3.12.56 \(\int (a+b x) (c+d x)^3 \, dx\)

Optimal. Leaf size=38 \[ \frac {b (c+d x)^5}{5 d^2}-\frac {(c+d x)^4 (b c-a d)}{4 d^2} \]

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Rubi [A]  time = 0.02, 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 {b (c+d x)^5}{5 d^2}-\frac {(c+d x)^4 (b c-a d)}{4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x)^3,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 67, normalized size = 1.76 \begin {gather*} \frac {1}{2} c^2 x^2 (3 a d+b c)+\frac {1}{4} d^2 x^4 (a d+3 b c)+c d x^3 (a d+b c)+a c^3 x+\frac {1}{5} b d^3 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x)^3,x]

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.09, size = 72, normalized size = 1.89 \begin {gather*} \frac {1}{5} x^{5} d^{3} b + \frac {3}{4} x^{4} d^{2} c b + \frac {1}{4} x^{4} d^{3} a + x^{3} d c^{2} b + x^{3} d^{2} c a + \frac {1}{2} x^{2} c^{3} b + \frac {3}{2} x^{2} d c^{2} a + x c^{3} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.88, size = 72, normalized size = 1.89 \begin {gather*} \frac {1}{5} \, b d^{3} x^{5} + \frac {3}{4} \, b c d^{2} x^{4} + \frac {1}{4} \, a d^{3} x^{4} + b c^{2} d x^{3} + a c d^{2} x^{3} + \frac {1}{2} \, b c^{3} x^{2} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 73, normalized size = 1.92 \begin {gather*} \frac {b \,d^{3} x^{5}}{5}+a \,c^{3} x +\frac {\left (a \,d^{3}+3 b c \,d^{2}\right ) x^{4}}{4}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) x^{3}}{3}+\frac {\left (3 a \,c^{2} d +b \,c^{3}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.39, size = 69, normalized size = 1.82 \begin {gather*} \frac {1}{5} \, b d^{3} x^{5} + a c^{3} x + \frac {1}{4} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{4} + {\left (b c^{2} d + a c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 65, normalized size = 1.71 \begin {gather*} x^2\,\left (\frac {b\,c^3}{2}+\frac {3\,a\,d\,c^2}{2}\right )+x^4\,\left (\frac {a\,d^3}{4}+\frac {3\,b\,c\,d^2}{4}\right )+\frac {b\,d^3\,x^5}{5}+a\,c^3\,x+c\,d\,x^3\,\left (a\,d+b\,c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(c + d*x)^3,x)

[Out]

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

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sympy [B]  time = 0.08, size = 73, normalized size = 1.92 \begin {gather*} a c^{3} x + \frac {b d^{3} x^{5}}{5} + x^{4} \left (\frac {a d^{3}}{4} + \frac {3 b c d^{2}}{4}\right ) + x^{3} \left (a c d^{2} + b c^{2} d\right ) + x^{2} \left (\frac {3 a c^{2} d}{2} + \frac {b c^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)**3,x)

[Out]

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

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