3.2911 \(\int (c+d x)^3 (a+b (c+d x)^4) \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (a+b (c+d x)^4\right )^2}{8 b d} \]

[Out]

(a + b*(c + d*x)^4)^2/(8*b*d)

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Rubi [A]  time = 0.030693, antiderivative size = 31, normalized size of antiderivative = 1.35, 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} \[ \frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^8}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*(a + b*(c + d*x)^4),x]

[Out]

(a*(c + d*x)^4)/(4*d) + (b*(c + d*x)^8)/(8*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)^4\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^4\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a x^3+b x^7\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^8}{8 d}\\ \end{align*}

Mathematica [B]  time = 0.0165293, size = 80, normalized size = 3.48 \[ \frac{1}{8} x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \left (2 a+b \left (6 c^2 d^2 x^2+4 c^3 d x+2 c^4+4 c d^3 x^3+d^4 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*(a + b*(c + d*x)^4),x]

[Out]

(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(2*a + b*(2*c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4
*x^4)))/8

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Maple [B]  time = 0.001, size = 136, normalized size = 5.9 \begin{align*}{\frac{{d}^{7}b{x}^{8}}{8}}+c{d}^{6}b{x}^{7}+{\frac{7\,{c}^{2}{d}^{5}b{x}^{6}}{2}}+7\,{c}^{3}b{d}^{4}{x}^{5}+{\frac{ \left ( 34\,{c}^{4}b{d}^{3}+{d}^{3} \left ( b{c}^{4}+a \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 18\,{c}^{5}{d}^{2}b+3\,c{d}^{2} \left ( b{c}^{4}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{c}^{6}db+3\,{c}^{2}d \left ( b{c}^{4}+a \right ) \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{4}+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)^4),x)

[Out]

1/8*d^7*b*x^8+c*d^6*b*x^7+7/2*c^2*d^5*b*x^6+7*c^3*b*d^4*x^5+1/4*(34*c^4*b*d^3+d^3*(b*c^4+a))*x^4+1/3*(18*c^5*d
^2*b+3*c*d^2*(b*c^4+a))*x^3+1/2*(4*c^6*d*b+3*c^2*d*(b*c^4+a))*x^2+c^3*(b*c^4+a)*x

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Maxima [A]  time = 1.22403, size = 28, normalized size = 1.22 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{2}}{8 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^4),x, algorithm="maxima")

[Out]

1/8*((d*x + c)^4*b + a)^2/(b*d)

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Fricas [B]  time = 1.13192, size = 259, normalized size = 11.26 \begin{align*} \frac{1}{8} x^{8} d^{7} b + x^{7} d^{6} c b + \frac{7}{2} x^{6} d^{5} c^{2} b + 7 x^{5} d^{4} c^{3} b + \frac{35}{4} x^{4} d^{3} c^{4} b + 7 x^{3} d^{2} c^{5} b + \frac{7}{2} x^{2} d c^{6} b + x c^{7} 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)^4),x, algorithm="fricas")

[Out]

1/8*x^8*d^7*b + x^7*d^6*c*b + 7/2*x^6*d^5*c^2*b + 7*x^5*d^4*c^3*b + 35/4*x^4*d^3*c^4*b + 7*x^3*d^2*c^5*b + 7/2
*x^2*d*c^6*b + x*c^7*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 = 0.081968, size = 126, normalized size = 5.48 \begin{align*} 7 b c^{3} d^{4} x^{5} + \frac{7 b c^{2} d^{5} x^{6}}{2} + b c d^{6} x^{7} + \frac{b d^{7} x^{8}}{8} + x^{4} \left (\frac{a d^{3}}{4} + \frac{35 b c^{4} d^{3}}{4}\right ) + x^{3} \left (a c d^{2} + 7 b c^{5} d^{2}\right ) + x^{2} \left (\frac{3 a c^{2} d}{2} + \frac{7 b c^{6} d}{2}\right ) + x \left (a c^{3} + b c^{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*(a+b*(d*x+c)**4),x)

[Out]

7*b*c**3*d**4*x**5 + 7*b*c**2*d**5*x**6/2 + b*c*d**6*x**7 + b*d**7*x**8/8 + x**4*(a*d**3/4 + 35*b*c**4*d**3/4)
+ x**3*(a*c*d**2 + 7*b*c**5*d**2) + x**2*(3*a*c**2*d/2 + 7*b*c**6*d/2) + x*(a*c**3 + b*c**7)

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Giac [A]  time = 1.21171, size = 34, normalized size = 1.48 \begin{align*} \frac{{\left (d x + c\right )}^{8} b + 2 \,{\left (d x + c\right )}^{4} a}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^4),x, algorithm="giac")

[Out]

1/8*((d*x + c)^8*b + 2*(d*x + c)^4*a)/d