Optimal. Leaf size=269 \[ -\frac {b^2 i (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac {d^2 i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b d i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B i (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^3}-\frac {B d^2 i (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b B d i (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^3} \]
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Rubi [A] time = 0.39, antiderivative size = 257, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 44} \[ -\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^5 (a+b x)^3}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b^2 g^5 (a+b x)^4}-\frac {B d^3 i}{12 b^2 g^5 (a+b x) (b c-a d)^2}+\frac {B d^2 i}{24 b^2 g^5 (a+b x)^2 (b c-a d)}-\frac {B d^4 i \log (a+b x)}{12 b^2 g^5 (b c-a d)^3}+\frac {B d^4 i \log (c+d x)}{12 b^2 g^5 (b c-a d)^3}-\frac {B i (b c-a d)}{16 b^2 g^5 (a+b x)^4}-\frac {B d i}{36 b^2 g^5 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(9 c+9 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^5 (a+b x)^5}+\frac {9 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^5 (a+b x)^4}\right ) \, dx\\ &=\frac {(9 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^5}+\frac {(9 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b g^5}\\ &=-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {(3 B d) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac {(9 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^2 g^5}\\ &=-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {(3 B d (b c-a d)) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac {\left (9 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^2 g^5}\\ &=-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {(3 B d (b c-a d)) \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^2 g^5}+\frac {\left (9 B (b c-a d)^2\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^2 g^5}\\ &=-\frac {9 B (b c-a d)}{16 b^2 g^5 (a+b x)^4}-\frac {B d}{4 b^2 g^5 (a+b x)^3}+\frac {3 B d^2}{8 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {3 B d^3}{4 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {3 B d^4 \log (a+b x)}{4 b^2 (b c-a d)^3 g^5}-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {3 B d^4 \log (c+d x)}{4 b^2 (b c-a d)^3 g^5}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 210, normalized size = 0.78 \[ -\frac {i \left (\frac {36 A b c}{(a+b x)^4}+\frac {48 A d}{(a+b x)^3}-\frac {36 a A d}{(a+b x)^4}+\frac {12 B d^4 \log (a+b x)}{(b c-a d)^3}-\frac {12 B d^4 \log (c+d x)}{(b c-a d)^3}+\frac {12 B d^3}{(a+b x) (b c-a d)^2}-\frac {6 B d^2}{(a+b x)^2 (b c-a d)}+\frac {12 B (a d+3 b c+4 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4}+\frac {9 b B c}{(a+b x)^4}+\frac {4 B d}{(a+b x)^3}-\frac {9 a B d}{(a+b x)^4}\right )}{144 b^2 g^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 602, normalized size = 2.24 \[ -\frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i x^{2} + 4 \, {\left ({\left (12 \, A + B\right )} b^{4} c^{3} d - 6 \, {\left (6 \, A + B\right )} a b^{3} c^{2} d^{2} + 18 \, {\left (2 \, A + B\right )} a^{2} b^{2} c d^{3} - {\left (12 \, A + 13 \, B\right )} a^{3} b d^{4}\right )} i x + {\left (9 \, {\left (4 \, A + B\right )} b^{4} c^{4} - 32 \, {\left (3 \, A + B\right )} a b^{3} c^{3} d + 36 \, {\left (2 \, A + B\right )} a^{2} b^{2} c^{2} d^{2} - {\left (12 \, A + 13 \, B\right )} a^{4} d^{4}\right )} i + 12 \, {\left (B b^{4} d^{4} i x^{4} + 4 \, B a b^{3} d^{4} i x^{3} + 6 \, B a^{2} b^{2} d^{4} i x^{2} + 4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3}\right )} i x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.44, size = 391, normalized size = 1.45 \[ -\frac {{\left (36 \, B b^{2} i e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {96 \, {\left (b x e + a e\right )} B b d i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {72 \, {\left (b x e + a e\right )}^{2} B d^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + 36 \, A b^{2} i e^{5} + 9 \, B b^{2} i e^{5} - \frac {96 \, {\left (b x e + a e\right )} A b d i e^{4}}{d x + c} - \frac {32 \, {\left (b x e + a e\right )} B b d i e^{4}}{d x + c} + \frac {72 \, {\left (b x e + a e\right )}^{2} A d^{2} i e^{3}}{{\left (d x + c\right )}^{2}} + \frac {36 \, {\left (b x e + a e\right )}^{2} B d^{2} i e^{3}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{144 \, {\left (\frac {{\left (b x e + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x e + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x e + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1226, normalized size = 4.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.80, size = 1386, normalized size = 5.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.46, size = 590, normalized size = 2.19 \[ \frac {B\,d^4\,i\,\mathrm {atanh}\left (\frac {12\,a^3\,b^2\,d^3\,g^5-12\,a^2\,b^3\,c\,d^2\,g^5-12\,a\,b^4\,c^2\,d\,g^5+12\,b^5\,c^3\,g^5}{12\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{6\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{4\,b^2\,g^5}+\frac {B\,a\,d\,i}{12\,b^3\,g^5}+\frac {B\,d\,i\,x}{3\,b^2\,g^5}\right )}{4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3}-\frac {\frac {12\,A\,a^3\,d^3\,i+36\,A\,b^3\,c^3\,i+13\,B\,a^3\,d^3\,i+9\,B\,b^3\,c^3\,i-60\,A\,a\,b^2\,c^2\,d\,i+12\,A\,a^2\,b\,c\,d^2\,i-23\,B\,a\,b^2\,c^2\,d\,i+13\,B\,a^2\,b\,c\,d^2\,i}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i+13\,B\,a^2\,b\,d^3\,i+12\,A\,b^3\,c^2\,d\,i+B\,b^3\,c^2\,d\,i-24\,A\,a\,b^2\,c\,d^2\,i-5\,B\,a\,b^2\,c\,d^2\,i\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {d\,x^2\,\left (B\,b^3\,c\,d\,i-7\,B\,a\,b^2\,d^2\,i\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^3\,d^3\,i\,x^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{12\,a^4\,b^2\,g^5+48\,a^3\,b^3\,g^5\,x+72\,a^2\,b^4\,g^5\,x^2+48\,a\,b^5\,g^5\,x^3+12\,b^6\,g^5\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 18.73, size = 928, normalized size = 3.45 \[ - \frac {B d^{4} i \log {\left (x + \frac {- \frac {B a^{4} d^{8} i}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{7} i}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{6} i}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{5} i}{\left (a d - b c\right )^{3}} + B a d^{5} i - \frac {B b^{4} c^{4} d^{4} i}{\left (a d - b c\right )^{3}} + B b c d^{4} i}{2 B b d^{5} i} \right )}}{12 b^{2} g^{5} \left (a d - b c\right )^{3}} + \frac {B d^{4} i \log {\left (x + \frac {\frac {B a^{4} d^{8} i}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{7} i}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{6} i}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{5} i}{\left (a d - b c\right )^{3}} + B a d^{5} i + \frac {B b^{4} c^{4} d^{4} i}{\left (a d - b c\right )^{3}} + B b c d^{4} i}{2 B b d^{5} i} \right )}}{12 b^{2} g^{5} \left (a d - b c\right )^{3}} + \frac {\left (- B a d i - 3 B b c i - 4 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{12 a^{4} b^{2} g^{5} + 48 a^{3} b^{3} g^{5} x + 72 a^{2} b^{4} g^{5} x^{2} + 48 a b^{5} g^{5} x^{3} + 12 b^{6} g^{5} x^{4}} + \frac {- 12 A a^{3} d^{3} i - 12 A a^{2} b c d^{2} i + 60 A a b^{2} c^{2} d i - 36 A b^{3} c^{3} i - 13 B a^{3} d^{3} i - 13 B a^{2} b c d^{2} i + 23 B a b^{2} c^{2} d i - 9 B b^{3} c^{3} i - 12 B b^{3} d^{3} i x^{3} + x^{2} \left (- 42 B a b^{2} d^{3} i + 6 B b^{3} c d^{2} i\right ) + x \left (- 48 A a^{2} b d^{3} i + 96 A a b^{2} c d^{2} i - 48 A b^{3} c^{2} d i - 52 B a^{2} b d^{3} i + 20 B a b^{2} c d^{2} i - 4 B b^{3} c^{2} d i\right )}{144 a^{6} b^{2} d^{2} g^{5} - 288 a^{5} b^{3} c d g^{5} + 144 a^{4} b^{4} c^{2} g^{5} + x^{4} \left (144 a^{2} b^{6} d^{2} g^{5} - 288 a b^{7} c d g^{5} + 144 b^{8} c^{2} g^{5}\right ) + x^{3} \left (576 a^{3} b^{5} d^{2} g^{5} - 1152 a^{2} b^{6} c d g^{5} + 576 a b^{7} c^{2} g^{5}\right ) + x^{2} \left (864 a^{4} b^{4} d^{2} g^{5} - 1728 a^{3} b^{5} c d g^{5} + 864 a^{2} b^{6} c^{2} g^{5}\right ) + x \left (576 a^{5} b^{3} d^{2} g^{5} - 1152 a^{4} b^{4} c d g^{5} + 576 a^{3} b^{5} c^{2} g^{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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