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

Optimal. Leaf size=164 \[ \frac {a^3 (b c-a d)^3}{b^7 (a+b x)}+\frac {3 a^2 (b c-2 a d) (b c-a d)^2 \log (a+b x)}{b^7}-\frac {a x (2 b c-5 a d) (b c-a d)^2}{b^6}+\frac {x^2 (b c-4 a d) (b c-a d)^2}{2 b^5}+\frac {d x^3 (b c-a d)^2}{b^4}+\frac {d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac {d^3 x^5}{5 b^2} \]

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Rubi [A]  time = 0.16, antiderivative size = 164, 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 {a^3 (b c-a d)^3}{b^7 (a+b x)}+\frac {3 a^2 (b c-2 a d) (b c-a d)^2 \log (a+b x)}{b^7}+\frac {d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac {d x^3 (b c-a d)^2}{b^4}+\frac {x^2 (b c-4 a d) (b c-a d)^2}{2 b^5}-\frac {a x (2 b c-5 a d) (b c-a d)^2}{b^6}+\frac {d^3 x^5}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

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

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Mathematica [A]  time = 0.07, size = 160, normalized size = 0.98 \begin {gather*} \frac {-\frac {20 a^3 (a d-b c)^3}{a+b x}-60 a^2 (b c-a d)^2 (2 a d-b c) \log (a+b x)+5 b^4 d^2 x^4 (3 b c-2 a d)+20 b^3 d x^3 (b c-a d)^2+10 b^2 x^2 (b c-4 a d) (b c-a d)^2+20 a b x (b c-a d)^2 (5 a d-2 b c)+4 b^5 d^3 x^5}{20 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*a*b*(b*c - a*d)^2*(-2*b*c + 5*a*d)*x + 10*b^2*(b*c - 4*a*d)*(b*c - a*d)^2*x^2 + 20*b^3*d*(b*c - a*d)^2*x^3
+ 5*b^4*d^2*(3*b*c - 2*a*d)*x^4 + 4*b^5*d^3*x^5 - (20*a^3*(-(b*c) + a*d)^3)/(a + b*x) - 60*a^2*(b*c - a*d)^2*
(-(b*c) + 2*a*d)*Log[a + b*x])/(20*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)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.22, size = 366, normalized size = 2.23 \begin {gather*} \frac {4 \, b^{6} d^{3} x^{6} + 20 \, a^{3} b^{3} c^{3} - 60 \, a^{4} b^{2} c^{2} d + 60 \, a^{5} b c d^{2} - 20 \, a^{6} d^{3} + 3 \, {\left (5 \, b^{6} c d^{2} - 2 \, a b^{5} d^{3}\right )} x^{5} + 5 \, {\left (4 \, b^{6} c^{2} d - 5 \, a b^{5} c d^{2} + 2 \, a^{2} b^{4} d^{3}\right )} x^{4} + 10 \, {\left (b^{6} c^{3} - 4 \, a b^{5} c^{2} d + 5 \, a^{2} b^{4} c d^{2} - 2 \, a^{3} b^{3} d^{3}\right )} x^{3} - 30 \, {\left (a b^{5} c^{3} - 4 \, a^{2} b^{4} c^{2} d + 5 \, a^{3} b^{3} c d^{2} - 2 \, a^{4} b^{2} d^{3}\right )} x^{2} - 20 \, {\left (2 \, a^{2} b^{4} c^{3} - 9 \, a^{3} b^{3} c^{2} d + 12 \, a^{4} b^{2} c d^{2} - 5 \, a^{5} b d^{3}\right )} x + 60 \, {\left (a^{3} b^{3} c^{3} - 4 \, a^{4} b^{2} c^{2} d + 5 \, a^{5} b c d^{2} - 2 \, a^{6} d^{3} + {\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\right )} \log \left (b x + a\right )}{20 \, {\left (b^{8} x + a b^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*b^6*d^3*x^6 + 20*a^3*b^3*c^3 - 60*a^4*b^2*c^2*d + 60*a^5*b*c*d^2 - 20*a^6*d^3 + 3*(5*b^6*c*d^2 - 2*a*b
^5*d^3)*x^5 + 5*(4*b^6*c^2*d - 5*a*b^5*c*d^2 + 2*a^2*b^4*d^3)*x^4 + 10*(b^6*c^3 - 4*a*b^5*c^2*d + 5*a^2*b^4*c*
d^2 - 2*a^3*b^3*d^3)*x^3 - 30*(a*b^5*c^3 - 4*a^2*b^4*c^2*d + 5*a^3*b^3*c*d^2 - 2*a^4*b^2*d^3)*x^2 - 20*(2*a^2*
b^4*c^3 - 9*a^3*b^3*c^2*d + 12*a^4*b^2*c*d^2 - 5*a^5*b*d^3)*x + 60*(a^3*b^3*c^3 - 4*a^4*b^2*c^2*d + 5*a^5*b*c*
d^2 - 2*a^6*d^3 + (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)*log(b*x + a))/(b^8*x + a*
b^7)

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giac [B]  time = 1.08, size = 341, normalized size = 2.08 \begin {gather*} \frac {{\left (4 \, d^{3} + \frac {15 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {20 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {10 \, {\left (b^{6} c^{3} - 12 \, a b^{5} c^{2} d + 30 \, a^{2} b^{4} c d^{2} - 20 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} - \frac {60 \, {\left (a b^{7} c^{3} - 6 \, a^{2} b^{6} c^{2} d + 10 \, a^{3} b^{5} c d^{2} - 5 \, a^{4} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )} {\left (b x + a\right )}^{5}}{20 \, b^{7}} - \frac {3 \, {\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{7}} + \frac {\frac {a^{3} b^{8} c^{3}}{b x + a} - \frac {3 \, a^{4} b^{7} c^{2} d}{b x + a} + \frac {3 \, a^{5} b^{6} c d^{2}}{b x + a} - \frac {a^{6} b^{5} d^{3}}{b x + a}}{b^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*d^3 + 15*(b^2*c*d^2 - 2*a*b*d^3)/((b*x + a)*b) + 20*(b^4*c^2*d - 5*a*b^3*c*d^2 + 5*a^2*b^2*d^3)/((b*x
+ a)^2*b^2) + 10*(b^6*c^3 - 12*a*b^5*c^2*d + 30*a^2*b^4*c*d^2 - 20*a^3*b^3*d^3)/((b*x + a)^3*b^3) - 60*(a*b^7*
c^3 - 6*a^2*b^6*c^2*d + 10*a^3*b^5*c*d^2 - 5*a^4*b^4*d^3)/((b*x + a)^4*b^4))*(b*x + a)^5/b^7 - 3*(a^2*b^3*c^3
- 4*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 - 2*a^5*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^7 + (a^3*b^8*c^3/(b*x
+ a) - 3*a^4*b^7*c^2*d/(b*x + a) + 3*a^5*b^6*c*d^2/(b*x + a) - a^6*b^5*d^3/(b*x + a))/b^12

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maple [B]  time = 0.01, size = 318, normalized size = 1.94 \begin {gather*} \frac {d^{3} x^{5}}{5 b^{2}}-\frac {a \,d^{3} x^{4}}{2 b^{3}}+\frac {3 c \,d^{2} x^{4}}{4 b^{2}}+\frac {a^{2} d^{3} x^{3}}{b^{4}}-\frac {2 a c \,d^{2} x^{3}}{b^{3}}+\frac {c^{2} d \,x^{3}}{b^{2}}-\frac {2 a^{3} d^{3} x^{2}}{b^{5}}+\frac {9 a^{2} c \,d^{2} x^{2}}{2 b^{4}}-\frac {3 a \,c^{2} d \,x^{2}}{b^{3}}+\frac {c^{3} x^{2}}{2 b^{2}}-\frac {a^{6} d^{3}}{\left (b x +a \right ) b^{7}}+\frac {3 a^{5} c \,d^{2}}{\left (b x +a \right ) b^{6}}-\frac {6 a^{5} d^{3} \ln \left (b x +a \right )}{b^{7}}-\frac {3 a^{4} c^{2} d}{\left (b x +a \right ) b^{5}}+\frac {15 a^{4} c \,d^{2} \ln \left (b x +a \right )}{b^{6}}+\frac {5 a^{4} d^{3} x}{b^{6}}+\frac {a^{3} c^{3}}{\left (b x +a \right ) b^{4}}-\frac {12 a^{3} c^{2} d \ln \left (b x +a \right )}{b^{5}}-\frac {12 a^{3} c \,d^{2} x}{b^{5}}+\frac {3 a^{2} c^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {9 a^{2} c^{2} d x}{b^{4}}-\frac {2 a \,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)^2,x)

[Out]

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

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maxima [A]  time = 1.10, size = 270, normalized size = 1.65 \begin {gather*} \frac {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}}{b^{8} x + a b^{7}} + \frac {4 \, b^{4} d^{3} x^{5} + 5 \, {\left (3 \, b^{4} c d^{2} - 2 \, a b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3} + 10 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x^{2} - 20 \, {\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} x}{20 \, b^{6}} + \frac {3 \, {\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} 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)^2,x, algorithm="maxima")

[Out]

(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)/(b^8*x + a*b^7) + 1/20*(4*b^4*d^3*x^5 + 5*(3*b^4*c*d
^2 - 2*a*b^3*d^3)*x^4 + 20*(b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3)*x^3 + 10*(b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2
*b^2*c*d^2 - 4*a^3*b*d^3)*x^2 - 20*(2*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 - 5*a^4*d^3)*x)/b^6 + 3*(a^
2*b^3*c^3 - 4*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 - 2*a^5*d^3)*log(b*x + a)/b^7

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mupad [B]  time = 0.09, size = 438, normalized size = 2.67 \begin {gather*} x^2\,\left (\frac {c^3}{2\,b^2}-\frac {a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{2\,b^2}\right )-x^4\,\left (\frac {a\,d^3}{2\,b^3}-\frac {3\,c\,d^2}{4\,b^2}\right )-x\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b^2}+\frac {2\,a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )+x^3\,\left (\frac {c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{3\,b}-\frac {a^2\,d^3}{3\,b^4}\right )-\frac {\ln \left (a+b\,x\right )\,\left (6\,a^5\,d^3-15\,a^4\,b\,c\,d^2+12\,a^3\,b^2\,c^2\,d-3\,a^2\,b^3\,c^3\right )}{b^7}-\frac {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}{b\,\left (x\,b^7+a\,b^6\right )}+\frac {d^3\,x^5}{5\,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)^2,x)

[Out]

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

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sympy [A]  time = 0.93, size = 257, normalized size = 1.57 \begin {gather*} - \frac {3 a^{2} \left (a d - b c\right )^{2} \left (2 a d - b c\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{4} \left (- \frac {a d^{3}}{2 b^{3}} + \frac {3 c d^{2}}{4 b^{2}}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{b^{4}} - \frac {2 a c d^{2}}{b^{3}} + \frac {c^{2} d}{b^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} d^{3}}{b^{5}} + \frac {9 a^{2} c d^{2}}{2 b^{4}} - \frac {3 a c^{2} d}{b^{3}} + \frac {c^{3}}{2 b^{2}}\right ) + x \left (\frac {5 a^{4} d^{3}}{b^{6}} - \frac {12 a^{3} c d^{2}}{b^{5}} + \frac {9 a^{2} c^{2} d}{b^{4}} - \frac {2 a c^{3}}{b^{3}}\right ) + \frac {- 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}}{a b^{7} + b^{8} x} + \frac {d^{3} x^{5}}{5 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a**2*(a*d - b*c)**2*(2*a*d - b*c)*log(a + b*x)/b**7 + x**4*(-a*d**3/(2*b**3) + 3*c*d**2/(4*b**2)) + x**3*(a
**2*d**3/b**4 - 2*a*c*d**2/b**3 + c**2*d/b**2) + x**2*(-2*a**3*d**3/b**5 + 9*a**2*c*d**2/(2*b**4) - 3*a*c**2*d
/b**3 + c**3/(2*b**2)) + x*(5*a**4*d**3/b**6 - 12*a**3*c*d**2/b**5 + 9*a**2*c**2*d/b**4 - 2*a*c**3/b**3) + (-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)/(a*b**7 + b**8*x) + d**3*x**5/(5*b**2)

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