3.1262 \(\int (a+b x) (c+d x)^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0150808, antiderivative size = 38, normalized size of antiderivative = 1., 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} \[ \frac{b (c+d x)^5}{5 d^2}-\frac{(c+d x)^4 (b c-a d)}{4 d^2} \]

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*}

Mathematica [A]  time = 0.0083032, size = 67, normalized size = 1.76 \[ \frac{1}{2} c^2 x^2 (3 a d+b c)+\frac{1}{4} d^2 x^4 (a d+3 b c)+c d x^3 (a d+b c)+a c^3 x+\frac{1}{5} b d^3 x^5 \]

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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Maple [B]  time = 0., size = 73, normalized size = 1.9 \begin{align*}{\frac{b{d}^{3}{x}^{5}}{5}}+{\frac{ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,ac{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}}+a{c}^{3}x \end{align*}

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 = 0.973537, size = 93, normalized size = 2.45 \begin{align*} \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{align*}

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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Fricas [B]  time = 1.77785, size = 163, normalized size = 4.29 \begin{align*} \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{align*}

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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Sympy [B]  time = 0.070579, size = 73, normalized size = 1.92 \begin{align*} 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{align*}

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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Giac [B]  time = 1.04949, size = 97, normalized size = 2.55 \begin{align*} \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{align*}

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