3.2853 \(\int (c+d x)^3 (a+b (c+d x)^3) \, dx\)

Optimal. Leaf size=31 \[ \frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^7}{7 d} \]

[Out]

(a*(c + d*x)^4)/(4*d) + (b*(c + d*x)^7)/(7*d)

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Rubi [A]  time = 0.029149, 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} \[ \frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^7}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x)^4)/(4*d) + (b*(c + d*x)^7)/(7*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)^3\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a x^3+b x^6\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^7}{7 d}\\ \end{align*}

Mathematica [B]  time = 0.0133414, size = 98, normalized size = 3.16 \[ \frac{1}{4} d^3 x^4 \left (a+20 b c^3\right )+c d^2 x^3 \left (a+5 b c^3\right )+\frac{3}{2} c^2 d x^2 \left (a+2 b c^3\right )+c^3 x \left (a+b c^3\right )+3 b c^2 d^4 x^5+b c d^5 x^6+\frac{1}{7} b d^6 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(a + b*c^3)*x + (3*c^2*(a + 2*b*c^3)*d*x^2)/2 + c*(a + 5*b*c^3)*d^2*x^3 + ((a + 20*b*c^3)*d^3*x^4)/4 + 3*b
*c^2*d^4*x^5 + b*c*d^5*x^6 + (b*d^6*x^7)/7

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Maple [B]  time = 0., size = 124, normalized size = 4. \begin{align*}{\frac{{d}^{6}b{x}^{7}}{7}}+c{d}^{5}b{x}^{6}+3\,{c}^{2}{d}^{4}b{x}^{5}+{\frac{ \left ( 19\,{c}^{3}b{d}^{3}+{d}^{3} \left ( b{c}^{3}+a \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 12\,{c}^{4}b{d}^{2}+3\,c{d}^{2} \left ( b{c}^{3}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{c}^{5}bd+3\,{c}^{2}d \left ( b{c}^{3}+a \right ) \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{3}+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)^3),x)

[Out]

1/7*d^6*b*x^7+c*d^5*b*x^6+3*c^2*d^4*b*x^5+1/4*(19*c^3*b*d^3+d^3*(b*c^3+a))*x^4+1/3*(12*c^4*b*d^2+3*c*d^2*(b*c^
3+a))*x^3+1/2*(3*c^5*b*d+3*c^2*d*(b*c^3+a))*x^2+c^3*(b*c^3+a)*x

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Maxima [B]  time = 1.13889, size = 128, normalized size = 4.13 \begin{align*} \frac{1}{7} \, b d^{6} x^{7} + b c d^{5} x^{6} + 3 \, b c^{2} d^{4} x^{5} + \frac{1}{4} \,{\left (20 \, b c^{3} + a\right )} d^{3} x^{4} +{\left (5 \, b c^{4} + a c\right )} d^{2} x^{3} + \frac{3}{2} \,{\left (2 \, b c^{5} + a c^{2}\right )} d x^{2} +{\left (b c^{6} + 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)^3),x, algorithm="maxima")

[Out]

1/7*b*d^6*x^7 + b*c*d^5*x^6 + 3*b*c^2*d^4*x^5 + 1/4*(20*b*c^3 + a)*d^3*x^4 + (5*b*c^4 + a*c)*d^2*x^3 + 3/2*(2*
b*c^5 + a*c^2)*d*x^2 + (b*c^6 + a*c^3)*x

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Fricas [B]  time = 1.25805, size = 225, normalized size = 7.26 \begin{align*} \frac{1}{7} x^{7} d^{6} b + x^{6} d^{5} c b + 3 x^{5} d^{4} c^{2} b + 5 x^{4} d^{3} c^{3} b + 5 x^{3} d^{2} c^{4} b + 3 x^{2} d c^{5} b + x c^{6} b + \frac{1}{4} x^{4} d^{3} a + 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)^3),x, algorithm="fricas")

[Out]

1/7*x^7*d^6*b + x^6*d^5*c*b + 3*x^5*d^4*c^2*b + 5*x^4*d^3*c^3*b + 5*x^3*d^2*c^4*b + 3*x^2*d*c^5*b + x*c^6*b +
1/4*x^4*d^3*a + 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 = 0.085777, size = 107, normalized size = 3.45 \begin{align*} 3 b c^{2} d^{4} x^{5} + b c d^{5} x^{6} + \frac{b d^{6} x^{7}}{7} + x^{4} \left (\frac{a d^{3}}{4} + 5 b c^{3} d^{3}\right ) + x^{3} \left (a c d^{2} + 5 b c^{4} d^{2}\right ) + x^{2} \left (\frac{3 a c^{2} d}{2} + 3 b c^{5} d\right ) + x \left (a c^{3} + b c^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.09145, size = 142, normalized size = 4.58 \begin{align*} \frac{1}{7} \, b d^{6} x^{7} + b c d^{5} x^{6} + 3 \, b c^{2} d^{4} x^{5} + 5 \, b c^{3} d^{3} x^{4} + 5 \, b c^{4} d^{2} x^{3} + 3 \, b c^{5} d x^{2} + b c^{6} x + \frac{1}{4} \, a d^{3} x^{4} + 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)^3),x, algorithm="giac")

[Out]

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