3.1910 \(\int (a+b x) (d+e x) (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=38 \[ \frac {(a+b x)^6 (b d-a e)}{6 b^2}+\frac {e (a+b x)^7}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \[ \frac {(a+b x)^6 (b d-a e)}{6 b^2}+\frac {e (a+b x)^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)*(a + b*x)^6)/(6*b^2) + (e*(a + b*x)^7)/(7*b^2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 109, normalized size = 2.87 \[ a^5 d x+\frac {1}{2} a^4 x^2 (a e+5 b d)+\frac {5}{3} a^3 b x^3 (a e+2 b d)+\frac {5}{2} a^2 b^2 x^4 (a e+b d)+\frac {1}{6} b^4 x^6 (5 a e+b d)+a b^3 x^5 (2 a e+b d)+\frac {1}{7} b^5 e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.03, size = 121, normalized size = 3.18 \[ \frac {1}{7} x^{7} e b^{5} + \frac {1}{6} x^{6} d b^{5} + \frac {5}{6} x^{6} e b^{4} a + x^{5} d b^{4} a + 2 x^{5} e b^{3} a^{2} + \frac {5}{2} x^{4} d b^{3} a^{2} + \frac {5}{2} x^{4} e b^{2} a^{3} + \frac {10}{3} x^{3} d b^{2} a^{3} + \frac {5}{3} x^{3} e b a^{4} + \frac {5}{2} x^{2} d b a^{4} + \frac {1}{2} x^{2} e a^{5} + x d a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e*b^5 + 1/6*x^6*d*b^5 + 5/6*x^6*e*b^4*a + x^5*d*b^4*a + 2*x^5*e*b^3*a^2 + 5/2*x^4*d*b^3*a^2 + 5/2*x^4*
e*b^2*a^3 + 10/3*x^3*d*b^2*a^3 + 5/3*x^3*e*b*a^4 + 5/2*x^2*d*b*a^4 + 1/2*x^2*e*a^5 + x*d*a^5

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giac [B]  time = 0.15, size = 127, normalized size = 3.34 \[ \frac {1}{7} \, b^{5} x^{7} e + \frac {1}{6} \, b^{5} d x^{6} + \frac {5}{6} \, a b^{4} x^{6} e + a b^{4} d x^{5} + 2 \, a^{2} b^{3} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d x^{4} + \frac {5}{2} \, a^{3} b^{2} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d x^{3} + \frac {5}{3} \, a^{4} b x^{3} e + \frac {5}{2} \, a^{4} b d x^{2} + \frac {1}{2} \, a^{5} x^{2} e + a^{5} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*x^7*e + 1/6*b^5*d*x^6 + 5/6*a*b^4*x^6*e + a*b^4*d*x^5 + 2*a^2*b^3*x^5*e + 5/2*a^2*b^3*d*x^4 + 5/2*a^3*
b^2*x^4*e + 10/3*a^3*b^2*d*x^3 + 5/3*a^4*b*x^3*e + 5/2*a^4*b*d*x^2 + 1/2*a^5*x^2*e + a^5*d*x

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maple [B]  time = 0.04, size = 172, normalized size = 4.53 \[ \frac {b^{5} e \,x^{7}}{7}+a^{5} d x +\frac {\left (4 a \,b^{4} e +\left (a e +b d \right ) b^{4}\right ) x^{6}}{6}+\frac {\left (6 a^{2} b^{3} e +a \,b^{4} d +4 \left (a e +b d \right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 a^{3} b^{2} e +4 a^{2} b^{3} d +6 \left (a e +b d \right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (a^{4} b e +6 a^{3} b^{2} d +4 \left (a e +b d \right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b d +\left (a e +b d \right ) a^{4}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.60, size = 115, normalized size = 3.03 \[ \frac {1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac {1}{6} \, {\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} + {\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.02, size = 103, normalized size = 2.71 \[ x^2\,\left (\frac {e\,a^5}{2}+\frac {5\,b\,d\,a^4}{2}\right )+x^6\,\left (\frac {d\,b^5}{6}+\frac {5\,a\,e\,b^4}{6}\right )+\frac {b^5\,e\,x^7}{7}+a^5\,d\,x+\frac {5\,a^3\,b\,x^3\,\left (a\,e+2\,b\,d\right )}{3}+a\,b^3\,x^5\,\left (2\,a\,e+b\,d\right )+\frac {5\,a^2\,b^2\,x^4\,\left (a\,e+b\,d\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.10, size = 129, normalized size = 3.39 \[ a^{5} d x + \frac {b^{5} e x^{7}}{7} + x^{6} \left (\frac {5 a b^{4} e}{6} + \frac {b^{5} d}{6}\right ) + x^{5} \left (2 a^{2} b^{3} e + a b^{4} d\right ) + x^{4} \left (\frac {5 a^{3} b^{2} e}{2} + \frac {5 a^{2} b^{3} d}{2}\right ) + x^{3} \left (\frac {5 a^{4} b e}{3} + \frac {10 a^{3} b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{5} e}{2} + \frac {5 a^{4} b d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d*x + b**5*e*x**7/7 + x**6*(5*a*b**4*e/6 + b**5*d/6) + x**5*(2*a**2*b**3*e + a*b**4*d) + x**4*(5*a**3*b**
2*e/2 + 5*a**2*b**3*d/2) + x**3*(5*a**4*b*e/3 + 10*a**3*b**2*d/3) + x**2*(a**5*e/2 + 5*a**4*b*d/2)

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