d x^{8} e^{3} + \frac {6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac {2}{3} \, c^{2} d^{3} x^{6} e + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {1}{4} \, b c x^{8} e^{4} + \frac {8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac {8}{5} \, b c d^{3} x^{5} e + \frac {1}{2} \, b c d^{4} x^{4} + \frac {1}{7} \, b^{2} x^{7} e^{4} + \frac {2}{3} \, b^{2} d x^{6} e^{3} + \frac {6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac {1}{3} \, b^{2} d^{4} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*x^9*e^4 + 1/2*c^2*d*x^8*e^3 + 6/7*c^2*d^2*x^7*e^2 + 2/3*c^2*d^3*x^6*e + 1/5*c^2*d^4*x^5 + 1/4*b*c*x^8*
e^4 + 8/7*b*c*d*x^7*e^3 + 2*b*c*d^2*x^6*e^2 + 8/5*b*c*d^3*x^5*e + 1/2*b*c*d^4*x^4 + 1/7*b^2*x^7*e^4 + 2/3*b^2*
d*x^6*e^3 + 6/5*b^2*d^2*x^5*e^2 + b^2*d^3*x^4*e + 1/3*b^2*d^4*x^3

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maple [A]  time = 0.04, size = 166, normalized size = 1.21 \begin {gather*} \frac {c^{2} e^{4} x^{9}}{9}+\frac {b^{2} d^{4} x^{3}}{3}+\frac {\left (2 e^{4} b c +4 d \,e^{3} c^{2}\right ) x^{8}}{8}+\frac {\left (e^{4} b^{2}+8 d \,e^{3} b c +6 d^{2} e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (4 d \,e^{3} b^{2}+12 d^{2} e^{2} b c +4 d^{3} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (6 d^{2} e^{2} b^{2}+8 d^{3} e b c +c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (4 d^{3} e \,b^{2}+2 d^{4} b c \right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+b*x)^2,x)

[Out]

1/9*e^4*c^2*x^9+1/8*(2*b*c*e^4+4*c^2*d*e^3)*x^8+1/7*(b^2*e^4+8*b*c*d*e^3+6*c^2*d^2*e^2)*x^7+1/6*(4*b^2*d*e^3+1
2*b*c*d^2*e^2+4*c^2*d^3*e)*x^6+1/5*(6*b^2*d^2*e^2+8*b*c*d^3*e+c^2*d^4)*x^5+1/4*(4*b^2*d^3*e+2*b*c*d^4)*x^4+1/3
*d^4*b^2*x^3

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maxima [A]  time = 1.38, size = 161, normalized size = 1.18 \begin {gather*} \frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

1/9*c^2*e^4*x^9 + 1/3*b^2*d^4*x^3 + 1/4*(2*c^2*d*e^3 + b*c*e^4)*x^8 + 1/7*(6*c^2*d^2*e^2 + 8*b*c*d*e^3 + b^2*e
^4)*x^7 + 2/3*(c^2*d^3*e + 3*b*c*d^2*e^2 + b^2*d*e^3)*x^6 + 1/5*(c^2*d^4 + 8*b*c*d^3*e + 6*b^2*d^2*e^2)*x^5 +
1/2*(b*c*d^4 + 2*b^2*d^3*e)*x^4

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mupad [B]  time = 0.07, size = 149, normalized size = 1.09 \begin {gather*} x^5\,\left (\frac {6\,b^2\,d^2\,e^2}{5}+\frac {8\,b\,c\,d^3\,e}{5}+\frac {c^2\,d^4}{5}\right )+x^7\,\left (\frac {b^2\,e^4}{7}+\frac {8\,b\,c\,d\,e^3}{7}+\frac {6\,c^2\,d^2\,e^2}{7}\right )+\frac {b^2\,d^4\,x^3}{3}+\frac {c^2\,e^4\,x^9}{9}+\frac {b\,d^3\,x^4\,\left (2\,b\,e+c\,d\right )}{2}+\frac {c\,e^3\,x^8\,\left (b\,e+2\,c\,d\right )}{4}+\frac {2\,d\,e\,x^6\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(d + e*x)^4,x)

[Out]

x^5*((c^2*d^4)/5 + (6*b^2*d^2*e^2)/5 + (8*b*c*d^3*e)/5) + x^7*((b^2*e^4)/7 + (6*c^2*d^2*e^2)/7 + (8*b*c*d*e^3)
/7) + (b^2*d^4*x^3)/3 + (c^2*e^4*x^9)/9 + (b*d^3*x^4*(2*b*e + c*d))/2 + (c*e^3*x^8*(b*e + 2*c*d))/4 + (2*d*e*x
^6*(b^2*e^2 + c^2*d^2 + 3*b*c*d*e))/3

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sympy [A]  time = 0.11, size = 178, normalized size = 1.30 \begin {gather*} \frac {b^{2} d^{4} x^{3}}{3} + \frac {c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac {b c e^{4}}{4} + \frac {c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac {b^{2} e^{4}}{7} + \frac {8 b c d e^{3}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac {6 b^{2} d^{2} e^{2}}{5} + \frac {8 b c d^{3} e}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (b^{2} d^{3} e + \frac {b c d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(c*x**2+b*x)**2,x)

[Out]

b**2*d**4*x**3/3 + c**2*e**4*x**9/9 + x**8*(b*c*e**4/4 + c**2*d*e**3/2) + x**7*(b**2*e**4/7 + 8*b*c*d*e**3/7 +
6*c**2*d**2*e**2/7) + x**6*(2*b**2*d*e**3/3 + 2*b*c*d**2*e**2 + 2*c**2*d**3*e/3) + x**5*(6*b**2*d**2*e**2/5 +
8*b*c*d**3*e/5 + c**2*d**4/5) + x**4*(b**2*d**3*e + b*c*d**4/2)

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