3.3.8 \(\int \frac {x^3 (c+d x)^3}{a+b x} \, dx\)

Optimal. Leaf size=152 \[ -\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7}+\frac {a^2 x (b c-a d)^3}{b^6}+\frac {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac {a x^2 (b c-a d)^3}{2 b^5}+\frac {x^3 (b c-a d)^3}{3 b^4}+\frac {d^2 x^5 (3 b c-a d)}{5 b^2}+\frac {d^3 x^6}{6 b} \]

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Rubi [A]  time = 0.16, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \begin {gather*} \frac {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}+\frac {a^2 x (b c-a d)^3}{b^6}-\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7}+\frac {d^2 x^5 (3 b c-a d)}{5 b^2}+\frac {x^3 (b c-a d)^3}{3 b^4}-\frac {a x^2 (b c-a d)^3}{2 b^5}+\frac {d^3 x^6}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^3)/(a + b*x),x]

[Out]

(a^2*(b*c - a*d)^3*x)/b^6 - (a*(b*c - a*d)^3*x^2)/(2*b^5) + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*
b*c*d + a^2*d^2)*x^4)/(4*b^3) + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) + (d^3*x^6)/(6*b) - (a^3*(b*c - a*d)^3*Log[a +
b*x])/b^7

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 145, normalized size = 0.95 \begin {gather*} \frac {60 a^3 (a d-b c)^3 \log (a+b x)+15 b^4 d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )-60 a^2 b x (a d-b c)^3+12 b^5 d^2 x^5 (3 b c-a d)+20 b^3 x^3 (b c-a d)^3+30 a b^2 x^2 (a d-b c)^3+10 b^6 d^3 x^6}{60 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-60*a^2*b*(-(b*c) + a*d)^3*x + 30*a*b^2*(-(b*c) + a*d)^3*x^2 + 20*b^3*(b*c - a*d)^3*x^3 + 15*b^4*d*(3*b^2*c^2
- 3*a*b*c*d + a^2*d^2)*x^4 + 12*b^5*d^2*(3*b*c - a*d)*x^5 + 10*b^6*d^3*x^6 + 60*a^3*(-(b*c) + a*d)^3*Log[a +
b*x])/(60*b^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (c+d x)^3}{a+b x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.32, size = 267, normalized size = 1.76 \begin {gather*} \frac {10 \, b^{6} d^{3} x^{6} + 12 \, {\left (3 \, b^{6} c d^{2} - a b^{5} d^{3}\right )} x^{5} + 15 \, {\left (3 \, b^{6} c^{2} d - 3 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4} + 20 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} - 30 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x - 60 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left (b x + a\right )}{60 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*b^6*d^3*x^6 + 12*(3*b^6*c*d^2 - a*b^5*d^3)*x^5 + 15*(3*b^6*c^2*d - 3*a*b^5*c*d^2 + a^2*b^4*d^3)*x^4 +
20*(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^3 - 30*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^
3*c*d^2 - a^4*b^2*d^3)*x^2 + 60*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x - 60*(a^3*b^3*
c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*log(b*x + a))/b^7

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giac [B]  time = 0.95, size = 286, normalized size = 1.88 \begin {gather*} \frac {10 \, b^{5} d^{3} x^{6} + 36 \, b^{5} c d^{2} x^{5} - 12 \, a b^{4} d^{3} x^{5} + 45 \, b^{5} c^{2} d x^{4} - 45 \, a b^{4} c d^{2} x^{4} + 15 \, a^{2} b^{3} d^{3} x^{4} + 20 \, b^{5} c^{3} x^{3} - 60 \, a b^{4} c^{2} d x^{3} + 60 \, a^{2} b^{3} c d^{2} x^{3} - 20 \, a^{3} b^{2} d^{3} x^{3} - 30 \, a b^{4} c^{3} x^{2} + 90 \, a^{2} b^{3} c^{2} d x^{2} - 90 \, a^{3} b^{2} c d^{2} x^{2} + 30 \, a^{4} b d^{3} x^{2} + 60 \, a^{2} b^{3} c^{3} x - 180 \, a^{3} b^{2} c^{2} d x + 180 \, a^{4} b c d^{2} x - 60 \, a^{5} d^{3} x}{60 \, b^{6}} - \frac {{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*b^5*d^3*x^6 + 36*b^5*c*d^2*x^5 - 12*a*b^4*d^3*x^5 + 45*b^5*c^2*d*x^4 - 45*a*b^4*c*d^2*x^4 + 15*a^2*b^
3*d^3*x^4 + 20*b^5*c^3*x^3 - 60*a*b^4*c^2*d*x^3 + 60*a^2*b^3*c*d^2*x^3 - 20*a^3*b^2*d^3*x^3 - 30*a*b^4*c^3*x^2
+ 90*a^2*b^3*c^2*d*x^2 - 90*a^3*b^2*c*d^2*x^2 + 30*a^4*b*d^3*x^2 + 60*a^2*b^3*c^3*x - 180*a^3*b^2*c^2*d*x + 1
80*a^4*b*c*d^2*x - 60*a^5*d^3*x)/b^6 - (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*log(abs(b*x +
a))/b^7

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maple [B]  time = 0.00, size = 302, normalized size = 1.99 \begin {gather*} \frac {d^{3} x^{6}}{6 b}-\frac {a \,d^{3} x^{5}}{5 b^{2}}+\frac {3 c \,d^{2} x^{5}}{5 b}+\frac {a^{2} d^{3} x^{4}}{4 b^{3}}-\frac {3 a c \,d^{2} x^{4}}{4 b^{2}}+\frac {3 c^{2} d \,x^{4}}{4 b}-\frac {a^{3} d^{3} x^{3}}{3 b^{4}}+\frac {a^{2} c \,d^{2} x^{3}}{b^{3}}-\frac {a \,c^{2} d \,x^{3}}{b^{2}}+\frac {c^{3} x^{3}}{3 b}+\frac {a^{4} d^{3} x^{2}}{2 b^{5}}-\frac {3 a^{3} c \,d^{2} x^{2}}{2 b^{4}}+\frac {3 a^{2} c^{2} d \,x^{2}}{2 b^{3}}-\frac {a \,c^{3} x^{2}}{2 b^{2}}+\frac {a^{6} d^{3} \ln \left (b x +a \right )}{b^{7}}-\frac {3 a^{5} c \,d^{2} \ln \left (b x +a \right )}{b^{6}}-\frac {a^{5} d^{3} x}{b^{6}}+\frac {3 a^{4} c^{2} d \ln \left (b x +a \right )}{b^{5}}+\frac {3 a^{4} c \,d^{2} x}{b^{5}}-\frac {a^{3} c^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 a^{3} c^{2} d x}{b^{4}}+\frac {a^{2} c^{3} x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.02, size = 266, normalized size = 1.75 \begin {gather*} \frac {10 \, b^{5} d^{3} x^{6} + 12 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{5} + 15 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{3} - 30 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x}{60 \, b^{6}} - \frac {{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*b^5*d^3*x^6 + 12*(3*b^5*c*d^2 - a*b^4*d^3)*x^5 + 15*(3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b^3*d^3)*x^4 +
20*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^3 - 30*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^
2*c*d^2 - a^4*b*d^3)*x^2 + 60*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*x)/b^6 - (a^3*b^3*c^3
- 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*log(b*x + a)/b^7

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mupad [B]  time = 0.06, size = 288, normalized size = 1.89 \begin {gather*} x^3\,\left (\frac {c^3}{3\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^5\,\left (\frac {a\,d^3}{5\,b^2}-\frac {3\,c\,d^2}{5\,b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{4\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3\right )}{b^7}+\frac {d^3\,x^6}{6\,b}-\frac {a\,x^2\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{2\,b}+\frac {a^2\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(c + d*x)^3)/(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.65, size = 243, normalized size = 1.60 \begin {gather*} \frac {a^{3} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{7}} + x^{5} \left (- \frac {a d^{3}}{5 b^{2}} + \frac {3 c d^{2}}{5 b}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4 b^{3}} - \frac {3 a c d^{2}}{4 b^{2}} + \frac {3 c^{2} d}{4 b}\right ) + x^{3} \left (- \frac {a^{3} d^{3}}{3 b^{4}} + \frac {a^{2} c d^{2}}{b^{3}} - \frac {a c^{2} d}{b^{2}} + \frac {c^{3}}{3 b}\right ) + x^{2} \left (\frac {a^{4} d^{3}}{2 b^{5}} - \frac {3 a^{3} c d^{2}}{2 b^{4}} + \frac {3 a^{2} c^{2} d}{2 b^{3}} - \frac {a c^{3}}{2 b^{2}}\right ) + x \left (- \frac {a^{5} d^{3}}{b^{6}} + \frac {3 a^{4} c d^{2}}{b^{5}} - \frac {3 a^{3} c^{2} d}{b^{4}} + \frac {a^{2} c^{3}}{b^{3}}\right ) + \frac {d^{3} x^{6}}{6 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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