Optimal. Leaf size=169 \[ -\frac {a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac {a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac {c^4}{2 d^2 (c+d x)^2 (b c-a d)^3}-\frac {c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 169, 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^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac {a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}-\frac {c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4}+\frac {c^4}{2 d^2 (c+d x)^2 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx &=\int \left (\frac {a^4}{b (b c-a d)^3 (a+b x)^3}+\frac {a^3 (-4 b c+a d)}{b (b c-a d)^4 (a+b x)^2}+\frac {6 a^2 b c^2}{(b c-a d)^5 (a+b x)}+\frac {c^4}{d (-b c+a d)^3 (c+d x)^3}+\frac {c^3 (b c-4 a d)}{d (-b c+a d)^4 (c+d x)^2}+\frac {6 a^2 c^2 d}{(-b c+a d)^5 (c+d x)}\right ) \, dx\\ &=-\frac {a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {a^3 (4 b c-a d)}{b^2 (b c-a d)^4 (a+b x)}+\frac {c^4}{2 d^2 (b c-a d)^3 (c+d x)^2}-\frac {c^3 (b c-4 a d)}{d^2 (b c-a d)^4 (c+d x)}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 171, normalized size = 1.01 \begin {gather*} -\frac {a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac {4 a^3 b c-a^4 d}{b^2 (a+b x) (b c-a d)^4}-\frac {c^4}{2 d^2 (c+d x)^2 (a d-b c)^3}-\frac {c^3 (b c-4 a d)}{d^2 (c+d x) (a d-b c)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.50, size = 985, normalized size = 5.83 \begin {gather*} -\frac {a^{2} b^{4} c^{6} - 8 \, a^{3} b^{3} c^{5} d + 8 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4} + 2 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 4 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d - 13 \, a^{2} b^{4} c^{4} d^{2} + 13 \, a^{4} b^{2} c^{2} d^{4} + 4 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} + 7 \, a^{5} b c^{2} d^{4} - a^{6} c d^{5}\right )} x - 12 \, {\left (a^{2} b^{4} c^{2} d^{4} x^{4} + a^{4} b^{2} c^{4} d^{2} + 2 \, {\left (a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (a^{3} b^{3} c^{4} d^{2} + a^{4} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (a^{2} b^{4} c^{2} d^{4} x^{4} + a^{4} b^{2} c^{4} d^{2} + 2 \, {\left (a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (a^{3} b^{3} c^{4} d^{2} + a^{4} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{7} c^{7} d^{2} - 5 \, a^{3} b^{6} c^{6} d^{3} + 10 \, a^{4} b^{5} c^{5} d^{4} - 10 \, a^{5} b^{4} c^{4} d^{5} + 5 \, a^{6} b^{3} c^{3} d^{6} - a^{7} b^{2} c^{2} d^{7} + {\left (b^{9} c^{5} d^{4} - 5 \, a b^{8} c^{4} d^{5} + 10 \, a^{2} b^{7} c^{3} d^{6} - 10 \, a^{3} b^{6} c^{2} d^{7} + 5 \, a^{4} b^{5} c d^{8} - a^{5} b^{4} d^{9}\right )} x^{4} + 2 \, {\left (b^{9} c^{6} d^{3} - 4 \, a b^{8} c^{5} d^{4} + 5 \, a^{2} b^{7} c^{4} d^{5} - 5 \, a^{4} b^{5} c^{2} d^{7} + 4 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + {\left (b^{9} c^{7} d^{2} - a b^{8} c^{6} d^{3} - 9 \, a^{2} b^{7} c^{5} d^{4} + 25 \, a^{3} b^{6} c^{4} d^{5} - 25 \, a^{4} b^{5} c^{3} d^{6} + 9 \, a^{5} b^{4} c^{2} d^{7} + a^{6} b^{3} c d^{8} - a^{7} b^{2} d^{9}\right )} x^{2} + 2 \, {\left (a b^{8} c^{7} d^{2} - 4 \, a^{2} b^{7} c^{6} d^{3} + 5 \, a^{3} b^{6} c^{5} d^{4} - 5 \, a^{5} b^{4} c^{3} d^{6} + 4 \, a^{6} b^{3} c^{2} d^{7} - a^{7} b^{2} c d^{8}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.10, size = 494, normalized size = 2.92 \begin {gather*} \frac {6 \, a^{2} b c^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {6 \, a^{2} c^{2} d \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {2 \, b^{5} c^{4} d x^{3} - 8 \, a b^{4} c^{3} d^{2} x^{3} - 8 \, a^{3} b^{2} c d^{4} x^{3} + 2 \, a^{4} b d^{5} x^{3} + b^{5} c^{5} x^{2} - 3 \, a b^{4} c^{4} d x^{2} - 16 \, a^{2} b^{3} c^{3} d^{2} x^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 3 \, a^{4} b c d^{4} x^{2} + a^{5} d^{5} x^{2} + 2 \, a b^{4} c^{5} x - 12 \, a^{2} b^{3} c^{4} d x - 16 \, a^{3} b^{2} c^{3} d^{2} x - 12 \, a^{4} b c^{2} d^{3} x + 2 \, a^{5} c d^{4} x + a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}}{2 \, {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 204, normalized size = 1.21 \begin {gather*} -\frac {6 a^{2} c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {6 a^{2} c^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {a^{4} d}{\left (a d -b c \right )^{4} \left (b x +a \right ) b^{2}}+\frac {4 a^{3} c}{\left (a d -b c \right )^{4} \left (b x +a \right ) b}+\frac {4 a \,c^{3}}{\left (a d -b c \right )^{4} \left (d x +c \right ) d}-\frac {b \,c^{4}}{\left (a d -b c \right )^{4} \left (d x +c \right ) d^{2}}+\frac {a^{4}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2} b^{2}}-\frac {c^{4}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.12, size = 740, normalized size = 4.38 \begin {gather*} \frac {6 \, a^{2} c^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {6 \, a^{2} c^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3} + 2 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x^{3} + {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 16 \, a^{2} b^{3} c^{3} d^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{2} + 2 \, {\left (a b^{4} c^{5} - 6 \, a^{2} b^{3} c^{4} d - 8 \, a^{3} b^{2} c^{3} d^{2} - 6 \, a^{4} b c^{2} d^{3} + a^{5} c d^{4}\right )} x}{2 \, {\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6} + {\left (b^{8} c^{4} d^{4} - 4 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} - 4 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 2 \, {\left (b^{8} c^{5} d^{3} - 3 \, a b^{7} c^{4} d^{4} + 2 \, a^{2} b^{6} c^{3} d^{5} + 2 \, a^{3} b^{5} c^{2} d^{6} - 3 \, a^{4} b^{4} c d^{7} + a^{5} b^{3} d^{8}\right )} x^{3} + {\left (b^{8} c^{6} d^{2} - 9 \, a^{2} b^{6} c^{4} d^{4} + 16 \, a^{3} b^{5} c^{3} d^{5} - 9 \, a^{4} b^{4} c^{2} d^{6} + a^{6} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (a b^{7} c^{6} d^{2} - 3 \, a^{2} b^{6} c^{5} d^{3} + 2 \, a^{3} b^{5} c^{4} d^{4} + 2 \, a^{4} b^{4} c^{3} d^{5} - 3 \, a^{5} b^{3} c^{2} d^{6} + a^{6} b^{2} c d^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.72, size = 678, normalized size = 4.01 \begin {gather*} \frac {\frac {x^2\,\left (-a^5\,d^5+3\,a^4\,b\,c\,d^4+16\,a^3\,b^2\,c^2\,d^3+16\,a^2\,b^3\,c^3\,d^2+3\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,b^2\,d^2\,\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 )}-\frac {x^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{b\,d\,\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 )}-\frac {a^2\,c^2\,\left (a^3\,d^3-7\,a^2\,b\,c\,d^2-7\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{2\,b^2\,d^2\,\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 )}+\frac {a\,c\,x\,\left (-a^4\,d^4+6\,a^3\,b\,c\,d^3+8\,a^2\,b^2\,c^2\,d^2+6\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{b^2\,d^2\,\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 )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {12\,a^2\,c^2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\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 )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 3.23, size = 1046, normalized size = 6.19 \begin {gather*} \frac {6 a^{2} c^{2} \log {\left (x + \frac {- \frac {6 a^{8} c^{2} d^{6}}{\left (a d - b c\right )^{5}} + \frac {36 a^{7} b c^{3} d^{5}}{\left (a d - b c\right )^{5}} - \frac {90 a^{6} b^{2} c^{4} d^{4}}{\left (a d - b c\right )^{5}} + \frac {120 a^{5} b^{3} c^{5} d^{3}}{\left (a d - b c\right )^{5}} - \frac {90 a^{4} b^{4} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + \frac {36 a^{3} b^{5} c^{7} d}{\left (a d - b c\right )^{5}} + 6 a^{3} c^{2} d - \frac {6 a^{2} b^{6} c^{8}}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{3}}{12 a^{2} b c^{2} d} \right )}}{\left (a d - b c\right )^{5}} - \frac {6 a^{2} c^{2} \log {\left (x + \frac {\frac {6 a^{8} c^{2} d^{6}}{\left (a d - b c\right )^{5}} - \frac {36 a^{7} b c^{3} d^{5}}{\left (a d - b c\right )^{5}} + \frac {90 a^{6} b^{2} c^{4} d^{4}}{\left (a d - b c\right )^{5}} - \frac {120 a^{5} b^{3} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + \frac {90 a^{4} b^{4} c^{6} d^{2}}{\left (a d - b c\right )^{5}} - \frac {36 a^{3} b^{5} c^{7} d}{\left (a d - b c\right )^{5}} + 6 a^{3} c^{2} d + \frac {6 a^{2} b^{6} c^{8}}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{3}}{12 a^{2} b c^{2} d} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{5} c^{2} d^{3} + 7 a^{4} b c^{3} d^{2} + 7 a^{3} b^{2} c^{4} d - a^{2} b^{3} c^{5} + x^{3} \left (- 2 a^{4} b d^{5} + 8 a^{3} b^{2} c d^{4} + 8 a b^{4} c^{3} d^{2} - 2 b^{5} c^{4} d\right ) + x^{2} \left (- a^{5} d^{5} + 3 a^{4} b c d^{4} + 16 a^{3} b^{2} c^{2} d^{3} + 16 a^{2} b^{3} c^{3} d^{2} + 3 a b^{4} c^{4} d - b^{5} c^{5}\right ) + x \left (- 2 a^{5} c d^{4} + 12 a^{4} b c^{2} d^{3} + 16 a^{3} b^{2} c^{3} d^{2} + 12 a^{2} b^{3} c^{4} d - 2 a b^{4} c^{5}\right )}{2 a^{6} b^{2} c^{2} d^{6} - 8 a^{5} b^{3} c^{3} d^{5} + 12 a^{4} b^{4} c^{4} d^{4} - 8 a^{3} b^{5} c^{5} d^{3} + 2 a^{2} b^{6} c^{6} d^{2} + x^{4} \left (2 a^{4} b^{4} d^{8} - 8 a^{3} b^{5} c d^{7} + 12 a^{2} b^{6} c^{2} d^{6} - 8 a b^{7} c^{3} d^{5} + 2 b^{8} c^{4} d^{4}\right ) + x^{3} \left (4 a^{5} b^{3} d^{8} - 12 a^{4} b^{4} c d^{7} + 8 a^{3} b^{5} c^{2} d^{6} + 8 a^{2} b^{6} c^{3} d^{5} - 12 a b^{7} c^{4} d^{4} + 4 b^{8} c^{5} d^{3}\right ) + x^{2} \left (2 a^{6} b^{2} d^{8} - 18 a^{4} b^{4} c^{2} d^{6} + 32 a^{3} b^{5} c^{3} d^{5} - 18 a^{2} b^{6} c^{4} d^{4} + 2 b^{8} c^{6} d^{2}\right ) + x \left (4 a^{6} b^{2} c d^{7} - 12 a^{5} b^{3} c^{2} d^{6} + 8 a^{4} b^{4} c^{3} d^{5} + 8 a^{3} b^{5} c^{4} d^{4} - 12 a^{2} b^{6} c^{5} d^{3} + 4 a b^{7} c^{6} d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________