3.16.31 \(\int \frac {(c+d x)^3}{(a-b x) (a+b x)} \, dx\) [1531]

Optimal. Leaf size=76 \[ -\frac {3 c d^2 x}{b^2}-\frac {d^3 x^2}{2 b^2}-\frac {(b c+a d)^3 \log (a-b x)}{2 a b^4}+\frac {(b c-a d)^3 \log (a+b x)}{2 a b^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {84} \begin {gather*} -\frac {(a d+b c)^3 \log (a-b x)}{2 a b^4}+\frac {(b c-a d)^3 \log (a+b x)}{2 a b^4}-\frac {3 c d^2 x}{b^2}-\frac {d^3 x^2}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 62, normalized size = 0.82 \begin {gather*} -\frac {a b^2 d^2 x (6 c+d x)+(b c+a d)^3 \log (a-b x)-(b c-a d)^3 \log (a+b x)}{2 a b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 123, normalized size = 1.62

method	result	size
default	\(-\frac {d^{2} \left (\frac {1}{2} d \,x^{2}+3 c x \right )}{b^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{2 b^{4} a}+\frac {\left (-a^{3} d^{3}-3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (-b x +a \right )}{2 b^{4} a}\)	\(123\)
norman	\(-\frac {d^{3} x^{2}}{2 b^{2}}-\frac {3 c \,d^{2} x}{b^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{2 a \,b^{4}}-\frac {\left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (-b x +a \right )}{2 a \,b^{4}}\)	\(123\)
risch	\(-\frac {d^{3} x^{2}}{2 b^{2}}-\frac {3 c \,d^{2} x}{b^{2}}-\frac {a^{2} \ln \left (b x +a \right ) d^{3}}{2 b^{4}}+\frac {3 a \ln \left (b x +a \right ) c \,d^{2}}{2 b^{3}}-\frac {3 \ln \left (b x +a \right ) c^{2} d}{2 b^{2}}+\frac {\ln \left (b x +a \right ) c^{3}}{2 a b}-\frac {a^{2} \ln \left (-b x +a \right ) d^{3}}{2 b^{4}}-\frac {3 a \ln \left (-b x +a \right ) c \,d^{2}}{2 b^{3}}-\frac {3 \ln \left (-b x +a \right ) c^{2} d}{2 b^{2}}-\frac {\ln \left (-b x +a \right ) c^{3}}{2 a b}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 122, normalized size = 1.61 \begin {gather*} -\frac {d^{3} x^{2} + 6 \, c d^{2} x}{2 \, b^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{2 \, a b^{4}} - \frac {{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (b x - a\right )}{2 \, a b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.73, size = 119, normalized size = 1.57 \begin {gather*} -\frac {a b^{2} d^{3} x^{2} + 6 \, a b^{2} c d^{2} x - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right ) + {\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (b x - a\right )}{2 \, a b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (66) = 132\).
time = 0.58, size = 163, normalized size = 2.14 \begin {gather*} - \frac {3 c d^{2} x}{b^{2}} - \frac {d^{3} x^{2}}{2 b^{2}} - \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} d^{3} + 3 a^{2} b^{2} c^{2} d - a \left (a d - b c\right )^{3}}{3 a^{2} b^{2} c d^{2} + b^{4} c^{3}} \right )}}{2 a b^{4}} - \frac {\left (a d + b c\right )^{3} \log {\left (x + \frac {a^{4} d^{3} + 3 a^{2} b^{2} c^{2} d - a \left (a d + b c\right )^{3}}{3 a^{2} b^{2} c d^{2} + b^{4} c^{3}} \right )}}{2 a b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*c*d**2*x/b**2 - d**3*x**2/(2*b**2) - (a*d - b*c)**3*log(x + (a**4*d**3 + 3*a**2*b**2*c**2*d - a*(a*d - b*c)
**3)/(3*a**2*b**2*c*d**2 + b**4*c**3))/(2*a*b**4) - (a*d + b*c)**3*log(x + (a**4*d**3 + 3*a**2*b**2*c**2*d - a
*(a*d + b*c)**3)/(3*a**2*b**2*c*d**2 + b**4*c**3))/(2*a*b**4)

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Giac [A]
time = 1.68, size = 130, normalized size = 1.71 \begin {gather*} -\frac {b^{2} d^{3} x^{2} + 6 \, b^{2} c d^{2} x}{2 \, b^{4}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{4}} - \frac {{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.18, size = 122, normalized size = 1.61 \begin {gather*} -\frac {d^3\,x^2}{2\,b^2}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a\,b^4}-\frac {\ln \left (a-b\,x\right )\,\left (a^3\,d^3+3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{2\,a\,b^4}-\frac {3\,c\,d^2\,x}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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