3.7.10 \(\int (d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4) \, dx\) [610]

Optimal. Leaf size=55 \[ \frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e} \]

[Out]

1/4*a*f^3*(e*x+d)^4/e+1/6*b*f^3*(e*x+d)^6/e+1/8*c*f^3*(e*x+d)^8/e

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1156, 14} \begin {gather*} \frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

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

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 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rubi steps

\begin {align*} \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx &=\frac {f^3 \text {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac {f^3 \text {Subst}\left (\int \left (a x^3+b x^5+c x^7\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(55)=110\).
time = 0.01, size = 154, normalized size = 2.80 \begin {gather*} f^3 \left (d^3 \left (a+b d^2+c d^4\right ) x+\frac {1}{2} d^2 \left (3 a+5 b d^2+7 c d^4\right ) e x^2+\frac {1}{3} d \left (3 a+10 b d^2+21 c d^4\right ) e^2 x^3+\frac {1}{4} \left (a+10 b d^2+35 c d^4\right ) e^3 x^4+d \left (b+7 c d^2\right ) e^4 x^5+\frac {1}{6} \left (b+21 c d^2\right ) e^5 x^6+c d e^6 x^7+\frac {1}{8} c e^7 x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

f^3*(d^3*(a + b*d^2 + c*d^4)*x + (d^2*(3*a + 5*b*d^2 + 7*c*d^4)*e*x^2)/2 + (d*(3*a + 10*b*d^2 + 21*c*d^4)*e^2*
x^3)/3 + ((a + 10*b*d^2 + 35*c*d^4)*e^3*x^4)/4 + d*(b + 7*c*d^2)*e^4*x^5 + ((b + 21*c*d^2)*e^5*x^6)/6 + c*d*e^
6*x^7 + (c*e^7*x^8)/8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs. \(2(49)=98\).
time = 0.20, size = 349, normalized size = 6.35

method	result	size
gosper	\(\frac {f^{3} x \left (3 e^{7} c \,x^{7}+24 d \,e^{6} c \,x^{6}+84 x^{5} d^{2} e^{5} c +168 c \,d^{3} e^{4} x^{4}+4 x^{5} b \,e^{5}+210 x^{3} d^{4} e^{3} c +24 b d \,e^{4} x^{4}+168 x^{2} c \,d^{5} e^{2}+60 x^{3} b \,d^{2} e^{3}+84 x c \,d^{6} e +80 x^{2} b \,d^{3} e^{2}+24 c \,d^{7}+6 a \,e^{3} x^{3}+60 x b \,d^{4} e +24 x^{2} d \,e^{2} a +24 b \,d^{5}+36 x a \,d^{2} e +24 d^{3} a \right )}{24}\)	\(179\)
norman	\(\left (\frac {7}{2} d^{2} f^{3} e^{5} c +\frac {1}{6} b \,e^{5} f^{3}\right ) x^{6}+\left (7 c \,d^{5} e^{2} f^{3}+\frac {10}{3} b \,d^{3} e^{2} f^{3}+a d \,e^{2} f^{3}\right ) x^{3}+\left (\frac {7}{2} c \,d^{6} e \,f^{3}+\frac {5}{2} b \,d^{4} e \,f^{3}+\frac {3}{2} a \,d^{2} e \,f^{3}\right ) x^{2}+\left (\frac {35}{4} d^{4} f^{3} e^{3} c +\frac {5}{2} b \,d^{2} e^{3} f^{3}+\frac {1}{4} a \,e^{3} f^{3}\right ) x^{4}+\left (7 d^{3} f^{3} e^{4} c +b d \,e^{4} f^{3}\right ) x^{5}+\left (c \,d^{7} f^{3}+b \,d^{5} f^{3}+a \,d^{3} f^{3}\right ) x +d \,f^{3} e^{6} c \,x^{7}+\frac {e^{7} f^{3} c \,x^{8}}{8}\)	\(216\)
risch	\(\frac {1}{8} e^{7} f^{3} c \,x^{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {7}{2} f^{3} x^{6} d^{2} e^{5} c +\frac {1}{6} f^{3} x^{6} b \,e^{5}+7 f^{3} c \,d^{3} e^{4} x^{5}+f^{3} b d \,e^{4} x^{5}+\frac {35}{4} f^{3} x^{4} d^{4} e^{3} c +\frac {5}{2} f^{3} x^{4} b \,d^{2} e^{3}+\frac {1}{4} f^{3} x^{4} a \,e^{3}+7 f^{3} x^{3} c \,d^{5} e^{2}+\frac {10}{3} f^{3} x^{3} b \,d^{3} e^{2}+f^{3} x^{3} d \,e^{2} a +\frac {7}{2} f^{3} x^{2} c \,d^{6} e +\frac {5}{2} f^{3} x^{2} b \,d^{4} e +\frac {3}{2} f^{3} x^{2} a \,d^{2} e +f^{3} c \,d^{7} x +f^{3} b \,d^{5} x +f^{3} a \,d^{3} x\)	\(230\)
default	\(\frac {e^{7} f^{3} c \,x^{8}}{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {\left (15 d^{2} f^{3} e^{5} c +e^{3} f^{3} \left (6 d^{2} e^{2} c +e^{2} b \right )\right ) x^{6}}{6}+\frac {\left (13 d^{3} f^{3} e^{4} c +3 d \,f^{3} e^{2} \left (6 d^{2} e^{2} c +e^{2} b \right )+e^{3} f^{3} \left (4 d^{3} e c +2 d e b \right )\right ) x^{5}}{5}+\frac {\left (4 d^{4} f^{3} e^{3} c +3 d^{2} f^{3} e \left (6 d^{2} e^{2} c +e^{2} b \right )+3 d \,f^{3} e^{2} \left (4 d^{3} e c +2 d e b \right )+e^{3} f^{3} \left (d^{4} c +d^{2} b +a \right )\right ) x^{4}}{4}+\frac {\left (d^{3} f^{3} \left (6 d^{2} e^{2} c +e^{2} b \right )+3 d^{2} f^{3} e \left (4 d^{3} e c +2 d e b \right )+3 d \,f^{3} e^{2} \left (d^{4} c +d^{2} b +a \right )\right ) x^{3}}{3}+\frac {\left (d^{3} f^{3} \left (4 d^{3} e c +2 d e b \right )+3 d^{2} f^{3} e \left (d^{4} c +d^{2} b +a \right )\right ) x^{2}}{2}+d^{3} f^{3} \left (d^{4} c +d^{2} b +a \right ) x\)	\(349\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x,method=_RETURNVERBOSE)

[Out]

1/8*e^7*f^3*c*x^8+d*f^3*e^6*c*x^7+1/6*(15*d^2*f^3*e^5*c+e^3*f^3*(6*c*d^2*e^2+b*e^2))*x^6+1/5*(13*d^3*f^3*e^4*c
+3*d*f^3*e^2*(6*c*d^2*e^2+b*e^2)+e^3*f^3*(4*c*d^3*e+2*b*d*e))*x^5+1/4*(4*d^4*f^3*e^3*c+3*d^2*f^3*e*(6*c*d^2*e^
2+b*e^2)+3*d*f^3*e^2*(4*c*d^3*e+2*b*d*e)+e^3*f^3*(c*d^4+b*d^2+a))*x^4+1/3*(d^3*f^3*(6*c*d^2*e^2+b*e^2)+3*d^2*f
^3*e*(4*c*d^3*e+2*b*d*e)+3*d*f^3*e^2*(c*d^4+b*d^2+a))*x^3+1/2*(d^3*f^3*(4*c*d^3*e+2*b*d*e)+3*d^2*f^3*e*(c*d^4+
b*d^2+a))*x^2+d^3*f^3*(c*d^4+b*d^2+a)*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (49) = 98\).
time = 0.29, size = 179, normalized size = 3.25 \begin {gather*} \frac {1}{8} \, c f^{3} x^{8} e^{7} + c d f^{3} x^{7} e^{6} + \frac {1}{6} \, {\left (21 \, c d^{2} e^{5} + b e^{5}\right )} f^{3} x^{6} + {\left (7 \, c d^{3} e^{4} + b d e^{4}\right )} f^{3} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} e^{3} + 10 \, b d^{2} e^{3} + a e^{3}\right )} f^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} e^{2} + 10 \, b d^{3} e^{2} + 3 \, a d e^{2}\right )} f^{3} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} e + 5 \, b d^{4} e + 3 \, a d^{2} e\right )} f^{3} x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

1/8*c*f^3*x^8*e^7 + c*d*f^3*x^7*e^6 + 1/6*(21*c*d^2*e^5 + b*e^5)*f^3*x^6 + (7*c*d^3*e^4 + b*d*e^4)*f^3*x^5 + 1
/4*(35*c*d^4*e^3 + 10*b*d^2*e^3 + a*e^3)*f^3*x^4 + 1/3*(21*c*d^5*e^2 + 10*b*d^3*e^2 + 3*a*d*e^2)*f^3*x^3 + 1/2
*(7*c*d^6*e + 5*b*d^4*e + 3*a*d^2*e)*f^3*x^2 + (c*d^7 + b*d^5 + a*d^3)*f^3*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (49) = 98\).
time = 0.40, size = 161, normalized size = 2.93 \begin {gather*} \frac {1}{8} \, c f^{3} x^{8} e^{7} + c d f^{3} x^{7} e^{6} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} f^{3} x^{6} e^{5} + {\left (7 \, c d^{3} + b d\right )} f^{3} x^{5} e^{4} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} f^{3} x^{4} e^{3} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} f^{3} x^{3} e^{2} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} f^{3} x^{2} e + {\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

1/8*c*f^3*x^8*e^7 + c*d*f^3*x^7*e^6 + 1/6*(21*c*d^2 + b)*f^3*x^6*e^5 + (7*c*d^3 + b*d)*f^3*x^5*e^4 + 1/4*(35*c
*d^4 + 10*b*d^2 + a)*f^3*x^4*e^3 + 1/3*(21*c*d^5 + 10*b*d^3 + 3*a*d)*f^3*x^3*e^2 + 1/2*(7*c*d^6 + 5*b*d^4 + 3*
a*d^2)*f^3*x^2*e + (c*d^7 + b*d^5 + a*d^3)*f^3*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (44) = 88\).
time = 0.03, size = 240, normalized size = 4.36 \begin {gather*} c d e^{6} f^{3} x^{7} + \frac {c e^{7} f^{3} x^{8}}{8} + x^{6} \left (\frac {b e^{5} f^{3}}{6} + \frac {7 c d^{2} e^{5} f^{3}}{2}\right ) + x^{5} \left (b d e^{4} f^{3} + 7 c d^{3} e^{4} f^{3}\right ) + x^{4} \left (\frac {a e^{3} f^{3}}{4} + \frac {5 b d^{2} e^{3} f^{3}}{2} + \frac {35 c d^{4} e^{3} f^{3}}{4}\right ) + x^{3} \left (a d e^{2} f^{3} + \frac {10 b d^{3} e^{2} f^{3}}{3} + 7 c d^{5} e^{2} f^{3}\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e f^{3}}{2} + \frac {5 b d^{4} e f^{3}}{2} + \frac {7 c d^{6} e f^{3}}{2}\right ) + x \left (a d^{3} f^{3} + b d^{5} f^{3} + c d^{7} f^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

c*d*e**6*f**3*x**7 + c*e**7*f**3*x**8/8 + x**6*(b*e**5*f**3/6 + 7*c*d**2*e**5*f**3/2) + x**5*(b*d*e**4*f**3 +
7*c*d**3*e**4*f**3) + x**4*(a*e**3*f**3/4 + 5*b*d**2*e**3*f**3/2 + 35*c*d**4*e**3*f**3/4) + x**3*(a*d*e**2*f**
3 + 10*b*d**3*e**2*f**3/3 + 7*c*d**5*e**2*f**3) + x**2*(3*a*d**2*e*f**3/2 + 5*b*d**4*e*f**3/2 + 7*c*d**6*e*f**
3/2) + x*(a*d**3*f**3 + b*d**5*f**3 + c*d**7*f**3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (49) = 98\).
time = 3.76, size = 213, normalized size = 3.87 \begin {gather*} \frac {1}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} c d^{6} f^{2} + \frac {1}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} b d^{4} f^{2} + \frac {1}{2} \, {\left (f x^{2} e + 2 \, d f x\right )} a d^{2} f^{2} + \frac {18 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} c d^{4} f^{2} e + 12 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} c d^{2} f e^{2} + 12 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} b d^{2} f^{2} e + 3 \, {\left (f x^{2} e + 2 \, d f x\right )}^{4} c e^{3} + 4 \, {\left (f x^{2} e + 2 \, d f x\right )}^{3} b f e^{2} + 6 \, {\left (f x^{2} e + 2 \, d f x\right )}^{2} a f^{2} e}{24 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

1/2*(f*x^2*e + 2*d*f*x)*c*d^6*f^2 + 1/2*(f*x^2*e + 2*d*f*x)*b*d^4*f^2 + 1/2*(f*x^2*e + 2*d*f*x)*a*d^2*f^2 + 1/
24*(18*(f*x^2*e + 2*d*f*x)^2*c*d^4*f^2*e + 12*(f*x^2*e + 2*d*f*x)^3*c*d^2*f*e^2 + 12*(f*x^2*e + 2*d*f*x)^2*b*d
^2*f^2*e + 3*(f*x^2*e + 2*d*f*x)^4*c*e^3 + 4*(f*x^2*e + 2*d*f*x)^3*b*f*e^2 + 6*(f*x^2*e + 2*d*f*x)^2*a*f^2*e)/
f

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Mupad [B]
time = 0.08, size = 164, normalized size = 2.98 \begin {gather*} \frac {e^5\,f^3\,x^6\,\left (21\,c\,d^2+b\right )}{6}+\frac {c\,e^7\,f^3\,x^8}{8}+d^3\,f^3\,x\,\left (c\,d^4+b\,d^2+a\right )+\frac {e^3\,f^3\,x^4\,\left (35\,c\,d^4+10\,b\,d^2+a\right )}{4}+\frac {d^2\,e\,f^3\,x^2\,\left (7\,c\,d^4+5\,b\,d^2+3\,a\right )}{2}+\frac {d\,e^2\,f^3\,x^3\,\left (21\,c\,d^4+10\,b\,d^2+3\,a\right )}{3}+d\,e^4\,f^3\,x^5\,\left (7\,c\,d^2+b\right )+c\,d\,e^6\,f^3\,x^7 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x)

[Out]

(e^5*f^3*x^6*(b + 21*c*d^2))/6 + (c*e^7*f^3*x^8)/8 + d^3*f^3*x*(a + b*d^2 + c*d^4) + (e^3*f^3*x^4*(a + 10*b*d^
2 + 35*c*d^4))/4 + (d^2*e*f^3*x^2*(3*a + 5*b*d^2 + 7*c*d^4))/2 + (d*e^2*f^3*x^3*(3*a + 10*b*d^2 + 21*c*d^4))/3
+ d*e^4*f^3*x^5*(b + 7*c*d^2) + c*d*e^6*f^3*x^7

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