3.14.55 \(\int \frac {(a+b x)^3}{(c+d x)^3} \, dx\) [1355]

Optimal. Leaf size=78 \[ \frac {b^3 x}{d^3}+\frac {(b c-a d)^3}{2 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2}{d^4 (c+d x)}-\frac {3 b^2 (b c-a d) \log (c+d x)}{d^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} -\frac {3 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac {3 b (b c-a d)^2}{d^4 (c+d x)}+\frac {(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac {b^3 x}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^3,x]

[Out]

(b^3*x)/d^3 + (b*c - a*d)^3/(2*d^4*(c + d*x)^2) - (3*b*(b*c - a*d)^2)/(d^4*(c + d*x)) - (3*b^2*(b*c - a*d)*Log
[c + d*x])/d^4

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 114, normalized size = 1.46 \begin {gather*} \frac {-a^3 d^3-3 a^2 b d^2 (c+2 d x)+3 a b^2 c d (3 c+4 d x)+b^3 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )-6 b^2 (b c-a d) (c+d x)^2 \log (c+d x)}{2 d^4 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(c + d*x)^3,x]

[Out]

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

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Mathics [A]
time = 3.20, size = 144, normalized size = 1.85 \begin {gather*} \frac {-a^3 d^3-3 a^2 b c d^2+6 b^2 \text {Log}\left [c+d x\right ] \left (a d-b c\right ) \left (c^2+2 c d x+d^2 x^2\right )+9 a b^2 c^2 d-5 b^3 c^3-6 b d x \left (a^2 d^2-2 a b c d+b^2 c^2\right )+2 b^3 d x \left (c^2+2 c d x+d^2 x^2\right )}{2 d^4 \left (c^2+2 c d x+d^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 114, normalized size = 1.46

method	result	size
default	\(\frac {b^{3} x}{d^{3}}-\frac {3 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{4} \left (d x +c \right )}+\frac {3 b^{2} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 d^{4} \left (d x +c \right )^{2}}\)	\(114\)
norman	\(\frac {\frac {b^{3} x^{3}}{d}-\frac {a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +9 b^{3} c^{3}}{2 d^{4}}-\frac {\left (3 a^{2} b \,d^{2}-6 a \,b^{2} c d +6 b^{3} c^{2}\right ) x}{d^{3}}}{\left (d x +c \right )^{2}}+\frac {3 b^{2} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{4}}\)	\(116\)
risch	\(\frac {b^{3} x}{d^{3}}+\frac {\left (-3 a^{2} b \,d^{2}+6 a \,b^{2} c d -3 b^{3} c^{2}\right ) x -\frac {a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}}{2 d}}{d^{3} \left (d x +c \right )^{2}}+\frac {3 b^{2} \ln \left (d x +c \right ) a}{d^{3}}-\frac {3 b^{3} \ln \left (d x +c \right ) c}{d^{4}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 125, normalized size = 1.60 \begin {gather*} \frac {b^{3} x}{d^{3}} - \frac {5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} - \frac {3 \, {\left (b^{3} c - a b^{2} d\right )} \log \left (d x + c\right )}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (76) = 152\).
time = 0.29, size = 188, normalized size = 2.41 \begin {gather*} \frac {2 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c d^{2} x^{2} - 5 \, b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - a^{3} d^{3} - 2 \, {\left (2 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.49, size = 128, normalized size = 1.64 \begin {gather*} \frac {b^{3} x}{d^{3}} + \frac {3 b^{2} \left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{4}} + \frac {- a^{3} d^{3} - 3 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 5 b^{3} c^{3} + x \left (- 6 a^{2} b d^{3} + 12 a b^{2} c d^{2} - 6 b^{3} c^{2} d\right )}{2 c^{2} d^{4} + 4 c d^{5} x + 2 d^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 125, normalized size = 1.60 \begin {gather*} \frac {x b^{3}}{d^{3}}+\frac {\frac {1}{2} \left (-5 b^{3} c^{3}+9 b^{2} d c^{2} a-3 b d^{2} c a^{2}-d^{3} a^{3}+\left (-6 b^{3} d c^{2}+12 b^{2} d^{2} c a-6 b d^{3} a^{2}\right ) x\right )}{d^{4} \left (x d+c\right )^{2}}+\frac {\left (-3 b^{3} c+3 b^{2} a d\right ) \ln \left |x d+c\right |}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.11, size = 130, normalized size = 1.67 \begin {gather*} \frac {b^3\,x}{d^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,b^3\,c-3\,a\,b^2\,d\right )}{d^4}-\frac {\frac {a^3\,d^3+3\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+5\,b^3\,c^3}{2\,d}+x\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{c^2\,d^3+2\,c\,d^4\,x+d^5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/(c + d*x)^3,x)

[Out]

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

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