3.278 \(\int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx\)

Optimal. Leaf size=162 \[ \frac {b \log (x) (5 b c-2 a d) (b c-a d)^2}{a^6}-\frac {b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6}+\frac {(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac {b (b c-a d)^3}{a^5 (a+b x)}-\frac {3 c (b c-a d)^2}{2 a^4 x^2}+\frac {c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac {c^3}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 162, 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} \[ \frac {c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac {3 c (b c-a d)^2}{2 a^4 x^2}+\frac {(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac {b (b c-a d)^3}{a^5 (a+b x)}+\frac {b \log (x) (5 b c-2 a d) (b c-a d)^2}{a^6}-\frac {b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6}-\frac {c^3}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^5*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]  time = 0.11, size = 155, normalized size = 0.96 \[ -\frac {\frac {3 a^4 c^3}{x^4}+\frac {4 a^3 c^2 (3 a d-2 b c)}{x^3}+\frac {18 a^2 c (b c-a d)^2}{x^2}+\frac {12 a (b c-a d)^2 (a d-4 b c)}{x}+\frac {12 a b (a d-b c)^3}{a+b x}-12 b \log (x) (5 b c-2 a d) (b c-a d)^2+12 b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^5*(a + b*x)^2),x]

[Out]

-1/12*((3*a^4*c^3)/x^4 + (4*a^3*c^2*(-2*b*c + 3*a*d))/x^3 + (18*a^2*c*(b*c - a*d)^2)/x^2 + (12*a*(b*c - a*d)^2
*(-4*b*c + a*d))/x + (12*a*b*(-(b*c) + a*d)^3)/(a + b*x) - 12*b*(5*b*c - 2*a*d)*(b*c - a*d)^2*Log[x] + 12*b*(5
*b*c - 2*a*d)*(b*c - a*d)^2*Log[a + b*x])/a^6

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fricas [B]  time = 0.68, size = 382, normalized size = 2.36 \[ -\frac {3 \, a^{5} c^{3} - 12 \, {\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4} - 6 \, {\left (5 \, a^{2} b^{3} c^{3} - 12 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} x^{3} + 2 \, {\left (5 \, a^{3} b^{2} c^{3} - 12 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{3} - 12 \, a^{5} c^{2} d\right )} x + 12 \, {\left ({\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} x^{5} + {\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4}\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} x^{5} + {\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4}\right )} \log \relax (x)}{12 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^5/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.94, size = 373, normalized size = 2.30 \[ \frac {{\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{6} b} + \frac {\frac {b^{9} c^{3}}{b x + a} - \frac {3 \, a b^{8} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{7} c d^{2}}{b x + a} - \frac {a^{3} b^{6} d^{3}}{b x + a}}{a^{5} b^{5}} + \frac {77 \, b^{4} c^{3} - 156 \, a b^{3} c^{2} d + 90 \, a^{2} b^{2} c d^{2} - 12 \, a^{3} b d^{3} - \frac {4 \, {\left (65 \, a b^{5} c^{3} - 129 \, a^{2} b^{4} c^{2} d + 72 \, a^{3} b^{3} c d^{2} - 9 \, a^{4} b^{2} d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (50 \, a^{2} b^{6} c^{3} - 96 \, a^{3} b^{5} c^{2} d + 51 \, a^{4} b^{4} c d^{2} - 6 \, a^{5} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {12 \, {\left (10 \, a^{3} b^{7} c^{3} - 18 \, a^{4} b^{6} c^{2} d + 9 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, a^{6} {\left (\frac {a}{b x + a} - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^5/(b*x+a)^2,x, algorithm="giac")

[Out]

(5*b^5*c^3 - 12*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*log(abs(-a/(b*x + a) + 1))/(a^6*b) + (b^9*c^3/(
b*x + a) - 3*a*b^8*c^2*d/(b*x + a) + 3*a^2*b^7*c*d^2/(b*x + a) - a^3*b^6*d^3/(b*x + a))/(a^5*b^5) + 1/12*(77*b
^4*c^3 - 156*a*b^3*c^2*d + 90*a^2*b^2*c*d^2 - 12*a^3*b*d^3 - 4*(65*a*b^5*c^3 - 129*a^2*b^4*c^2*d + 72*a^3*b^3*
c*d^2 - 9*a^4*b^2*d^3)/((b*x + a)*b) + 6*(50*a^2*b^6*c^3 - 96*a^3*b^5*c^2*d + 51*a^4*b^4*c*d^2 - 6*a^5*b^3*d^3
)/((b*x + a)^2*b^2) - 12*(10*a^3*b^7*c^3 - 18*a^4*b^6*c^2*d + 9*a^5*b^5*c*d^2 - a^6*b^4*d^3)/((b*x + a)^3*b^3)
)/(a^6*(a/(b*x + a) - 1)^4)

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maple [B]  time = 0.01, size = 320, normalized size = 1.98 \[ -\frac {b \,d^{3}}{\left (b x +a \right ) a^{2}}+\frac {3 b^{2} c \,d^{2}}{\left (b x +a \right ) a^{3}}-\frac {2 b \,d^{3} \ln \relax (x )}{a^{3}}+\frac {2 b \,d^{3} \ln \left (b x +a \right )}{a^{3}}-\frac {3 b^{3} c^{2} d}{\left (b x +a \right ) a^{4}}+\frac {9 b^{2} c \,d^{2} \ln \relax (x )}{a^{4}}-\frac {9 b^{2} c \,d^{2} \ln \left (b x +a \right )}{a^{4}}+\frac {b^{4} c^{3}}{\left (b x +a \right ) a^{5}}-\frac {12 b^{3} c^{2} d \ln \relax (x )}{a^{5}}+\frac {12 b^{3} c^{2} d \ln \left (b x +a \right )}{a^{5}}+\frac {5 b^{4} c^{3} \ln \relax (x )}{a^{6}}-\frac {5 b^{4} c^{3} \ln \left (b x +a \right )}{a^{6}}-\frac {d^{3}}{a^{2} x}+\frac {6 b c \,d^{2}}{a^{3} x}-\frac {9 b^{2} c^{2} d}{a^{4} x}+\frac {4 b^{3} c^{3}}{a^{5} x}-\frac {3 c \,d^{2}}{2 a^{2} x^{2}}+\frac {3 b \,c^{2} d}{a^{3} x^{2}}-\frac {3 b^{2} c^{3}}{2 a^{4} x^{2}}-\frac {c^{2} d}{a^{2} x^{3}}+\frac {2 b \,c^{3}}{3 a^{3} x^{3}}-\frac {c^{3}}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^5/(b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 1.19, size = 275, normalized size = 1.70 \[ -\frac {3 \, a^{4} c^{3} - 12 \, {\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{4} - 6 \, {\left (5 \, a b^{3} c^{3} - 12 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{3} + 2 \, {\left (5 \, a^{2} b^{2} c^{3} - 12 \, a^{3} b c^{2} d + 9 \, a^{4} c d^{2}\right )} x^{2} - {\left (5 \, a^{3} b c^{3} - 12 \, a^{4} c^{2} d\right )} x}{12 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac {{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac {{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} \log \relax (x)}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^5/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.51, size = 263, normalized size = 1.62 \[ -\frac {\frac {c^3}{4\,a}+\frac {x^3\,\left (2\,a^3\,d^3-9\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{2\,a^4}+\frac {c^2\,x\,\left (12\,a\,d-5\,b\,c\right )}{12\,a^2}+\frac {c\,x^2\,\left (9\,a^2\,d^2-12\,a\,b\,c\,d+5\,b^2\,c^2\right )}{6\,a^3}+\frac {b\,x^4\,\left (2\,a^3\,d^3-9\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{a^5}}{b\,x^5+a\,x^4}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-5\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (-2\,a^3\,b\,d^3+9\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+5\,b^4\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-5\,b\,c\right )}{a^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^5*(a + b*x)^2),x)

[Out]

- (c^3/(4*a) + (x^3*(2*a^3*d^3 - 5*b^3*c^3 + 12*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(2*a^4) + (c^2*x*(12*a*d - 5*b*c
))/(12*a^2) + (c*x^2*(9*a^2*d^2 + 5*b^2*c^2 - 12*a*b*c*d))/(6*a^3) + (b*x^4*(2*a^3*d^3 - 5*b^3*c^3 + 12*a*b^2*
c^2*d - 9*a^2*b*c*d^2))/a^5)/(a*x^4 + b*x^5) - (2*b*atanh((b*(a*d - b*c)^2*(2*a*d - 5*b*c)*(a + 2*b*x))/(a*(5*
b^4*c^3 - 2*a^3*b*d^3 + 9*a^2*b^2*c*d^2 - 12*a*b^3*c^2*d)))*(a*d - b*c)^2*(2*a*d - 5*b*c))/a^6

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sympy [B]  time = 2.02, size = 466, normalized size = 2.88 \[ \frac {- 3 a^{4} c^{3} + x^{4} \left (- 24 a^{3} b d^{3} + 108 a^{2} b^{2} c d^{2} - 144 a b^{3} c^{2} d + 60 b^{4} c^{3}\right ) + x^{3} \left (- 12 a^{4} d^{3} + 54 a^{3} b c d^{2} - 72 a^{2} b^{2} c^{2} d + 30 a b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{4} c d^{2} + 24 a^{3} b c^{2} d - 10 a^{2} b^{2} c^{3}\right ) + x \left (- 12 a^{4} c^{2} d + 5 a^{3} b c^{3}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} - \frac {b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right ) \log {\left (x + \frac {2 a^{4} b d^{3} - 9 a^{3} b^{2} c d^{2} + 12 a^{2} b^{3} c^{2} d - 5 a b^{4} c^{3} - a b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right )}{4 a^{3} b^{2} d^{3} - 18 a^{2} b^{3} c d^{2} + 24 a b^{4} c^{2} d - 10 b^{5} c^{3}} \right )}}{a^{6}} + \frac {b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right ) \log {\left (x + \frac {2 a^{4} b d^{3} - 9 a^{3} b^{2} c d^{2} + 12 a^{2} b^{3} c^{2} d - 5 a b^{4} c^{3} + a b \left (a d - b c\right )^{2} \left (2 a d - 5 b c\right )}{4 a^{3} b^{2} d^{3} - 18 a^{2} b^{3} c d^{2} + 24 a b^{4} c^{2} d - 10 b^{5} c^{3}} \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**5/(b*x+a)**2,x)

[Out]

(-3*a**4*c**3 + x**4*(-24*a**3*b*d**3 + 108*a**2*b**2*c*d**2 - 144*a*b**3*c**2*d + 60*b**4*c**3) + x**3*(-12*a
**4*d**3 + 54*a**3*b*c*d**2 - 72*a**2*b**2*c**2*d + 30*a*b**3*c**3) + x**2*(-18*a**4*c*d**2 + 24*a**3*b*c**2*d
- 10*a**2*b**2*c**3) + x*(-12*a**4*c**2*d + 5*a**3*b*c**3))/(12*a**6*x**4 + 12*a**5*b*x**5) - b*(a*d - b*c)**
2*(2*a*d - 5*b*c)*log(x + (2*a**4*b*d**3 - 9*a**3*b**2*c*d**2 + 12*a**2*b**3*c**2*d - 5*a*b**4*c**3 - a*b*(a*d
- b*c)**2*(2*a*d - 5*b*c))/(4*a**3*b**2*d**3 - 18*a**2*b**3*c*d**2 + 24*a*b**4*c**2*d - 10*b**5*c**3))/a**6 +
b*(a*d - b*c)**2*(2*a*d - 5*b*c)*log(x + (2*a**4*b*d**3 - 9*a**3*b**2*c*d**2 + 12*a**2*b**3*c**2*d - 5*a*b**4
*c**3 + a*b*(a*d - b*c)**2*(2*a*d - 5*b*c))/(4*a**3*b**2*d**3 - 18*a**2*b**3*c*d**2 + 24*a*b**4*c**2*d - 10*b*
*5*c**3))/a**6

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