Optimal. Leaf size=423 \[ \frac {B (b c-a d)^5 g^3 i^2 x}{60 b^2 d^3}+\frac {B (b c-a d)^4 g^3 i^2 (c+d x)^2}{120 b d^4}-\frac {19 B (b c-a d)^3 g^3 i^2 (c+d x)^3}{180 d^4}+\frac {13 b B (b c-a d)^2 g^3 i^2 (c+d x)^4}{120 d^4}-\frac {b^2 B (b c-a d) g^3 i^2 (c+d x)^5}{30 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}-\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^4}-\frac {3 b^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log (c+d x)}{60 b^3 d^4} \]
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Rubi [A]
time = 0.30, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 45, 2382,
12, 1634} \begin {gather*} \frac {b^3 g^3 i^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4}-\frac {3 b^2 g^3 i^2 (c+d x)^5 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4}-\frac {g^3 i^2 (c+d x)^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^4}+\frac {3 b g^3 i^2 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4}+\frac {B g^3 i^2 (b c-a d)^6 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}+\frac {B g^3 i^2 (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}-\frac {b^2 B g^3 i^2 (c+d x)^5 (b c-a d)}{30 d^4}+\frac {B g^3 i^2 x (b c-a d)^5}{60 b^2 d^3}+\frac {B g^3 i^2 (c+d x)^2 (b c-a d)^4}{120 b d^4}-\frac {19 B g^3 i^2 (c+d x)^3 (b c-a d)^3}{180 d^4}+\frac {13 b B g^3 i^2 (c+d x)^4 (b c-a d)^2}{120 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 1634
Rule 2382
Rule 2562
Rubi steps
\begin {align*} \int (10 c+10 d x)^2 (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac {100 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {200 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {100 d^2 (a g+b g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}\right ) \, dx\\ &=\frac {\left (100 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2}+\frac {\left (100 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g^2}+\frac {(200 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g}\\ &=\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {\left (50 B d^2\right ) \int \frac {(b c-a d) g^6 (a+b x)^5}{c+d x} \, dx}{3 b^3 g^3}-\frac {(40 B d (b c-a d)) \int \frac {(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{b^3 g^2}-\frac {\left (25 B (b c-a d)^2\right ) \int \frac {(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{b^3 g}\\ &=\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {\left (50 B d^2 (b c-a d) g^3\right ) \int \frac {(a+b x)^5}{c+d x} \, dx}{3 b^3}-\frac {\left (40 B d (b c-a d)^2 g^3\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{b^3}-\frac {\left (25 B (b c-a d)^3 g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{b^3}\\ &=\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {\left (50 B d^2 (b c-a d) g^3\right ) \int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{3 b^3}-\frac {\left (40 B d (b c-a d)^2 g^3\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{b^3}-\frac {\left (25 B (b c-a d)^3 g^3\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{b^3}\\ &=-\frac {5 B (b c-a d)^5 g^3 x}{3 b^2 d^3}+\frac {5 B (b c-a d)^4 g^3 (a+b x)^2}{6 b^3 d^2}-\frac {5 B (b c-a d)^3 g^3 (a+b x)^3}{9 b^3 d}-\frac {35 B (b c-a d)^2 g^3 (a+b x)^4}{6 b^3}-\frac {10 B d (b c-a d) g^3 (a+b x)^5}{3 b^3}+\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}+\frac {5 B (b c-a d)^6 g^3 \log (c+d x)}{3 b^3 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 429, normalized size = 1.01 \begin {gather*} \frac {g^3 i^2 \left (90 d^4 (b c-a d)^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+144 d^5 (b c-a d) (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+60 d^6 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-15 B (b c-a d)^3 \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^2 \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )-B (b c-a d) \left (60 b d (b c-a d)^4 x+30 d^2 (-b c+a d)^3 (a+b x)^2+20 d^3 (b c-a d)^2 (a+b x)^3+15 d^4 (-b c+a d) (a+b x)^4+12 d^5 (a+b x)^5-60 (b c-a d)^5 \log (c+d x)\right )\right )}{360 b^3 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(14442\) vs.
\(2(401)=802\).
time = 0.76, size = 14443, normalized size = 34.14
method | result | size |
risch | \(\frac {17 i^{2} g^{3} d B \,a^{3} c \,x^{2}}{60}-\frac {i^{2} g^{3} b B \,a^{2} c^{2} x^{2}}{4}-\frac {i^{2} g^{3} b^{2} B a \,c^{3} x^{2}}{20 d}+i^{2} g^{3} A \,a^{3} c^{2} x -\frac {i^{2} g^{3} b B \,a^{2} c^{3} x}{4 d}+\frac {3 i^{2} g^{3} b^{2} d A a c \,x^{4}}{2}-\frac {i^{2} g^{3} b^{2} d B a c \,x^{4}}{20}+2 i^{2} g^{3} b d A \,a^{2} c \,x^{3}+i^{2} g^{3} b^{2} A a \,c^{2} x^{3}+\frac {7 i^{2} g^{3} b d B \,a^{2} c \,x^{3}}{60}-\frac {13 i^{2} g^{3} b^{2} B a \,c^{2} x^{3}}{60}+i^{2} g^{3} d A \,a^{3} c \,x^{2}+\frac {3 i^{2} g^{3} b A \,a^{2} c^{2} x^{2}}{2}-\frac {i^{2} g^{3} d B \ln \left (-b x -a \right ) a^{5} c}{10 b^{2}}+\frac {i^{2} g^{3} b^{2} B a \,c^{4} x}{10 d^{2}}+\frac {i^{2} g^{3} B \,a^{3} c^{2} x}{12}+\frac {i^{2} g^{3} d B \,a^{4} c x}{10 b}+\frac {i^{2} g^{3} b B \ln \left (d x +c \right ) a^{2} c^{4}}{4 d^{2}}-\frac {i^{2} g^{3} b^{2} B \ln \left (d x +c \right ) a \,c^{5}}{10 d^{3}}+\frac {i^{2} g^{3} B x \left (10 d^{2} b^{3} x^{5}+36 a \,b^{2} d^{2} x^{4}+24 b^{3} c d \,x^{4}+45 a^{2} b \,d^{2} x^{3}+90 a \,b^{2} c d \,x^{3}+15 b^{3} c^{2} x^{3}+20 a^{3} d^{2} x^{2}+120 a^{2} b c d \,x^{2}+60 a \,b^{2} c^{2} x^{2}+60 a^{3} c d x +90 a^{2} b \,c^{2} x +60 c^{2} a^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{60}+\frac {i^{2} g^{3} b^{2} d^{2} B a \,x^{5}}{30}-\frac {i^{2} g^{3} b^{3} d B c \,x^{5}}{30}+\frac {3 i^{2} g^{3} b \,d^{2} A \,a^{2} x^{4}}{4}+\frac {3 i^{2} g^{3} b^{2} d^{2} A a \,x^{5}}{5}+\frac {2 i^{2} g^{3} b^{3} d A c \,x^{5}}{5}+\frac {i^{2} g^{3} b^{3} d^{2} A \,x^{6}}{6}+\frac {i^{2} g^{3} b^{3} A \,c^{2} x^{4}}{4}+\frac {13 i^{2} g^{3} b \,d^{2} B \,a^{2} x^{4}}{120}-\frac {7 i^{2} g^{3} b^{3} B \,c^{2} x^{4}}{120}+\frac {i^{2} g^{3} d^{2} A \,a^{3} x^{3}}{3}+\frac {19 i^{2} g^{3} d^{2} B \,a^{3} x^{3}}{180}-\frac {i^{2} g^{3} b^{3} B \,c^{3} x^{3}}{180 d}+\frac {i^{2} g^{3} d^{2} B \,a^{4} x^{2}}{120 b}+\frac {i^{2} g^{3} b^{3} B \,c^{4} x^{2}}{120 d^{2}}-\frac {i^{2} g^{3} d^{2} B \,a^{5} x}{60 b^{2}}-\frac {i^{2} g^{3} b^{3} B \,c^{5} x}{60 d^{3}}+\frac {i^{2} g^{3} B \ln \left (-b x -a \right ) a^{4} c^{2}}{4 b}-\frac {i^{2} g^{3} B \ln \left (d x +c \right ) a^{3} c^{3}}{3 d}+\frac {i^{2} g^{3} d^{2} B \ln \left (-b x -a \right ) a^{6}}{60 b^{3}}+\frac {i^{2} g^{3} b^{3} B \ln \left (d x +c \right ) c^{6}}{60 d^{4}}\) | \(925\) |
derivativedivides | \(\text {Expression too large to display}\) | \(14443\) |
default | \(\text {Expression too large to display}\) | \(14443\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1747 vs.
\(2 (372) = 744\).
time = 0.33, size = 1747, normalized size = 4.13 \begin {gather*} -\frac {1}{6} \, A b^{3} d^{2} g^{3} x^{6} - \frac {2}{5} \, A b^{3} c d g^{3} x^{5} - \frac {3}{5} \, A a b^{2} d^{2} g^{3} x^{5} - \frac {1}{4} \, A b^{3} c^{2} g^{3} x^{4} - \frac {3}{2} \, A a b^{2} c d g^{3} x^{4} - \frac {3}{4} \, A a^{2} b d^{2} g^{3} x^{4} - A a b^{2} c^{2} g^{3} x^{3} - 2 \, A a^{2} b c d g^{3} x^{3} - \frac {1}{3} \, A a^{3} d^{2} g^{3} x^{3} - \frac {3}{2} \, A a^{2} b c^{2} g^{3} x^{2} - A a^{3} c d g^{3} x^{2} - {\left (x \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{3} c^{2} g^{3} - \frac {3}{2} \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{2} b c^{2} g^{3} - \frac {1}{2} \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b^{2} c^{2} g^{3} - \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{3} c^{2} g^{3} - {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{3} c d g^{3} - {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a^{2} b c d g^{3} - \frac {1}{4} \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B a b^{2} c d g^{3} - \frac {1}{30} \, {\left (12 \, x^{5} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{3} c d g^{3} - \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a^{3} d^{2} g^{3} - \frac {1}{8} \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B a^{2} b d^{2} g^{3} - \frac {1}{20} \, {\left (12 \, x^{5} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B a b^{2} d^{2} g^{3} - \frac {1}{360} \, {\left (60 \, x^{6} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {60 \, a^{6} \log \left (b x + a\right )}{b^{6}} + \frac {60 \, c^{6} \log \left (d x + c\right )}{d^{6}} - \frac {12 \, {\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{5} - 15 \, {\left (b^{5} c^{2} d^{3} - a^{2} b^{3} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} d^{2} - a^{3} b^{2} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{4} d - a^{4} b d^{5}\right )} x^{2} + 60 \, {\left (b^{5} c^{5} - a^{5} d^{5}\right )} x}{b^{5} d^{5}}\right )} B b^{3} d^{2} g^{3} - A a^{3} c^{2} g^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 689, normalized size = 1.63 \begin {gather*} -\frac {60 \, A b^{6} d^{6} g^{3} x^{6} + 12 \, {\left ({\left (12 \, A - B\right )} b^{6} c d^{5} + {\left (18 \, A + B\right )} a b^{5} d^{6}\right )} g^{3} x^{5} + 3 \, {\left ({\left (30 \, A - 7 \, B\right )} b^{6} c^{2} d^{4} + 6 \, {\left (30 \, A - B\right )} a b^{5} c d^{5} + {\left (90 \, A + 13 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{3} x^{4} - 2 \, {\left (B b^{6} c^{3} d^{3} - 3 \, {\left (60 \, A - 13 \, B\right )} a b^{5} c^{2} d^{4} - 3 \, {\left (120 \, A + 7 \, B\right )} a^{2} b^{4} c d^{5} - {\left (60 \, A + 19 \, B\right )} a^{3} b^{3} d^{6}\right )} g^{3} x^{3} + 3 \, {\left (B b^{6} c^{4} d^{2} - 6 \, B a b^{5} c^{3} d^{3} + 30 \, {\left (6 \, A - B\right )} a^{2} b^{4} c^{2} d^{4} + 2 \, {\left (60 \, A + 17 \, B\right )} a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{3} x^{2} - 6 \, {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 15 \, B a^{2} b^{4} c^{3} d^{3} - 5 \, {\left (12 \, A + B\right )} a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{3} x + 6 \, {\left (15 \, B a^{4} b^{2} c^{2} d^{4} - 6 \, B a^{5} b c d^{5} + B a^{6} d^{6}\right )} g^{3} \log \left (\frac {b x + a}{b}\right ) + 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2} - 20 \, B a^{3} b^{3} c^{3} d^{3}\right )} g^{3} \log \left (\frac {d x + c}{d}\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{3} x^{6} + 60 \, B a^{3} b^{3} c^{2} d^{4} g^{3} x + 12 \, {\left (2 \, B b^{6} c d^{5} + 3 \, B a b^{5} d^{6}\right )} g^{3} x^{5} + 15 \, {\left (B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + 3 \, B a^{2} b^{4} d^{6}\right )} g^{3} x^{4} + 20 \, {\left (3 \, B a b^{5} c^{2} d^{4} + 6 \, B a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{3} x^{3} + 30 \, {\left (3 \, B a^{2} b^{4} c^{2} d^{4} + 2 \, B a^{3} b^{3} c d^{5}\right )} g^{3} x^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{360 \, b^{3} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1727 vs.
\(2 (398) = 796\).
time = 8.94, size = 1727, normalized size = 4.08 \begin {gather*} \frac {A b^{3} d^{2} g^{3} i^{2} x^{6}}{6} + \frac {B a^{4} g^{3} i^{2} \left (a^{2} d^{2} - 6 a b c d + 15 b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{6} c d^{5} g^{3} i^{2} - 6 B a^{5} b c^{2} d^{4} g^{3} i^{2} + \frac {B a^{5} d^{4} g^{3} i^{2} \left (a^{2} d^{2} - 6 a b c d + 15 b^{2} c^{2}\right )}{b} + 35 B a^{4} b^{2} c^{3} d^{3} g^{3} i^{2} - B a^{4} c d^{3} g^{3} i^{2} \left (a^{2} d^{2} - 6 a b c d + 15 b^{2} c^{2}\right ) - 15 B a^{3} b^{3} c^{4} d^{2} g^{3} i^{2} + 6 B a^{2} b^{4} c^{5} d g^{3} i^{2} - B a b^{5} c^{6} g^{3} i^{2}}{B a^{6} d^{6} g^{3} i^{2} - 6 B a^{5} b c d^{5} g^{3} i^{2} + 15 B a^{4} b^{2} c^{2} d^{4} g^{3} i^{2} + 20 B a^{3} b^{3} c^{3} d^{3} g^{3} i^{2} - 15 B a^{2} b^{4} c^{4} d^{2} g^{3} i^{2} + 6 B a b^{5} c^{5} d g^{3} i^{2} - B b^{6} c^{6} g^{3} i^{2}} \right )}}{60 b^{3}} - \frac {B c^{3} g^{3} i^{2} \cdot \left (20 a^{3} d^{3} - 15 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right ) \log {\left (x + \frac {B a^{6} c d^{5} g^{3} i^{2} - 6 B a^{5} b c^{2} d^{4} g^{3} i^{2} + 35 B a^{4} b^{2} c^{3} d^{3} g^{3} i^{2} - 15 B a^{3} b^{3} c^{4} d^{2} g^{3} i^{2} + 6 B a^{2} b^{4} c^{5} d g^{3} i^{2} - B a b^{5} c^{6} g^{3} i^{2} - B a b^{2} c^{3} g^{3} i^{2} \cdot \left (20 a^{3} d^{3} - 15 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right ) + \frac {B b^{3} c^{4} g^{3} i^{2} \cdot \left (20 a^{3} d^{3} - 15 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right )}{d}}{B a^{6} d^{6} g^{3} i^{2} - 6 B a^{5} b c d^{5} g^{3} i^{2} + 15 B a^{4} b^{2} c^{2} d^{4} g^{3} i^{2} + 20 B a^{3} b^{3} c^{3} d^{3} g^{3} i^{2} - 15 B a^{2} b^{4} c^{4} d^{2} g^{3} i^{2} + 6 B a b^{5} c^{5} d g^{3} i^{2} - B b^{6} c^{6} g^{3} i^{2}} \right )}}{60 d^{4}} + x^{5} \cdot \left (\frac {3 A a b^{2} d^{2} g^{3} i^{2}}{5} + \frac {2 A b^{3} c d g^{3} i^{2}}{5} + \frac {B a b^{2} d^{2} g^{3} i^{2}}{30} - \frac {B b^{3} c d g^{3} i^{2}}{30}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b d^{2} g^{3} i^{2}}{4} + \frac {3 A a b^{2} c d g^{3} i^{2}}{2} + \frac {A b^{3} c^{2} g^{3} i^{2}}{4} + \frac {13 B a^{2} b d^{2} g^{3} i^{2}}{120} - \frac {B a b^{2} c d g^{3} i^{2}}{20} - \frac {7 B b^{3} c^{2} g^{3} i^{2}}{120}\right ) + x^{3} \left (\frac {A a^{3} d^{2} g^{3} i^{2}}{3} + 2 A a^{2} b c d g^{3} i^{2} + A a b^{2} c^{2} g^{3} i^{2} + \frac {19 B a^{3} d^{2} g^{3} i^{2}}{180} + \frac {7 B a^{2} b c d g^{3} i^{2}}{60} - \frac {13 B a b^{2} c^{2} g^{3} i^{2}}{60} - \frac {B b^{3} c^{3} g^{3} i^{2}}{180 d}\right ) + x^{2} \left (A a^{3} c d g^{3} i^{2} + \frac {3 A a^{2} b c^{2} g^{3} i^{2}}{2} + \frac {B a^{4} d^{2} g^{3} i^{2}}{120 b} + \frac {17 B a^{3} c d g^{3} i^{2}}{60} - \frac {B a^{2} b c^{2} g^{3} i^{2}}{4} - \frac {B a b^{2} c^{3} g^{3} i^{2}}{20 d} + \frac {B b^{3} c^{4} g^{3} i^{2}}{120 d^{2}}\right ) + x \left (A a^{3} c^{2} g^{3} i^{2} - \frac {B a^{5} d^{2} g^{3} i^{2}}{60 b^{2}} + \frac {B a^{4} c d g^{3} i^{2}}{10 b} + \frac {B a^{3} c^{2} g^{3} i^{2}}{12} - \frac {B a^{2} b c^{3} g^{3} i^{2}}{4 d} + \frac {B a b^{2} c^{4} g^{3} i^{2}}{10 d^{2}} - \frac {B b^{3} c^{5} g^{3} i^{2}}{60 d^{3}}\right ) + \left (B a^{3} c^{2} g^{3} i^{2} x + B a^{3} c d g^{3} i^{2} x^{2} + \frac {B a^{3} d^{2} g^{3} i^{2} x^{3}}{3} + \frac {3 B a^{2} b c^{2} g^{3} i^{2} x^{2}}{2} + 2 B a^{2} b c d g^{3} i^{2} x^{3} + \frac {3 B a^{2} b d^{2} g^{3} i^{2} x^{4}}{4} + B a b^{2} c^{2} g^{3} i^{2} x^{3} + \frac {3 B a b^{2} c d g^{3} i^{2} x^{4}}{2} + \frac {3 B a b^{2} d^{2} g^{3} i^{2} x^{5}}{5} + \frac {B b^{3} c^{2} g^{3} i^{2} x^{4}}{4} + \frac {2 B b^{3} c d g^{3} i^{2} x^{5}}{5} + \frac {B b^{3} d^{2} g^{3} i^{2} x^{6}}{6}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7651 vs.
\(2 (372) = 744\).
time = 4.06, size = 7651, normalized size = 18.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.89, size = 2473, normalized size = 5.85 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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