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

Optimal. Leaf size=92 \[ \frac {e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac {3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac {(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac {e^3 (a+b x)^7}{7 b^4} \]

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Rubi [A]  time = 0.10, antiderivative size = 92, 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} \begin {gather*} \frac {e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac {3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac {(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac {e^3 (a+b x)^7}{7 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 161, normalized size = 1.75 \begin {gather*} a^3 d^3 x+\frac {3}{5} b e x^5 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a d x^3 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac {3}{2} a^2 d^2 x^2 (a e+b d)+\frac {1}{4} x^4 \left (a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{2} b^2 e^2 x^6 (a e+b d)+\frac {1}{7} b^3 e^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.39, size = 188, normalized size = 2.04 \begin {gather*} \frac {1}{7} x^{7} e^{3} b^{3} + \frac {1}{2} x^{6} e^{2} d b^{3} + \frac {1}{2} x^{6} e^{3} b^{2} a + \frac {3}{5} x^{5} e d^{2} b^{3} + \frac {9}{5} x^{5} e^{2} d b^{2} a + \frac {3}{5} x^{5} e^{3} b a^{2} + \frac {1}{4} x^{4} d^{3} b^{3} + \frac {9}{4} x^{4} e d^{2} b^{2} a + \frac {9}{4} x^{4} e^{2} d b a^{2} + \frac {1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} b^{2} a + 3 x^{3} e d^{2} b a^{2} + x^{3} e^{2} d a^{3} + \frac {3}{2} x^{2} d^{3} b a^{2} + \frac {3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 184, normalized size = 2.00 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} e^{3} + \frac {1}{2} \, b^{3} d x^{6} e^{2} + \frac {3}{5} \, b^{3} d^{2} x^{5} e + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {1}{2} \, a b^{2} x^{6} e^{3} + \frac {9}{5} \, a b^{2} d x^{5} e^{2} + \frac {9}{4} \, a b^{2} d^{2} x^{4} e + a b^{2} d^{3} x^{3} + \frac {3}{5} \, a^{2} b x^{5} e^{3} + \frac {9}{4} \, a^{2} b d x^{4} e^{2} + 3 \, a^{2} b d^{2} x^{3} e + \frac {3}{2} \, a^{2} b d^{3} x^{2} + \frac {1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac {3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 244, normalized size = 2.65 \begin {gather*} \frac {b^{3} e^{3} x^{7}}{7}+a^{3} d^{3} x +\frac {\left (2 a \,b^{2} e^{3}+\left (a \,e^{3}+3 b d \,e^{2}\right ) b^{2}\right ) x^{6}}{6}+\frac {\left (a^{2} b \,e^{3}+2 \left (a \,e^{3}+3 b d \,e^{2}\right ) a b +\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (\left (a \,e^{3}+3 b d \,e^{2}\right ) a^{2}+2 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a b +\left (3 a \,d^{2} e +b \,d^{3}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (a \,b^{2} d^{3}+\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{2}+2 \left (3 a \,d^{2} e +b \,d^{3}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a^{2} b \,d^{3}+\left (3 a \,d^{2} e +b \,d^{3}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 167, normalized size = 1.82 \begin {gather*} \frac {1}{7} \, b^{3} e^{3} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} + {\left (a b^{2} d^{3} + 3 \, a^{2} b d^{2} e + a^{3} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 152, normalized size = 1.65 \begin {gather*} x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,a^2\,b\,d\,e^2}{4}+\frac {9\,a\,b^2\,d^2\,e}{4}+\frac {b^3\,d^3}{4}\right )+a^3\,d^3\,x+\frac {b^3\,e^3\,x^7}{7}+a\,d\,x^3\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )+\frac {3\,b\,e\,x^5\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )}{5}+\frac {3\,a^2\,d^2\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {b^2\,e^2\,x^6\,\left (a\,e+b\,d\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.10, size = 190, normalized size = 2.07 \begin {gather*} a^{3} d^{3} x + \frac {b^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac {a b^{2} e^{3}}{2} + \frac {b^{3} d e^{2}}{2}\right ) + x^{5} \left (\frac {3 a^{2} b e^{3}}{5} + \frac {9 a b^{2} d e^{2}}{5} + \frac {3 b^{3} d^{2} e}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} b d e^{2}}{4} + \frac {9 a b^{2} d^{2} e}{4} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a b^{2} d^{3}\right ) + x^{2} \left (\frac {3 a^{3} d^{2} e}{2} + \frac {3 a^{2} b d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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