Optimal. Leaf size=181 \[ \frac {B d i^3 (c+d x)^4}{16 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b B i^3 (c+d x)^5}{25 (b c-a d)^2 g^6 (a+b x)^5}+\frac {d i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^2 g^6 (a+b x)^5} \]
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Rubi [A]
time = 0.11, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 45, 2372,
12} \begin {gather*} -\frac {b i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^2}+\frac {d i^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g^6 (a+b x)^4 (b c-a d)^2}-\frac {b B i^3 (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^2}+\frac {B d i^3 (c+d x)^4}{16 g^6 (a+b x)^4 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2372
Rule 2562
Rubi steps
\begin {align*} \int \frac {(29 c+29 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx &=\int \left (\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^6}+\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^5}+\frac {73167 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^4}+\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^3}\right ) \, dx\\ &=\frac {\left (24389 d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^6}+\frac {\left (73167 d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^6}+\frac {\left (73167 d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^3 g^6}+\frac {\left (24389 (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^6} \, dx}{b^3 g^6}\\ &=-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac {\left (24389 B d^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^6}+\frac {\left (24389 B d^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^6}+\frac {\left (73167 B d (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^4 g^6}+\frac {\left (24389 B (b c-a d)^3\right ) \int \frac {b c-a d}{(a+b x)^6 (c+d x)} \, dx}{5 b^4 g^6}\\ &=-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac {\left (24389 B d^3 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^6}+\frac {\left (24389 B d^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^6}+\frac {\left (73167 B d (b c-a d)^3\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^4 g^6}+\frac {\left (24389 B (b c-a d)^4\right ) \int \frac {1}{(a+b x)^6 (c+d x)} \, dx}{5 b^4 g^6}\\ &=-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac {\left (24389 B d^3 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^4 g^6}+\frac {\left (24389 B d^2 (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^4 g^6}+\frac {\left (73167 B d (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b^4 g^6}+\frac {\left (24389 B (b c-a d)^4\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^6}-\frac {b d}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5}{(b c-a d)^6 (a+b x)}+\frac {d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^4 g^6}\\ &=-\frac {24389 B (b c-a d)^3}{25 b^4 g^6 (a+b x)^5}-\frac {268279 B d (b c-a d)^2}{80 b^4 g^6 (a+b x)^4}-\frac {73167 B d^2 (b c-a d)}{20 b^4 g^6 (a+b x)^3}-\frac {24389 B d^3}{40 b^4 g^6 (a+b x)^2}+\frac {24389 B d^4}{20 b^4 (b c-a d) g^6 (a+b x)}+\frac {24389 B d^5 \log (a+b x)}{20 b^4 (b c-a d)^2 g^6}-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}-\frac {24389 B d^5 \log (c+d x)}{20 b^4 (b c-a d)^2 g^6}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(181)=362\).
time = 0.39, size = 608, normalized size = 3.36 \begin {gather*} -\frac {i^3 \left (80 A b^5 c^5+16 b^5 B c^5-100 a A b^4 c^4 d-25 a b^4 B c^4 d+20 a^5 A d^5+9 a^5 B d^5+300 A b^5 c^4 d x+55 b^5 B c^4 d x-400 a A b^4 c^3 d^2 x-100 a b^4 B c^3 d^2 x+100 a^4 A b d^5 x+45 a^4 b B d^5 x+400 A b^5 c^3 d^2 x^2+60 b^5 B c^3 d^2 x^2-600 a A b^4 c^2 d^3 x^2-150 a b^4 B c^2 d^3 x^2+200 a^3 A b^2 d^5 x^2+90 a^3 b^2 B d^5 x^2+200 A b^5 c^2 d^3 x^3+10 b^5 B c^2 d^3 x^3-400 a A b^4 c d^4 x^3-100 a b^4 B c d^4 x^3+200 a^2 A b^3 d^5 x^3+90 a^2 b^3 B d^5 x^3-20 b^5 B c d^4 x^4+20 a b^4 B d^5 x^4-20 B d^5 (a+b x)^5 \log (a+b x)+20 B (b c-a d)^2 \left (a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+20 a^5 B d^5 \log (c+d x)+100 a^4 b B d^5 x \log (c+d x)+200 a^3 b^2 B d^5 x^2 \log (c+d x)+200 a^2 b^3 B d^5 x^3 \log (c+d x)+100 a b^4 B d^5 x^4 \log (c+d x)+20 b^5 B d^5 x^5 \log (c+d x)\right )}{400 b^4 (b c-a d)^2 g^6 (a+b x)^5} \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. \(356\) vs.
\(2(173)=346\).
time = 0.75, size = 357, normalized size = 1.97 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 4202 vs. \(2 (163) = 326\).
time = 0.63, size = 4202, normalized size = 23.22 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 620 vs. \(2 (163) = 326\).
time = 0.41, size = 620, normalized size = 3.43 \begin {gather*} -\frac {16 \, {\left (-5 i \, A - i \, B\right )} b^{5} c^{5} + 25 \, {\left (4 i \, A + i \, B\right )} a b^{4} c^{4} d - {\left (20 i \, A + 9 i \, B\right )} a^{5} d^{5} + 20 \, {\left (i \, B b^{5} c d^{4} - i \, B a b^{4} d^{5}\right )} x^{4} + 10 \, {\left ({\left (-20 i \, A - i \, B\right )} b^{5} c^{2} d^{3} + 10 \, {\left (4 i \, A + i \, B\right )} a b^{4} c d^{4} + {\left (-20 i \, A - 9 i \, B\right )} a^{2} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (2 \, {\left (-20 i \, A - 3 i \, B\right )} b^{5} c^{3} d^{2} + 15 \, {\left (4 i \, A + i \, B\right )} a b^{4} c^{2} d^{3} + {\left (-20 i \, A - 9 i \, B\right )} a^{3} b^{2} d^{5}\right )} x^{2} + 5 \, {\left ({\left (-60 i \, A - 11 i \, B\right )} b^{5} c^{4} d + 20 \, {\left (4 i \, A + i \, B\right )} a b^{4} c^{3} d^{2} + {\left (-20 i \, A - 9 i \, B\right )} a^{4} b d^{5}\right )} x + 20 \, {\left (i \, B b^{5} d^{5} x^{5} + 5 i \, B a b^{4} d^{5} x^{4} - 4 i \, B b^{5} c^{5} + 5 i \, B a b^{4} c^{4} d + 10 \, {\left (-i \, B b^{5} c^{2} d^{3} + 2 i \, B a b^{4} c d^{4}\right )} x^{3} + 10 \, {\left (-2 i \, B b^{5} c^{3} d^{2} + 3 i \, B a b^{4} c^{2} d^{3}\right )} x^{2} + 5 \, {\left (-3 i \, B b^{5} c^{4} d + 4 i \, B a b^{4} c^{3} d^{2}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{400 \, {\left ({\left (b^{11} c^{2} - 2 \, a b^{10} c d + a^{2} b^{9} d^{2}\right )} g^{6} x^{5} + 5 \, {\left (a b^{10} c^{2} - 2 \, a^{2} b^{9} c d + a^{3} b^{8} d^{2}\right )} g^{6} x^{4} + 10 \, {\left (a^{2} b^{9} c^{2} - 2 \, a^{3} b^{8} c d + a^{4} b^{7} d^{2}\right )} g^{6} x^{3} + 10 \, {\left (a^{3} b^{8} c^{2} - 2 \, a^{4} b^{7} c d + a^{5} b^{6} d^{2}\right )} g^{6} x^{2} + 5 \, {\left (a^{4} b^{7} c^{2} - 2 \, a^{5} b^{6} c d + a^{6} b^{5} d^{2}\right )} g^{6} x + {\left (a^{5} b^{6} c^{2} - 2 \, a^{6} b^{5} c d + a^{7} b^{4} d^{2}\right )} g^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.00, size = 238, normalized size = 1.31 \begin {gather*} -\frac {{\left (-80 i \, B b e^{6} \log \left (\frac {b x e + a e}{d x + c}\right ) + \frac {100 i \, {\left (b x e + a e\right )} B d e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - 80 i \, A b e^{6} - 16 i \, B b e^{6} + \frac {100 i \, {\left (b x e + a e\right )} A d e^{5}}{d x + c} + \frac {25 i \, {\left (b x e + a e\right )} B d e^{5}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{400 \, {\left (\frac {{\left (b x e + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b x e + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.31, size = 1053, normalized size = 5.82 \begin {gather*} -\frac {\frac {20\,A\,a^4\,d^4\,i^3-80\,A\,b^4\,c^4\,i^3+9\,B\,a^4\,d^4\,i^3-16\,B\,b^4\,c^4\,i^3+20\,A\,a^2\,b^2\,c^2\,d^2\,i^3+9\,B\,a^2\,b^2\,c^2\,d^2\,i^3+20\,A\,a\,b^3\,c^3\,d\,i^3+20\,A\,a^3\,b\,c\,d^3\,i^3+9\,B\,a\,b^3\,c^3\,d\,i^3+9\,B\,a^3\,b\,c\,d^3\,i^3}{20\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (20\,A\,a^2\,b^2\,d^4\,i^3+9\,B\,a^2\,b^2\,d^4\,i^3-40\,A\,b^4\,c^2\,d^2\,i^3-6\,B\,b^4\,c^2\,d^2\,i^3+20\,A\,a\,b^3\,c\,d^3\,i^3+9\,B\,a\,b^3\,c\,d^3\,i^3\right )}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (20\,A\,a^3\,b\,d^4\,i^3+9\,B\,a^3\,b\,d^4\,i^3-60\,A\,b^4\,c^3\,d\,i^3-11\,B\,b^4\,c^3\,d\,i^3+20\,A\,a\,b^3\,c^2\,d^2\,i^3+20\,A\,a^2\,b^2\,c\,d^3\,i^3+9\,B\,a\,b^3\,c^2\,d^2\,i^3+9\,B\,a^2\,b^2\,c\,d^3\,i^3\right )}{4\,\left (a\,d-b\,c\right )}+\frac {x^3\,\left (20\,A\,a\,b^3\,d^4\,i^3+9\,B\,a\,b^3\,d^4\,i^3-20\,A\,b^4\,c\,d^3\,i^3-B\,b^4\,c\,d^3\,i^3\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^4\,d^4\,i^3\,x^4}{a\,d-b\,c}}{20\,a^5\,b^4\,g^6+100\,a^4\,b^5\,g^6\,x+200\,a^3\,b^6\,g^6\,x^2+200\,a^2\,b^7\,g^6\,x^3+100\,a\,b^8\,g^6\,x^4+20\,b^9\,g^6\,x^5}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x^2\,\left (b\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,a\,d^3\,i^3}{20\,b^4\,g^6}+\frac {3\,B\,c\,d^2\,i^3}{10\,b^3\,g^6}\right )+\frac {3\,B\,a\,d^3\,i^3}{10\,b^3\,g^6}+\frac {3\,B\,c\,d^2\,i^3}{5\,b^2\,g^6}\right )+x\,\left (b\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,c^2\,d\,i^3}{20\,b^3\,g^6}\right )+a\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,a\,d^3\,i^3}{20\,b^4\,g^6}+\frac {3\,B\,c\,d^2\,i^3}{10\,b^3\,g^6}\right )+\frac {3\,B\,c^2\,d\,i^3}{5\,b^2\,g^6}\right )+a\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,c^2\,d\,i^3}{20\,b^3\,g^6}\right )+\frac {B\,c^3\,i^3}{5\,b^2\,g^6}+\frac {B\,d^3\,i^3\,x^3}{2\,b^2\,g^6}\right )}{5\,a^4\,x+\frac {a^5}{b}+b^4\,x^5+10\,a^3\,b\,x^2+5\,a\,b^3\,x^4+10\,a^2\,b^2\,x^3}-\frac {B\,d^5\,i^3\,\mathrm {atanh}\left (\frac {20\,b^6\,c^2\,g^6-20\,a^2\,b^4\,d^2\,g^6}{20\,b^4\,g^6\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{10\,b^4\,g^6\,{\left (a\,d-b\,c\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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