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]

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

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Rubi [A]  time = 0.03, 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 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)^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*}

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Mathematica [B]  time = 0.02, size = 80, normalized size = 3.48 \[ \frac {1}{8} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \left (2 a+b \left (2 c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )\right ) \]

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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fricas [B]  time = 0.82, size = 117, normalized size = 5.09 \[ \frac {1}{8} x^{8} d^{7} b + x^{7} d^{6} c b + \frac {7}{2} x^{6} d^{5} c^{2} b + 7 x^{5} d^{4} c^{3} b + \frac {35}{4} x^{4} d^{3} c^{4} b + 7 x^{3} d^{2} c^{5} b + \frac {7}{2} x^{2} d c^{6} b + x c^{7} b + \frac {1}{4} x^{4} d^{3} a + x^{3} d^{2} c a + \frac {3}{2} x^{2} d c^{2} a + x c^{3} a \]

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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giac [A]  time = 0.16, size = 25, normalized size = 1.09 \[ \frac {{\left (d x + c\right )}^{8} b + 2 \, {\left (d x + c\right )}^{4} a}{8 \, d} \]

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

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maple [B]  time = 0.00, size = 136, normalized size = 5.91 \[ \frac {b \,d^{7} x^{8}}{8}+b c \,d^{6} x^{7}+\frac {7 b \,c^{2} d^{5} x^{6}}{2}+7 b \,c^{3} d^{4} x^{5}+\left (b \,c^{4}+a \right ) c^{3} x +\frac {\left (34 b \,c^{4} d^{3}+\left (b \,c^{4}+a \right ) d^{3}\right ) x^{4}}{4}+\frac {\left (18 b \,c^{5} d^{2}+3 \left (b \,c^{4}+a \right ) c \,d^{2}\right ) x^{3}}{3}+\frac {\left (4 b \,c^{6} d +3 \left (b \,c^{4}+a \right ) c^{2} d \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*(a+b*(d*x+c)^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*d^4*b*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 = 0.48, size = 21, normalized size = 0.91 \[ \frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{2}}{8 \, b d} \]

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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mupad [B]  time = 0.06, size = 107, normalized size = 4.65 \[ x\,\left (b\,c^7+a\,c^3\right )+\frac {d^3\,x^4\,\left (35\,b\,c^4+a\right )}{4}+\frac {b\,d^7\,x^8}{8}+\frac {c^2\,d\,x^2\,\left (7\,b\,c^4+3\,a\right )}{2}+7\,b\,c^3\,d^4\,x^5+\frac {7\,b\,c^2\,d^5\,x^6}{2}+c\,d^2\,x^3\,\left (7\,b\,c^4+a\right )+b\,c\,d^6\,x^7 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.10, size = 126, normalized size = 5.48 \[ 7 b c^{3} d^{4} x^{5} + \frac {7 b c^{2} d^{5} x^{6}}{2} + b c d^{6} x^{7} + \frac {b d^{7} x^{8}}{8} + x^{4} \left (\frac {a d^{3}}{4} + \frac {35 b c^{4} d^{3}}{4}\right ) + x^{3} \left (a c d^{2} + 7 b c^{5} d^{2}\right ) + x^{2} \left (\frac {3 a c^{2} d}{2} + \frac {7 b c^{6} d}{2}\right ) + x \left (a c^{3} + b c^{7}\right ) \]

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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