3.16.30 \(\int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx\) [1530]

Optimal. Leaf size=103 \[ -\frac {d^2 \left (6 b^2 c^2+a^2 d^2\right ) x}{b^4}-\frac {2 c d^3 x^2}{b^2}-\frac {d^4 x^3}{3 b^2}-\frac {(b c+a d)^4 \log (a-b x)}{2 a b^5}+\frac {(b c-a d)^4 \log (a+b x)}{2 a b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 103, 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 {d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac {(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac {(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac {2 c d^3 x^2}{b^2}-\frac {d^4 x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4/((a - b*x)*(a + b*x)),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 86, normalized size = 0.83 \begin {gather*} \frac {-2 a b d^2 x \left (3 a^2 d^2+b^2 \left (18 c^2+6 c d x+d^2 x^2\right )\right )-3 (b c+a d)^4 \log (a-b x)+3 (b c-a d)^4 \log (a+b x)}{6 a b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4/((a - b*x)*(a + b*x)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 178, normalized size = 1.73

method	result	size
norman	\(-\frac {d^{4} x^{3}}{3 b^{2}}-\frac {2 c \,d^{3} x^{2}}{b^{2}}-\frac {d^{2} \left (a^{2} d^{2}+6 b^{2} c^{2}\right ) x}{b^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{2 b^{5} a}-\frac {\left (a^{4} d^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (-b x +a \right )}{2 a \,b^{5}}\)	\(177\)
default	\(-\frac {d^{2} \left (\frac {1}{3} d^{2} x^{3} b^{2}+2 b^{2} c d \,x^{2}+a^{2} d^{2} x +6 b^{2} c^{2} x \right )}{b^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{2 b^{5} a}+\frac {\left (-a^{4} d^{4}-4 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \ln \left (-b x +a \right )}{2 b^{5} a}\)	\(178\)
risch	\(-\frac {d^{4} x^{3}}{3 b^{2}}-\frac {2 c \,d^{3} x^{2}}{b^{2}}-\frac {d^{4} a^{2} x}{b^{4}}-\frac {6 d^{2} c^{2} x}{b^{2}}-\frac {a^{3} \ln \left (b x -a \right ) d^{4}}{2 b^{5}}-\frac {2 a^{2} \ln \left (b x -a \right ) c \,d^{3}}{b^{4}}-\frac {3 a \ln \left (b x -a \right ) c^{2} d^{2}}{b^{3}}-\frac {2 \ln \left (b x -a \right ) c^{3} d}{b^{2}}-\frac {\ln \left (b x -a \right ) c^{4}}{2 b a}+\frac {a^{3} \ln \left (-b x -a \right ) d^{4}}{2 b^{5}}-\frac {2 a^{2} \ln \left (-b x -a \right ) c \,d^{3}}{b^{4}}+\frac {3 a \ln \left (-b x -a \right ) c^{2} d^{2}}{b^{3}}-\frac {2 \ln \left (-b x -a \right ) c^{3} d}{b^{2}}+\frac {\ln \left (-b x -a \right ) c^{4}}{2 b a}\)	\(244\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4/(-b*x+a)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 179, normalized size = 1.74 \begin {gather*} -\frac {b^{2} d^{4} x^{3} + 6 \, b^{2} c d^{3} x^{2} + 3 \, {\left (6 \, b^{2} c^{2} d^{2} + a^{2} d^{4}\right )} x}{3 \, b^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right )}{2 \, a b^{5}} - \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{2 \, a b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4/(-b*x+a)/(b*x+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.50, size = 174, normalized size = 1.69 \begin {gather*} -\frac {2 \, a b^{3} d^{4} x^{3} + 12 \, a b^{3} c d^{3} x^{2} + 6 \, {\left (6 \, a b^{3} c^{2} d^{2} + a^{3} b d^{4}\right )} x - 3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{6 \, a b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4/(-b*x+a)/(b*x+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (92) = 184\).
time = 0.78, size = 214, normalized size = 2.08 \begin {gather*} - x \left (\frac {a^{2} d^{4}}{b^{4}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) - \frac {2 c d^{3} x^{2}}{b^{2}} - \frac {d^{4} x^{3}}{3 b^{2}} + \frac {\left (a d - b c\right )^{4} \log {\left (x + \frac {4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d + \frac {a \left (a d - b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} - \frac {\left (a d + b c\right )^{4} \log {\left (x + \frac {4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - \frac {a \left (a d + b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4/(-b*x+a)/(b*x+a),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.73, size = 183, normalized size = 1.78 \begin {gather*} \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{5}} - \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{5}} - \frac {b^{4} d^{4} x^{3} + 6 \, b^{4} c d^{3} x^{2} + 18 \, b^{4} c^{2} d^{2} x + 3 \, a^{2} b^{2} d^{4} x}{3 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4/(-b*x+a)/(b*x+a),x, algorithm="giac")

[Out]

1/2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(abs(b*x + a))/(a*b^5) - 1/2*(b
^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + a^4*d^4)*log(abs(b*x - a))/(a*b^5) - 1/3*(b^4*d^4
*x^3 + 6*b^4*c*d^3*x^2 + 18*b^4*c^2*d^2*x + 3*a^2*b^2*d^4*x)/b^6

________________________________________________________________________________________

Mupad [B]
time = 1.27, size = 176, normalized size = 1.71 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,b^5}-\frac {d^4\,x^3}{3\,b^2}-\frac {2\,c\,d^3\,x^2}{b^2}-x\,\left (\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-\frac {\ln \left (a-b\,x\right )\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^4/((a + b*x)*(a - b*x)),x)

[Out]

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

________________________________________________________________________________________