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

Optimal. Leaf size=196 \[ \frac{x (b c-a d) \left (10 a^2 d^2-8 a b c d+b^2 c^2\right )}{b^6}-\frac{3 a (b c-a d) \left (5 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{b^7}-\frac{3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}+\frac{a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}+\frac{d^2 x^3 (b c-a d)}{b^4}+\frac{3 d x^2 (b c-2 a d) (b c-a d)}{2 b^5}+\frac{d^3 x^4}{4 b^3} \]

[Out]

((b*c - a*d)*(b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*x)/b^6 + (3*d*(b*c - 2*a*d)*(b*c - a*d)*x^2)/(2*b^5) + (d^2*(b
*c - a*d)*x^3)/b^4 + (d^3*x^4)/(4*b^3) + (a^3*(b*c - a*d)^3)/(2*b^7*(a + b*x)^2) - (3*a^2*(b*c - 2*a*d)*(b*c -
a*d)^2)/(b^7*(a + b*x)) - (3*a*(b*c - a*d)*(b^2*c^2 - 5*a*b*c*d + 5*a^2*d^2)*Log[a + b*x])/b^7

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Rubi [A]  time = 0.235019, antiderivative size = 196, 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} \[ \frac{x (b c-a d) \left (10 a^2 d^2-8 a b c d+b^2 c^2\right )}{b^6}-\frac{3 a (b c-a d) \left (5 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{b^7}-\frac{3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}+\frac{a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}+\frac{d^2 x^3 (b c-a d)}{b^4}+\frac{3 d x^2 (b c-2 a d) (b c-a d)}{2 b^5}+\frac{d^3 x^4}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^3)/(a + b*x)^3,x]

[Out]

((b*c - a*d)*(b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*x)/b^6 + (3*d*(b*c - 2*a*d)*(b*c - a*d)*x^2)/(2*b^5) + (d^2*(b
*c - a*d)*x^3)/b^4 + (d^3*x^4)/(4*b^3) + (a^3*(b*c - a*d)^3)/(2*b^7*(a + b*x)^2) - (3*a^2*(b*c - 2*a*d)*(b*c -
a*d)^2)/(b^7*(a + b*x)) - (3*a*(b*c - a*d)*(b^2*c^2 - 5*a*b*c*d + 5*a^2*d^2)*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)^3} \, dx &=\int \left (\frac{(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right )}{b^6}+\frac{3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac{3 d^2 (b c-a d) x^2}{b^4}+\frac{d^3 x^3}{b^3}+\frac{a^3 (-b c+a d)^3}{b^6 (a+b x)^3}-\frac{3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^6 (a+b x)^2}+\frac{3 a (b c-a d) \left (-b^2 c^2+5 a b c d-5 a^2 d^2\right )}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) x}{b^6}+\frac{3 d (b c-2 a d) (b c-a d) x^2}{2 b^5}+\frac{d^2 (b c-a d) x^3}{b^4}+\frac{d^3 x^4}{4 b^3}+\frac{a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}-\frac{3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}-\frac{3 a (b c-a d) \left (b^2 c^2-5 a b c d+5 a^2 d^2\right ) \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.139337, size = 207, normalized size = 1.06 \[ \frac{6 b^2 d x^2 \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+4 b x \left (18 a^2 b c d^2-10 a^3 d^3-9 a b^2 c^2 d+b^3 c^3\right )+12 a \left (-10 a^2 b c d^2+5 a^3 d^3+6 a b^2 c^2 d-b^3 c^3\right ) \log (a+b x)+\frac{12 a^2 (b c-a d)^2 (2 a d-b c)}{a+b x}+\frac{2 a^3 (b c-a d)^3}{(a+b x)^2}+4 b^3 d^2 x^3 (b c-a d)+b^4 d^3 x^4}{4 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*(b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 10*a^3*d^3)*x + 6*b^2*d*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^2
+ 4*b^3*d^2*(b*c - a*d)*x^3 + b^4*d^3*x^4 + (2*a^3*(b*c - a*d)^3)/(a + b*x)^2 + (12*a^2*(b*c - a*d)^2*(-(b*c)
+ 2*a*d))/(a + b*x) + 12*a*(-(b^3*c^3) + 6*a*b^2*c^2*d - 10*a^2*b*c*d^2 + 5*a^3*d^3)*Log[a + b*x])/(4*b^7)

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Maple [A]  time = 0.009, size = 335, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{x}^{4}}{4\,{b}^{3}}}-{\frac{{x}^{3}a{d}^{3}}{{b}^{4}}}+{\frac{c{x}^{3}{d}^{2}}{{b}^{3}}}+3\,{\frac{{a}^{2}{x}^{2}{d}^{3}}{{b}^{5}}}-{\frac{9\,a{x}^{2}c{d}^{2}}{2\,{b}^{4}}}+{\frac{3\,{x}^{2}{c}^{2}d}{2\,{b}^{3}}}-10\,{\frac{{a}^{3}{d}^{3}x}{{b}^{6}}}+18\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{5}}}-9\,{\frac{a{c}^{2}dx}{{b}^{4}}}+{\frac{{c}^{3}x}{{b}^{3}}}+6\,{\frac{{a}^{5}{d}^{3}}{{b}^{7} \left ( bx+a \right ) }}-15\,{\frac{{a}^{4}c{d}^{2}}{{b}^{6} \left ( bx+a \right ) }}+12\,{\frac{{a}^{3}{c}^{2}d}{{b}^{5} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}{c}^{3}}{{b}^{4} \left ( bx+a \right ) }}-{\frac{{a}^{6}{d}^{3}}{2\,{b}^{7} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{a}^{5}c{d}^{2}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{4}{c}^{2}d}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{3}{c}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+15\,{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{3}}{{b}^{7}}}-30\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{6}}}+18\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{5}}}-3\,{\frac{a\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)^3,x)

[Out]

1/4*d^3*x^4/b^3-1/b^4*x^3*a*d^3+1/b^3*x^3*c*d^2+3/b^5*x^2*a^2*d^3-9/2/b^4*x^2*a*c*d^2+3/2/b^3*x^2*c^2*d-10/b^6
*a^3*d^3*x+18/b^5*a^2*c*d^2*x-9/b^4*a*c^2*d*x+1/b^3*c^3*x+6*a^5/b^7/(b*x+a)*d^3-15*a^4/b^6/(b*x+a)*c*d^2+12*a^
3/b^5/(b*x+a)*c^2*d-3*a^2/b^4/(b*x+a)*c^3-1/2*a^6/b^7/(b*x+a)^2*d^3+3/2*a^5/b^6/(b*x+a)^2*c*d^2-3/2*a^4/b^5/(b
*x+a)^2*c^2*d+1/2*a^3/b^4/(b*x+a)^2*c^3+15*a^4/b^7*ln(b*x+a)*d^3-30*a^3/b^6*ln(b*x+a)*c*d^2+18*a^2/b^5*ln(b*x+
a)*c^2*d-3*a/b^4*ln(b*x+a)*c^3

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Maxima [A]  time = 1.22548, size = 374, normalized size = 1.91 \begin{align*} -\frac{5 \, a^{3} b^{3} c^{3} - 21 \, a^{4} b^{2} c^{2} d + 27 \, a^{5} b c d^{2} - 11 \, a^{6} d^{3} + 6 \,{\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x}{2 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac{b^{3} d^{3} x^{4} + 4 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} + 6 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} x}{4 \, b^{6}} - \frac{3 \,{\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 10 \, a^{3} b c d^{2} - 5 \, a^{4} 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)^3,x, algorithm="maxima")

[Out]

-1/2*(5*a^3*b^3*c^3 - 21*a^4*b^2*c^2*d + 27*a^5*b*c*d^2 - 11*a^6*d^3 + 6*(a^2*b^4*c^3 - 4*a^3*b^3*c^2*d + 5*a^
4*b^2*c*d^2 - 2*a^5*b*d^3)*x)/(b^9*x^2 + 2*a*b^8*x + a^2*b^7) + 1/4*(b^3*d^3*x^4 + 4*(b^3*c*d^2 - a*b^2*d^3)*x
^3 + 6*(b^3*c^2*d - 3*a*b^2*c*d^2 + 2*a^2*b*d^3)*x^2 + 4*(b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 10*a^3*d^
3)*x)/b^6 - 3*(a*b^3*c^3 - 6*a^2*b^2*c^2*d + 10*a^3*b*c*d^2 - 5*a^4*d^3)*log(b*x + a)/b^7

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Fricas [B]  time = 2.86468, size = 868, normalized size = 4.43 \begin{align*} \frac{b^{6} d^{3} x^{6} - 10 \, a^{3} b^{3} c^{3} + 42 \, a^{4} b^{2} c^{2} d - 54 \, a^{5} b c d^{2} + 22 \, a^{6} d^{3} + 2 \,{\left (2 \, b^{6} c d^{2} - a b^{5} d^{3}\right )} x^{5} +{\left (6 \, b^{6} c^{2} d - 10 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{4} + 4 \,{\left (b^{6} c^{3} - 6 \, a b^{5} c^{2} d + 10 \, a^{2} b^{4} c d^{2} - 5 \, a^{3} b^{3} d^{3}\right )} x^{3} + 2 \,{\left (4 \, a b^{5} c^{3} - 33 \, a^{2} b^{4} c^{2} d + 63 \, a^{3} b^{3} c d^{2} - 34 \, a^{4} b^{2} d^{3}\right )} x^{2} - 4 \,{\left (2 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 3 \, a^{4} b^{2} c d^{2} + 4 \, a^{5} b d^{3}\right )} x - 12 \,{\left (a^{3} b^{3} c^{3} - 6 \, a^{4} b^{2} c^{2} d + 10 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3} +{\left (a b^{5} c^{3} - 6 \, a^{2} b^{4} c^{2} d + 10 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a^{2} b^{4} c^{3} - 6 \, a^{3} b^{3} c^{2} d + 10 \, a^{4} b^{2} c d^{2} - 5 \, a^{5} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{4 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^6*d^3*x^6 - 10*a^3*b^3*c^3 + 42*a^4*b^2*c^2*d - 54*a^5*b*c*d^2 + 22*a^6*d^3 + 2*(2*b^6*c*d^2 - a*b^5*d^
3)*x^5 + (6*b^6*c^2*d - 10*a*b^5*c*d^2 + 5*a^2*b^4*d^3)*x^4 + 4*(b^6*c^3 - 6*a*b^5*c^2*d + 10*a^2*b^4*c*d^2 -
5*a^3*b^3*d^3)*x^3 + 2*(4*a*b^5*c^3 - 33*a^2*b^4*c^2*d + 63*a^3*b^3*c*d^2 - 34*a^4*b^2*d^3)*x^2 - 4*(2*a^2*b^4
*c^3 - 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 4*a^5*b*d^3)*x - 12*(a^3*b^3*c^3 - 6*a^4*b^2*c^2*d + 10*a^5*b*c*d^2
- 5*a^6*d^3 + (a*b^5*c^3 - 6*a^2*b^4*c^2*d + 10*a^3*b^3*c*d^2 - 5*a^4*b^2*d^3)*x^2 + 2*(a^2*b^4*c^3 - 6*a^3*b
^3*c^2*d + 10*a^4*b^2*c*d^2 - 5*a^5*b*d^3)*x)*log(b*x + a))/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)

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Sympy [A]  time = 1.93427, size = 277, normalized size = 1.41 \begin{align*} \frac{3 a \left (a d - b c\right ) \left (5 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) \log{\left (a + b x \right )}}{b^{7}} + \frac{11 a^{6} d^{3} - 27 a^{5} b c d^{2} + 21 a^{4} b^{2} c^{2} d - 5 a^{3} b^{3} c^{3} + x \left (12 a^{5} b d^{3} - 30 a^{4} b^{2} c d^{2} + 24 a^{3} b^{3} c^{2} d - 6 a^{2} b^{4} c^{3}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac{d^{3} x^{4}}{4 b^{3}} - \frac{x^{3} \left (a d^{3} - b c d^{2}\right )}{b^{4}} + \frac{x^{2} \left (6 a^{2} d^{3} - 9 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{5}} - \frac{x \left (10 a^{3} d^{3} - 18 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 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)**3,x)

[Out]

3*a*(a*d - b*c)*(5*a**2*d**2 - 5*a*b*c*d + b**2*c**2)*log(a + b*x)/b**7 + (11*a**6*d**3 - 27*a**5*b*c*d**2 + 2
1*a**4*b**2*c**2*d - 5*a**3*b**3*c**3 + x*(12*a**5*b*d**3 - 30*a**4*b**2*c*d**2 + 24*a**3*b**3*c**2*d - 6*a**2
*b**4*c**3))/(2*a**2*b**7 + 4*a*b**8*x + 2*b**9*x**2) + d**3*x**4/(4*b**3) - x**3*(a*d**3 - b*c*d**2)/b**4 + x
**2*(6*a**2*d**3 - 9*a*b*c*d**2 + 3*b**2*c**2*d)/(2*b**5) - x*(10*a**3*d**3 - 18*a**2*b*c*d**2 + 9*a*b**2*c**2
*d - b**3*c**3)/b**6

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Giac [A]  time = 1.25392, size = 374, normalized size = 1.91 \begin{align*} -\frac{3 \,{\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 10 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{5 \, a^{3} b^{3} c^{3} - 21 \, a^{4} b^{2} c^{2} d + 27 \, a^{5} b c d^{2} - 11 \, a^{6} d^{3} + 6 \,{\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{7}} + \frac{b^{9} d^{3} x^{4} + 4 \, b^{9} c d^{2} x^{3} - 4 \, a b^{8} d^{3} x^{3} + 6 \, b^{9} c^{2} d x^{2} - 18 \, a b^{8} c d^{2} x^{2} + 12 \, a^{2} b^{7} d^{3} x^{2} + 4 \, b^{9} c^{3} x - 36 \, a b^{8} c^{2} d x + 72 \, a^{2} b^{7} c d^{2} x - 40 \, a^{3} b^{6} d^{3} x}{4 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(a*b^3*c^3 - 6*a^2*b^2*c^2*d + 10*a^3*b*c*d^2 - 5*a^4*d^3)*log(abs(b*x + a))/b^7 - 1/2*(5*a^3*b^3*c^3 - 21*
a^4*b^2*c^2*d + 27*a^5*b*c*d^2 - 11*a^6*d^3 + 6*(a^2*b^4*c^3 - 4*a^3*b^3*c^2*d + 5*a^4*b^2*c*d^2 - 2*a^5*b*d^3
)*x)/((b*x + a)^2*b^7) + 1/4*(b^9*d^3*x^4 + 4*b^9*c*d^2*x^3 - 4*a*b^8*d^3*x^3 + 6*b^9*c^2*d*x^2 - 18*a*b^8*c*d
^2*x^2 + 12*a^2*b^7*d^3*x^2 + 4*b^9*c^3*x - 36*a*b^8*c^2*d*x + 72*a^2*b^7*c*d^2*x - 40*a^3*b^6*d^3*x)/b^12