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

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

[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

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Rubi [A]  time = 0.157212, antiderivative size = 152, normalized size of antiderivative = 1., 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} \[ \frac{d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}+\frac{a^2 x (b c-a d)^3}{b^6}-\frac{a^3 (b c-a d)^3 \log (a+b x)}{b^7}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{x^3 (b c-a d)^3}{3 b^4}-\frac{a x^2 (b c-a d)^3}{2 b^5}+\frac{d^3 x^6}{6 b} \]

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*}

Mathematica [A]  time = 0.0846817, size = 145, normalized size = 0.95 \[ \frac{15 b^4 d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )-60 a^2 b x (a d-b c)^3+60 a^3 (a d-b c)^3 \log (a+b x)+12 b^5 d^2 x^5 (3 b c-a d)+20 b^3 x^3 (b c-a d)^3+30 a b^2 x^2 (a d-b c)^3+10 b^6 d^3 x^6}{60 b^7} \]

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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Maple [B]  time = 0.003, size = 302, normalized size = 2. \begin{align*}{\frac{{d}^{3}{x}^{6}}{6\,b}}-{\frac{{x}^{5}a{d}^{3}}{5\,{b}^{2}}}+{\frac{3\,{x}^{5}c{d}^{2}}{5\,b}}+{\frac{{x}^{4}{a}^{2}{d}^{3}}{4\,{b}^{3}}}-{\frac{3\,{x}^{4}ac{d}^{2}}{4\,{b}^{2}}}+{\frac{3\,{x}^{4}{c}^{2}d}{4\,b}}-{\frac{{x}^{3}{a}^{3}{d}^{3}}{3\,{b}^{4}}}+{\frac{{x}^{3}{a}^{2}c{d}^{2}}{{b}^{3}}}-{\frac{{x}^{3}a{c}^{2}d}{{b}^{2}}}+{\frac{{x}^{3}{c}^{3}}{3\,b}}+{\frac{{x}^{2}{a}^{4}{d}^{3}}{2\,{b}^{5}}}-{\frac{3\,{x}^{2}{a}^{3}c{d}^{2}}{2\,{b}^{4}}}+{\frac{3\,{a}^{2}{x}^{2}{c}^{2}d}{2\,{b}^{3}}}-{\frac{a{x}^{2}{c}^{3}}{2\,{b}^{2}}}-{\frac{{a}^{5}{d}^{3}x}{{b}^{6}}}+3\,{\frac{{a}^{4}c{d}^{2}x}{{b}^{5}}}-3\,{\frac{{a}^{3}{c}^{2}dx}{{b}^{4}}}+{\frac{{a}^{2}{c}^{3}x}{{b}^{3}}}+{\frac{{a}^{6}\ln \left ( bx+a \right ){d}^{3}}{{b}^{7}}}-3\,{\frac{{a}^{5}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{6}}}+3\,{\frac{{a}^{4}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{5}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{3}}{{b}^{4}}} \end{align*}

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*a^5*d^3*x+3/b^5*a^4*c*d^2*x-3/b^4*a^3*c^2*d*x+1/b^3*a^2*c^3*x+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 = 1.00859, size = 359, normalized size = 2.36 \begin{align*} \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{align*}

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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Fricas [A]  time = 2.13599, size = 537, normalized size = 3.53 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.687105, size = 231, normalized size = 1.52 \begin{align*} \frac{a^{3} \left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{7}} + \frac{d^{3} x^{6}}{6 b} - \frac{x^{5} \left (a d^{3} - 3 b c d^{2}\right )}{5 b^{2}} + \frac{x^{4} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{4 b^{3}} - \frac{x^{3} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{3 b^{4}} + \frac{x^{2} \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{2 b^{5}} - \frac{x \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}} \end{align*}

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

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Giac [B]  time = 1.20193, size = 386, normalized size = 2.54 \begin{align*} \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{align*}

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