3.12.30 \(\int (a+b x)^4 (c+d x) \, dx\)

Optimal. Leaf size=38 \[ \frac {(a+b x)^5 (b c-a d)}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} \frac {(a+b x)^5 (b c-a d)}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x),x]

[Out]

((b*c - a*d)*(a + b*x)^5)/(5*b^2) + (d*(a + b*x)^6)/(6*b^2)

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

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Mathematica [B]  time = 0.02, size = 84, normalized size = 2.21 \begin {gather*} \frac {1}{30} x \left (15 a^4 (2 c+d x)+20 a^3 b x (3 c+2 d x)+15 a^2 b^2 x^2 (4 c+3 d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x),x]

[Out]

(x*(15*a^4*(2*c + d*x) + 20*a^3*b*x*(3*c + 2*d*x) + 15*a^2*b^2*x^2*(4*c + 3*d*x) + 6*a*b^3*x^3*(5*c + 4*d*x) +
b^4*x^4*(6*c + 5*d*x)))/30

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.20, size = 97, normalized size = 2.55 \begin {gather*} \frac {1}{6} x^{6} d b^{4} + \frac {1}{5} x^{5} c b^{4} + \frac {4}{5} x^{5} d b^{3} a + x^{4} c b^{3} a + \frac {3}{2} x^{4} d b^{2} a^{2} + 2 x^{3} c b^{2} a^{2} + \frac {4}{3} x^{3} d b a^{3} + 2 x^{2} c b a^{3} + \frac {1}{2} x^{2} d a^{4} + x c a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.03, size = 97, normalized size = 2.55 \begin {gather*} \frac {1}{6} \, b^{4} d x^{6} + \frac {1}{5} \, b^{4} c x^{5} + \frac {4}{5} \, a b^{3} d x^{5} + a b^{3} c x^{4} + \frac {3}{2} \, a^{2} b^{2} d x^{4} + 2 \, a^{2} b^{2} c x^{3} + \frac {4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 97, normalized size = 2.55 \begin {gather*} \frac {b^{4} d \,x^{6}}{6}+a^{4} c x +\frac {\left (4 a \,b^{3} d +b^{4} c \right ) x^{5}}{5}+\frac {\left (6 a^{2} b^{2} d +4 a \,b^{3} c \right ) x^{4}}{4}+\frac {\left (4 a^{3} b d +6 a^{2} b^{2} c \right ) x^{3}}{3}+\frac {\left (a^{4} d +4 a^{3} b c \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.38, size = 96, normalized size = 2.53 \begin {gather*} \frac {1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac {1}{5} \, {\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.19, size = 88, normalized size = 2.32 \begin {gather*} x^5\,\left (\frac {c\,b^4}{5}+\frac {4\,a\,d\,b^3}{5}\right )+x^2\,\left (\frac {d\,a^4}{2}+2\,b\,c\,a^3\right )+\frac {b^4\,d\,x^6}{6}+a^4\,c\,x+\frac {2\,a^2\,b\,x^3\,\left (2\,a\,d+3\,b\,c\right )}{3}+\frac {a\,b^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4*(c + d*x),x)

[Out]

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

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sympy [B]  time = 0.08, size = 100, normalized size = 2.63 \begin {gather*} a^{4} c x + \frac {b^{4} d x^{6}}{6} + x^{5} \left (\frac {4 a b^{3} d}{5} + \frac {b^{4} c}{5}\right ) + x^{4} \left (\frac {3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac {4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac {a^{4} d}{2} + 2 a^{3} b c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(d*x+c),x)

[Out]

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

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