3.2841 \(\int (c+d x)^3 (a+b (c+d x)^2) \, dx\)

Optimal. Leaf size=31 \[ \frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^6}{6 d} \]

[Out]

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

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Rubi [A]  time = 0.0256384, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {372, 14} \[ \frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^6}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^2\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a x^3+b x^5\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^6}{6 d}\\ \end{align*}

Mathematica [B]  time = 0.0188868, size = 77, normalized size = 2.48 \[ \frac{1}{12} x (2 c+d x) \left (3 a \left (2 c^2+2 c d x+d^2 x^2\right )+2 b \left (7 c^2 d^2 x^2+6 c^3 d x+3 c^4+4 c d^3 x^3+d^4 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(2*c + d*x)*(3*a*(2*c^2 + 2*c*d*x + d^2*x^2) + 2*b*(3*c^4 + 6*c^3*d*x + 7*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x
^4)))/12

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Maple [B]  time = 0.003, size = 112, normalized size = 3.6 \begin{align*}{\frac{{d}^{5}b{x}^{6}}{6}}+c{d}^{4}b{x}^{5}+{\frac{ \left ( 9\,{c}^{2}{d}^{3}b+{d}^{3} \left ( b{c}^{2}+a \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 7\,{c}^{3}b{d}^{2}+3\,c{d}^{2} \left ( b{c}^{2}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{c}^{4}bd+3\,{c}^{2}d \left ( b{c}^{2}+a \right ) \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{2}+a \right ) x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.08115, size = 116, normalized size = 3.74 \begin{align*} \frac{1}{6} \, b d^{5} x^{6} + b c d^{4} x^{5} + \frac{1}{4} \,{\left (10 \, b c^{2} + a\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (10 \, b c^{3} + 3 \, a c\right )} d^{2} x^{3} + \frac{1}{2} \,{\left (5 \, b c^{4} + 3 \, a c^{2}\right )} d x^{2} +{\left (b c^{5} + a c^{3}\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*d^5*x^6 + b*c*d^4*x^5 + 1/4*(10*b*c^2 + a)*d^3*x^4 + 1/3*(10*b*c^3 + 3*a*c)*d^2*x^3 + 1/2*(5*b*c^4 + 3*a
*c^2)*d*x^2 + (b*c^5 + a*c^3)*x

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Fricas [B]  time = 1.28411, size = 211, normalized size = 6.81 \begin{align*} \frac{1}{6} x^{6} d^{5} b + x^{5} d^{4} c b + \frac{5}{2} x^{4} d^{3} c^{2} b + \frac{10}{3} x^{3} d^{2} c^{3} b + \frac{5}{2} x^{2} d c^{4} b + \frac{1}{4} x^{4} d^{3} a + x c^{5} b + x^{3} d^{2} c a + \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((d*x+c)^3*(a+b*(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.078302, size = 99, normalized size = 3.19 \begin{align*} b c d^{4} x^{5} + \frac{b d^{5} x^{6}}{6} + x^{4} \left (\frac{a d^{3}}{4} + \frac{5 b c^{2} d^{3}}{2}\right ) + x^{3} \left (a c d^{2} + \frac{10 b c^{3} d^{2}}{3}\right ) + x^{2} \left (\frac{3 a c^{2} d}{2} + \frac{5 b c^{4} d}{2}\right ) + x \left (a c^{3} + b c^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.10035, size = 126, normalized size = 4.06 \begin{align*} \frac{1}{6} \, b d^{5} x^{6} + b c d^{4} x^{5} + \frac{5}{2} \, b c^{2} d^{3} x^{4} + \frac{10}{3} \, b c^{3} d^{2} x^{3} + \frac{5}{2} \, b c^{4} d x^{2} + \frac{1}{4} \, a d^{3} x^{4} + b c^{5} x + a c d^{2} x^{3} + \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((d*x+c)^3*(a+b*(d*x+c)^2),x, algorithm="giac")

[Out]

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