3.3.9 \(\int \frac {x^2 (c+d x)^3}{a+b x} \, dx\)

Optimal. Leaf size=129 \[ \frac {a^2 (b c-a d)^3 \log (a+b x)}{b^6}+\frac {d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {a x (b c-a d)^3}{b^5}+\frac {x^2 (b c-a d)^3}{2 b^4}+\frac {d^2 x^4 (3 b c-a d)}{4 b^2}+\frac {d^3 x^5}{5 b} \]

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Rubi [A]  time = 0.10, antiderivative size = 129, 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^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac {a^2 (b c-a d)^3 \log (a+b x)}{b^6}+\frac {d^2 x^4 (3 b c-a d)}{4 b^2}+\frac {x^2 (b c-a d)^3}{2 b^4}-\frac {a x (b c-a d)^3}{b^5}+\frac {d^3 x^5}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^3)/(a + b*x),x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 124, normalized size = 0.96 \begin {gather*} \frac {20 b^3 d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+60 a^2 (b c-a d)^3 \log (a+b x)+15 b^4 d^2 x^4 (3 b c-a d)+30 b^2 x^2 (b c-a d)^3+60 a b x (a d-b c)^3+12 b^5 d^3 x^5}{60 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 (c+d x)^3}{a+b x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^2*(c + d*x)^3)/(a + b*x),x]

[Out]

IntegrateAlgebraic[(x^2*(c + d*x)^3)/(a + b*x), x]

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fricas [A]  time = 1.17, size = 216, normalized size = 1.67 \begin {gather*} \frac {12 \, b^{5} d^{3} x^{5} + 15 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 20 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{3} + 30 \, {\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^{2} - 60 \, {\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 + 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 )} \log \left (b x + a\right )}{60 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.07, size = 227, normalized size = 1.76 \begin {gather*} \frac {12 \, b^{4} d^{3} x^{5} + 45 \, b^{4} c d^{2} x^{4} - 15 \, a b^{3} d^{3} x^{4} + 60 \, b^{4} c^{2} d x^{3} - 60 \, a b^{3} c d^{2} x^{3} + 20 \, a^{2} b^{2} d^{3} x^{3} + 30 \, b^{4} c^{3} x^{2} - 90 \, a b^{3} c^{2} d x^{2} + 90 \, a^{2} b^{2} c d^{2} x^{2} - 30 \, a^{3} b d^{3} x^{2} - 60 \, a b^{3} c^{3} x + 180 \, a^{2} b^{2} c^{2} d x - 180 \, a^{3} b c d^{2} x + 60 \, a^{4} d^{3} x}{60 \, b^{5}} + \frac {{\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 )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 244, normalized size = 1.89 \begin {gather*} \frac {d^{3} x^{5}}{5 b}-\frac {a \,d^{3} x^{4}}{4 b^{2}}+\frac {3 c \,d^{2} x^{4}}{4 b}+\frac {a^{2} d^{3} x^{3}}{3 b^{3}}-\frac {a c \,d^{2} x^{3}}{b^{2}}+\frac {c^{2} d \,x^{3}}{b}-\frac {a^{3} d^{3} x^{2}}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} x^{2}}{2 b^{3}}-\frac {3 a \,c^{2} d \,x^{2}}{2 b^{2}}+\frac {c^{3} x^{2}}{2 b}-\frac {a^{5} d^{3} \ln \left (b x +a \right )}{b^{6}}+\frac {3 a^{4} c \,d^{2} \ln \left (b x +a \right )}{b^{5}}+\frac {a^{4} d^{3} x}{b^{5}}-\frac {3 a^{3} c^{2} d \ln \left (b x +a \right )}{b^{4}}-\frac {3 a^{3} c \,d^{2} x}{b^{4}}+\frac {a^{2} c^{3} \ln \left (b x +a \right )}{b^{3}}+\frac {3 a^{2} c^{2} d x}{b^{3}}-\frac {a \,c^{3} x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 214, normalized size = 1.66 \begin {gather*} \frac {12 \, b^{4} d^{3} x^{5} + 15 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{4} + 20 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3} + 30 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 60 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{60 \, b^{5}} + \frac {{\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 )} \log \left (b x + a\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 231, normalized size = 1.79 \begin {gather*} x^2\,\left (\frac {c^3}{2\,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 )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )+x^3\,\left (\frac {c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{3\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{b^6}+\frac {d^3\,x^5}{5\,b}-\frac {a\,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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x)^3)/(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.95, size = 185, normalized size = 1.43 \begin {gather*} - \frac {a^{2} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{6}} + x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{3 b^{3}} - \frac {a c d^{2}}{b^{2}} + \frac {c^{2} d}{b}\right ) + x^{2} \left (- \frac {a^{3} d^{3}}{2 b^{4}} + \frac {3 a^{2} c d^{2}}{2 b^{3}} - \frac {3 a c^{2} d}{2 b^{2}} + \frac {c^{3}}{2 b}\right ) + x \left (\frac {a^{4} d^{3}}{b^{5}} - \frac {3 a^{3} c d^{2}}{b^{4}} + \frac {3 a^{2} c^{2} d}{b^{3}} - \frac {a c^{3}}{b^{2}}\right ) + \frac {d^{3} x^{5}}{5 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d*x+c)**3/(b*x+a),x)

[Out]

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

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