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

Optimal. Leaf size=156 \[ -\frac {a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac {\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}+\frac {a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac {3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac {3 d^2 x^2 (b c-a d)}{2 b^4}+\frac {d^3 x^3}{3 b^3} \]

[Out]

3*d*(-2*a*d+b*c)*(-a*d+b*c)*x/b^5+3/2*d^2*(-a*d+b*c)*x^2/b^4+1/3*d^3*x^3/b^3-1/2*a^2*(-a*d+b*c)^3/b^6/(b*x+a)^
2+a*(-5*a*d+2*b*c)*(-a*d+b*c)^2/b^6/(b*x+a)+(-a*d+b*c)*(10*a^2*d^2-8*a*b*c*d+b^2*c^2)*ln(b*x+a)/b^6

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Rubi [A]  time = 0.15, antiderivative size = 156, 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} \[ \frac {\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}-\frac {a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac {3 d^2 x^2 (b c-a d)}{2 b^4}+\frac {a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac {3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac {d^3 x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^3)/(a + b*x)^3,x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 160, normalized size = 1.03 \[ \frac {18 b d x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+\frac {3 a^2 (a d-b c)^3}{(a+b x)^2}+6 \left (-10 a^3 d^3+18 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right ) \log (a+b x)+9 b^2 d^2 x^2 (b c-a d)-\frac {6 a (b c-a d)^2 (5 a d-2 b c)}{a+b x}+2 b^3 d^3 x^3}{6 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.63, size = 361, normalized size = 2.31 \[ \frac {2 \, b^{5} d^{3} x^{5} + 9 \, a^{2} b^{3} c^{3} - 45 \, a^{3} b^{2} c^{2} d + 63 \, a^{4} b c d^{2} - 27 \, a^{5} d^{3} + {\left (9 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \, {\left (9 \, b^{5} c^{2} d - 18 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{3} + 9 \, {\left (4 \, a b^{4} c^{2} d - 11 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{2} + 6 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x + 6 \, {\left (a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2} - 10 \, a^{5} d^{3} + {\left (b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 18 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2} - 10 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^5*d^3*x^5 + 9*a^2*b^3*c^3 - 45*a^3*b^2*c^2*d + 63*a^4*b*c*d^2 - 27*a^5*d^3 + (9*b^5*c*d^2 - 5*a*b^4*d
^3)*x^4 + 2*(9*b^5*c^2*d - 18*a*b^4*c*d^2 + 10*a^2*b^3*d^3)*x^3 + 9*(4*a*b^4*c^2*d - 11*a^2*b^3*c*d^2 + 7*a^3*
b^2*d^3)*x^2 + 6*(2*a*b^4*c^3 - 6*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 + a^4*b*d^3)*x + 6*(a^2*b^3*c^3 - 9*a^3*b^2*
c^2*d + 18*a^4*b*c*d^2 - 10*a^5*d^3 + (b^5*c^3 - 9*a*b^4*c^2*d + 18*a^2*b^3*c*d^2 - 10*a^3*b^2*d^3)*x^2 + 2*(a
*b^4*c^3 - 9*a^2*b^3*c^2*d + 18*a^3*b^2*c*d^2 - 10*a^4*b*d^3)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)

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giac [A]  time = 1.10, size = 222, normalized size = 1.42 \[ \frac {{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} d^{3} x^{3} + 9 \, b^{6} c d^{2} x^{2} - 9 \, a b^{5} d^{3} x^{2} + 18 \, b^{6} c^{2} d x - 54 \, a b^{5} c d^{2} x + 36 \, a^{2} b^{4} d^{3} x}{6 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 10*a^3*d^3)*log(abs(b*x + a))/b^6 + 1/2*(3*a^2*b^3*c^3 - 15*a^3*b^
2*c^2*d + 21*a^4*b*c*d^2 - 9*a^5*d^3 + 2*(2*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 12*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x)/(
(b*x + a)^2*b^6) + 1/6*(2*b^6*d^3*x^3 + 9*b^6*c*d^2*x^2 - 9*a*b^5*d^3*x^2 + 18*b^6*c^2*d*x - 54*a*b^5*c*d^2*x
+ 36*a^2*b^4*d^3*x)/b^9

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maple [A]  time = 0.01, size = 280, normalized size = 1.79 \[ \frac {d^{3} x^{3}}{3 b^{3}}+\frac {a^{5} d^{3}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {3 a^{4} c \,d^{2}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {3 a^{3} c^{2} d}{2 \left (b x +a \right )^{2} b^{4}}-\frac {a^{2} c^{3}}{2 \left (b x +a \right )^{2} b^{3}}-\frac {3 a \,d^{3} x^{2}}{2 b^{4}}+\frac {3 c \,d^{2} x^{2}}{2 b^{3}}-\frac {5 a^{4} d^{3}}{\left (b x +a \right ) b^{6}}+\frac {12 a^{3} c \,d^{2}}{\left (b x +a \right ) b^{5}}-\frac {10 a^{3} d^{3} \ln \left (b x +a \right )}{b^{6}}-\frac {9 a^{2} c^{2} d}{\left (b x +a \right ) b^{4}}+\frac {18 a^{2} c \,d^{2} \ln \left (b x +a \right )}{b^{5}}+\frac {6 a^{2} d^{3} x}{b^{5}}+\frac {2 a \,c^{3}}{\left (b x +a \right ) b^{3}}-\frac {9 a \,c^{2} d \ln \left (b x +a \right )}{b^{4}}-\frac {9 a c \,d^{2} x}{b^{4}}+\frac {c^{3} \ln \left (b x +a \right )}{b^{3}}+\frac {3 c^{2} d x}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.02, size = 227, normalized size = 1.46 \[ \frac {3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, b^{2} d^{3} x^{3} + 9 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 18 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x}{6 \, b^{5}} + \frac {{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 21*a^4*b*c*d^2 - 9*a^5*d^3 + 2*(2*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 12*a^3
*b^2*c*d^2 - 5*a^4*b*d^3)*x)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) + 1/6*(2*b^2*d^3*x^3 + 9*(b^2*c*d^2 - a*b*d^3)*x^
2 + 18*(b^2*c^2*d - 3*a*b*c*d^2 + 2*a^2*d^3)*x)/b^5 + (b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 10*a^3*d^3)*
log(b*x + a)/b^6

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mupad [B]  time = 0.39, size = 248, normalized size = 1.59 \[ x\,\left (\frac {3\,c^2\,d}{b^3}+\frac {3\,a\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{b}-\frac {3\,a^2\,d^3}{b^5}\right )-x^2\,\left (\frac {3\,a\,d^3}{2\,b^4}-\frac {3\,c\,d^2}{2\,b^3}\right )-\frac {x\,\left (5\,a^4\,d^3-12\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3\right )+\frac {3\,\left (3\,a^5\,d^3-7\,a^4\,b\,c\,d^2+5\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{2\,b}}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}+\frac {d^3\,x^3}{3\,b^3}-\frac {\ln \left (a+b\,x\right )\,\left (10\,a^3\,d^3-18\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x)^3)/(a + b*x)^3,x)

[Out]

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

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sympy [A]  time = 1.58, size = 235, normalized size = 1.51 \[ x^{2} \left (- \frac {3 a d^{3}}{2 b^{4}} + \frac {3 c d^{2}}{2 b^{3}}\right ) + x \left (\frac {6 a^{2} d^{3}}{b^{5}} - \frac {9 a c d^{2}}{b^{4}} + \frac {3 c^{2} d}{b^{3}}\right ) + \frac {- 9 a^{5} d^{3} + 21 a^{4} b c d^{2} - 15 a^{3} b^{2} c^{2} d + 3 a^{2} b^{3} c^{3} + x \left (- 10 a^{4} b d^{3} + 24 a^{3} b^{2} c d^{2} - 18 a^{2} b^{3} c^{2} d + 4 a b^{4} c^{3}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac {d^{3} x^{3}}{3 b^{3}} - \frac {\left (a d - b c\right ) \left (10 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \log {\left (a + b x \right )}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*(-3*a*d**3/(2*b**4) + 3*c*d**2/(2*b**3)) + x*(6*a**2*d**3/b**5 - 9*a*c*d**2/b**4 + 3*c**2*d/b**3) + (-9*a
**5*d**3 + 21*a**4*b*c*d**2 - 15*a**3*b**2*c**2*d + 3*a**2*b**3*c**3 + x*(-10*a**4*b*d**3 + 24*a**3*b**2*c*d**
2 - 18*a**2*b**3*c**2*d + 4*a*b**4*c**3))/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**2) + d**3*x**3/(3*b**3) - (a*d
- b*c)*(10*a**2*d**2 - 8*a*b*c*d + b**2*c**2)*log(a + b*x)/b**6

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