Optimal. Leaf size=231 \[ \frac {a^7}{b^5 (a+b x) (b c-a d)^3}+\frac {a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac {c^5 \left (21 a^2 d^2-28 a b c d+10 b^2 c^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}+\frac {3 x \left (a^2 d^2+2 a b c d+2 b^2 c^2\right )}{b^4 d^5}-\frac {x^2 (2 a d+3 b c)}{2 b^3 d^4}+\frac {c^7}{2 d^6 (c+d x)^2 (b c-a d)^2}-\frac {c^6 (5 b c-7 a d)}{d^6 (c+d x) (b c-a d)^3}+\frac {x^3}{3 b^2 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 231, 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} \[ \frac {3 x \left (a^2 d^2+2 a b c d+2 b^2 c^2\right )}{b^4 d^5}-\frac {c^5 \left (21 a^2 d^2-28 a b c d+10 b^2 c^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}+\frac {a^7}{b^5 (a+b x) (b c-a d)^3}+\frac {a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac {x^2 (2 a d+3 b c)}{2 b^3 d^4}-\frac {c^6 (5 b c-7 a d)}{d^6 (c+d x) (b c-a d)^3}+\frac {c^7}{2 d^6 (c+d x)^2 (b c-a d)^2}+\frac {x^3}{3 b^2 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rubi steps
\begin {align*} \int \frac {x^7}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac {3 \left (2 b^2 c^2+2 a b c d+a^2 d^2\right )}{b^4 d^5}-\frac {(3 b c+2 a d) x}{b^3 d^4}+\frac {x^2}{b^2 d^3}-\frac {a^7}{b^4 (b c-a d)^3 (a+b x)^2}-\frac {a^6 (-7 b c+4 a d)}{b^4 (b c-a d)^4 (a+b x)}-\frac {c^7}{d^5 (-b c+a d)^2 (c+d x)^3}-\frac {c^6 (5 b c-7 a d)}{d^5 (-b c+a d)^3 (c+d x)^2}-\frac {c^5 \left (10 b^2 c^2-28 a b c d+21 a^2 d^2\right )}{d^5 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=\frac {3 \left (2 b^2 c^2+2 a b c d+a^2 d^2\right ) x}{b^4 d^5}-\frac {(3 b c+2 a d) x^2}{2 b^3 d^4}+\frac {x^3}{3 b^2 d^3}+\frac {a^7}{b^5 (b c-a d)^3 (a+b x)}+\frac {c^7}{2 d^6 (b c-a d)^2 (c+d x)^2}-\frac {c^6 (5 b c-7 a d)}{d^6 (b c-a d)^3 (c+d x)}+\frac {a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac {c^5 \left (10 b^2 c^2-28 a b c d+21 a^2 d^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.37, size = 230, normalized size = 1.00 \[ \frac {a^7}{b^5 (a+b x) (b c-a d)^3}+\frac {a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac {c^5 \left (21 a^2 d^2-28 a b c d+10 b^2 c^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}+\frac {3 x \left (a^2 d^2+2 a b c d+2 b^2 c^2\right )}{b^4 d^5}-\frac {x^2 (2 a d+3 b c)}{2 b^3 d^4}+\frac {c^7}{2 d^6 (c+d x)^2 (b c-a d)^2}+\frac {c^6 (5 b c-7 a d)}{d^6 (c+d x) (a d-b c)^3}+\frac {x^3}{3 b^2 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.26, size = 1201, normalized size = 5.20 \[ -\frac {27 \, a b^{7} c^{9} - 66 \, a^{2} b^{6} c^{8} d + 39 \, a^{3} b^{5} c^{7} d^{2} - 6 \, a^{7} b c^{3} d^{6} + 6 \, a^{8} c^{2} d^{7} - 2 \, {\left (b^{8} c^{4} d^{5} - 4 \, a b^{7} c^{3} d^{6} + 6 \, a^{2} b^{6} c^{2} d^{7} - 4 \, a^{3} b^{5} c d^{8} + a^{4} b^{4} d^{9}\right )} x^{6} + {\left (5 \, b^{8} c^{5} d^{4} - 16 \, a b^{7} c^{4} d^{5} + 14 \, a^{2} b^{6} c^{3} d^{6} + 4 \, a^{3} b^{5} c^{2} d^{7} - 11 \, a^{4} b^{4} c d^{8} + 4 \, a^{5} b^{3} d^{9}\right )} x^{5} - {\left (20 \, b^{8} c^{6} d^{3} - 61 \, a b^{7} c^{5} d^{4} + 56 \, a^{2} b^{6} c^{4} d^{5} - 14 \, a^{3} b^{5} c^{3} d^{6} + 16 \, a^{4} b^{4} c^{2} d^{7} - 29 \, a^{5} b^{3} c d^{8} + 12 \, a^{6} b^{2} d^{9}\right )} x^{4} - {\left (63 \, b^{8} c^{7} d^{2} - 166 \, a b^{7} c^{6} d^{3} + 94 \, a^{2} b^{6} c^{5} d^{4} + 42 \, a^{3} b^{5} c^{4} d^{5} + 7 \, a^{4} b^{4} c^{3} d^{6} - 46 \, a^{5} b^{3} c^{2} d^{7} - 12 \, a^{6} b^{2} c d^{8} + 18 \, a^{7} b d^{9}\right )} x^{3} - 3 \, {\left (2 \, b^{8} c^{8} d + 9 \, a b^{7} c^{7} d^{2} - 46 \, a^{2} b^{6} c^{6} d^{3} + 50 \, a^{3} b^{5} c^{5} d^{4} - 7 \, a^{5} b^{3} c^{3} d^{6} - 20 \, a^{6} b^{2} c^{2} d^{7} + 14 \, a^{7} b c d^{8} - 2 \, a^{8} d^{9}\right )} x^{2} + 3 \, {\left (9 \, b^{8} c^{9} - 24 \, a b^{7} c^{8} d + 25 \, a^{2} b^{6} c^{7} d^{2} - 16 \, a^{3} b^{5} c^{6} d^{3} + 12 \, a^{6} b^{2} c^{3} d^{6} - 10 \, a^{7} b c^{2} d^{7} + 4 \, a^{8} c d^{8}\right )} x - 6 \, {\left (7 \, a^{7} b c^{3} d^{6} - 4 \, a^{8} c^{2} d^{7} + {\left (7 \, a^{6} b^{2} c d^{8} - 4 \, a^{7} b d^{9}\right )} x^{3} + {\left (14 \, a^{6} b^{2} c^{2} d^{7} - a^{7} b c d^{8} - 4 \, a^{8} d^{9}\right )} x^{2} + {\left (7 \, a^{6} b^{2} c^{3} d^{6} + 10 \, a^{7} b c^{2} d^{7} - 8 \, a^{8} c d^{8}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (10 \, a b^{7} c^{9} - 28 \, a^{2} b^{6} c^{8} d + 21 \, a^{3} b^{5} c^{7} d^{2} + {\left (10 \, b^{8} c^{7} d^{2} - 28 \, a b^{7} c^{6} d^{3} + 21 \, a^{2} b^{6} c^{5} d^{4}\right )} x^{3} + {\left (20 \, b^{8} c^{8} d - 46 \, a b^{7} c^{7} d^{2} + 14 \, a^{2} b^{6} c^{6} d^{3} + 21 \, a^{3} b^{5} c^{5} d^{4}\right )} x^{2} + {\left (10 \, b^{8} c^{9} - 8 \, a b^{7} c^{8} d - 35 \, a^{2} b^{6} c^{7} d^{2} + 42 \, a^{3} b^{5} c^{6} d^{3}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (a b^{9} c^{6} d^{6} - 4 \, a^{2} b^{8} c^{5} d^{7} + 6 \, a^{3} b^{7} c^{4} d^{8} - 4 \, a^{4} b^{6} c^{3} d^{9} + a^{5} b^{5} c^{2} d^{10} + {\left (b^{10} c^{4} d^{8} - 4 \, a b^{9} c^{3} d^{9} + 6 \, a^{2} b^{8} c^{2} d^{10} - 4 \, a^{3} b^{7} c d^{11} + a^{4} b^{6} d^{12}\right )} x^{3} + {\left (2 \, b^{10} c^{5} d^{7} - 7 \, a b^{9} c^{4} d^{8} + 8 \, a^{2} b^{8} c^{3} d^{9} - 2 \, a^{3} b^{7} c^{2} d^{10} - 2 \, a^{4} b^{6} c d^{11} + a^{5} b^{5} d^{12}\right )} x^{2} + {\left (b^{10} c^{6} d^{6} - 2 \, a b^{9} c^{5} d^{7} - 2 \, a^{2} b^{8} c^{4} d^{8} + 8 \, a^{3} b^{7} c^{3} d^{9} - 7 \, a^{4} b^{6} c^{2} d^{10} + 2 \, a^{5} b^{5} c d^{11}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.14, size = 744, normalized size = 3.22 \[ \frac {a^{7} b^{6}}{{\left (b^{14} c^{3} - 3 \, a b^{13} c^{2} d + 3 \, a^{2} b^{12} c d^{2} - a^{3} b^{11} d^{3}\right )} {\left (b x + a\right )}} - \frac {{\left (10 \, b^{3} c^{7} - 28 \, a b^{2} c^{6} d + 21 \, a^{2} b c^{5} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{6} - 4 \, a b^{4} c^{3} d^{7} + 6 \, a^{2} b^{3} c^{2} d^{8} - 4 \, a^{3} b^{2} c d^{9} + a^{4} b d^{10}} + \frac {{\left (10 \, b^{3} c^{3} + 12 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + 4 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5} d^{6}} + \frac {{\left (2 \, b^{4} c^{4} d^{5} - 8 \, a b^{3} c^{3} d^{6} + 12 \, a^{2} b^{2} c^{2} d^{7} - 8 \, a^{3} b c d^{8} + 2 \, a^{4} d^{9} - \frac {5 \, b^{6} c^{5} d^{4} - 4 \, a b^{5} c^{4} d^{5} - 34 \, a^{2} b^{4} c^{3} d^{6} + 76 \, a^{3} b^{3} c^{2} d^{7} - 59 \, a^{4} b^{2} c d^{8} + 16 \, a^{5} b d^{9}}{{\left (b x + a\right )} b} + \frac {2 \, {\left (10 \, b^{8} c^{6} d^{3} - 18 \, a b^{7} c^{5} d^{4} + 3 \, a^{2} b^{6} c^{4} d^{5} - 32 \, a^{3} b^{5} c^{3} d^{6} + 108 \, a^{4} b^{4} c^{2} d^{7} - 102 \, a^{5} b^{3} c d^{8} + 31 \, a^{6} b^{2} d^{9}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {3 \, {\left (30 \, b^{10} c^{7} d^{2} - 84 \, a b^{9} c^{6} d^{3} + 63 \, a^{2} b^{8} c^{5} d^{4} + 35 \, a^{4} b^{6} c^{3} d^{6} - 126 \, a^{5} b^{5} c^{2} d^{7} + 105 \, a^{6} b^{4} c d^{8} - 28 \, a^{7} b^{3} d^{9}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {6 \, {\left (10 \, b^{12} c^{8} d - 38 \, a b^{11} c^{7} d^{2} + 49 \, a^{2} b^{10} c^{6} d^{3} - 21 \, a^{3} b^{9} c^{5} d^{4} - 21 \, a^{5} b^{7} c^{3} d^{6} + 42 \, a^{6} b^{6} c^{2} d^{7} - 27 \, a^{7} b^{5} c d^{8} + 6 \, a^{8} b^{4} d^{9}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )} {\left (b x + a\right )}^{3}}{6 \, {\left (b c - a d\right )}^{4} b^{5} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 304, normalized size = 1.32 \[ -\frac {4 a^{7} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{5}}+\frac {7 a^{6} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{4}}-\frac {21 a^{2} c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{4}}+\frac {28 a b \,c^{6} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{5}}-\frac {10 b^{2} c^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{6}}-\frac {a^{7}}{\left (a d -b c \right )^{3} \left (b x +a \right ) b^{5}}-\frac {7 a \,c^{6}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{5}}+\frac {5 b \,c^{7}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{6}}+\frac {c^{7}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{6}}+\frac {x^{3}}{3 b^{2} d^{3}}-\frac {a \,x^{2}}{b^{3} d^{3}}-\frac {3 c \,x^{2}}{2 b^{2} d^{4}}+\frac {3 a^{2} x}{b^{4} d^{3}}+\frac {6 a c x}{b^{3} d^{4}}+\frac {6 c^{2} x}{b^{2} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.27, size = 584, normalized size = 2.53 \[ \frac {{\left (7 \, a^{6} b c - 4 \, a^{7} d\right )} \log \left (b x + a\right )}{b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}} - \frac {{\left (10 \, b^{2} c^{7} - 28 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}} - \frac {9 \, a b^{6} c^{8} - 13 \, a^{2} b^{5} c^{7} d - 2 \, a^{7} c^{2} d^{6} + 2 \, {\left (5 \, b^{7} c^{7} d - 7 \, a b^{6} c^{6} d^{2} - a^{7} d^{8}\right )} x^{2} + {\left (9 \, b^{7} c^{8} - 3 \, a b^{6} c^{7} d - 14 \, a^{2} b^{5} c^{6} d^{2} - 4 \, a^{7} c d^{7}\right )} x}{2 \, {\left (a b^{8} c^{5} d^{6} - 3 \, a^{2} b^{7} c^{4} d^{7} + 3 \, a^{3} b^{6} c^{3} d^{8} - a^{4} b^{5} c^{2} d^{9} + {\left (b^{9} c^{3} d^{8} - 3 \, a b^{8} c^{2} d^{9} + 3 \, a^{2} b^{7} c d^{10} - a^{3} b^{6} d^{11}\right )} x^{3} + {\left (2 \, b^{9} c^{4} d^{7} - 5 \, a b^{8} c^{3} d^{8} + 3 \, a^{2} b^{7} c^{2} d^{9} + a^{3} b^{6} c d^{10} - a^{4} b^{5} d^{11}\right )} x^{2} + {\left (b^{9} c^{5} d^{6} - a b^{8} c^{4} d^{7} - 3 \, a^{2} b^{7} c^{3} d^{8} + 5 \, a^{3} b^{6} c^{2} d^{9} - 2 \, a^{4} b^{5} c d^{10}\right )} x\right )}} + \frac {2 \, b^{2} d^{2} x^{3} - 3 \, {\left (3 \, b^{2} c d + 2 \, a b d^{2}\right )} x^{2} + 18 \, {\left (2 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{4} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.62, size = 543, normalized size = 2.35 \[ \frac {x^3}{3\,b^2\,d^3}-\frac {\frac {2\,a^7\,c^2\,d^6+13\,a^2\,b^5\,c^7\,d-9\,a\,b^6\,c^8}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x^2\,\left (a^7\,d^7+7\,a\,b^6\,c^6\,d-5\,b^7\,c^7\right )}{b\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (4\,a^7\,c\,d^7+14\,a^2\,b^5\,c^6\,d^2+3\,a\,b^6\,c^7\,d-9\,b^7\,c^8\right )}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^2\,\left (2\,c\,b^5\,d^6+a\,b^4\,d^7\right )+x\,\left (b^5\,c^2\,d^5+2\,a\,b^4\,c\,d^6\right )+b^5\,d^7\,x^3+a\,b^4\,c^2\,d^5}-x\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{b^4\,d^6}-\frac {{\left (2\,a\,d+3\,b\,c\right )}^2}{b^4\,d^5}\right )-\ln \left (c+d\,x\right )\,\left (\frac {21\,c^5}{d^6\,{\left (a\,d-b\,c\right )}^2}+\frac {14\,b\,c^6}{d^6\,{\left (a\,d-b\,c\right )}^3}+\frac {3\,b^2\,c^7}{d^6\,{\left (a\,d-b\,c\right )}^4}\right )-\frac {\ln \left (a+b\,x\right )\,\left (4\,a^7\,d-7\,a^6\,b\,c\right )}{a^4\,b^5\,d^4-4\,a^3\,b^6\,c\,d^3+6\,a^2\,b^7\,c^2\,d^2-4\,a\,b^8\,c^3\,d+b^9\,c^4}-\frac {x^2\,\left (2\,a\,d+3\,b\,c\right )}{2\,b^3\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________