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]

1/4*a*(d*x+c)^4/d+1/6*b*(d*x+c)^6/d

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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]]

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]

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*}

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Mathematica [B]  time = 0.03, size = 77, normalized size = 2.48 \[ \frac {1}{12} x (2 c+d x) \left (3 a \left (2 c^2+2 c d x+d^2 x^2\right )+2 b \left (3 c^4+6 c^3 d x+7 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )\right ) \]

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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fricas [B]  time = 0.66, size = 93, normalized size = 3.00 \[ \frac {1}{6} x^{6} d^{5} b + x^{5} d^{4} c b + \frac {5}{2} x^{4} d^{3} c^{2} b + \frac {10}{3} x^{3} d^{2} c^{3} b + \frac {5}{2} x^{2} d c^{4} b + \frac {1}{4} x^{4} d^{3} a + x c^{5} b + x^{3} d^{2} c a + \frac {3}{2} x^{2} d c^{2} a + x c^{3} a \]

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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giac [B]  time = 0.17, size = 86, normalized size = 2.77 \[ \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} b c^{4} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{2} b c^{2} d + \frac {1}{6} \, {\left (d x^{2} + 2 \, c x\right )}^{3} b d^{2} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} a c^{2} + \frac {1}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a d \]

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]

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

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maple [B]  time = 0.00, size = 112, normalized size = 3.61 \[ \frac {b \,d^{5} x^{6}}{6}+b c \,d^{4} x^{5}+\left (b \,c^{2}+a \right ) c^{3} x +\frac {\left (9 b \,c^{2} d^{3}+\left (b \,c^{2}+a \right ) d^{3}\right ) x^{4}}{4}+\frac {\left (7 b \,c^{3} d^{2}+3 \left (b \,c^{2}+a \right ) c \,d^{2}\right ) x^{3}}{3}+\frac {\left (2 b \,c^{4} d +3 \left (b \,c^{2}+a \right ) c^{2} d \right ) x^{2}}{2} \]

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 = 0.53, size = 86, normalized size = 2.77 \[ \frac {1}{6} \, b d^{5} x^{6} + b c d^{4} x^{5} + \frac {1}{4} \, {\left (10 \, b c^{2} + a\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (10 \, b c^{3} + 3 \, a c\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (5 \, b c^{4} + 3 \, a c^{2}\right )} d x^{2} + {\left (b c^{5} + a c^{3}\right )} x \]

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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mupad [B]  time = 0.06, size = 86, normalized size = 2.77 \[ x\,\left (b\,c^5+a\,c^3\right )+\frac {d^3\,x^4\,\left (10\,b\,c^2+a\right )}{4}+\frac {b\,d^5\,x^6}{6}+\frac {c^2\,d\,x^2\,\left (5\,b\,c^2+3\,a\right )}{2}+\frac {c\,d^2\,x^3\,\left (10\,b\,c^2+3\,a\right )}{3}+b\,c\,d^4\,x^5 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.11, size = 99, normalized size = 3.19 \[ b c d^{4} x^{5} + \frac {b d^{5} x^{6}}{6} + x^{4} \left (\frac {a d^{3}}{4} + \frac {5 b c^{2} d^{3}}{2}\right ) + x^{3} \left (a c d^{2} + \frac {10 b c^{3} d^{2}}{3}\right ) + x^{2} \left (\frac {3 a c^{2} d}{2} + \frac {5 b c^{4} d}{2}\right ) + x \left (a c^{3} + b c^{5}\right ) \]

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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