Optimal. Leaf size=206 \[ \frac {B}{16 b g^5 (a+b x)^4}-\frac {B d}{12 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4} \]
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Rubi [A]
time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2548, 21, 46}
\begin {gather*} -\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b g^5 (a+b x)^4}-\frac {B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}+\frac {B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}-\frac {B d^3}{4 b g^5 (a+b x) (b c-a d)^3}+\frac {B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}-\frac {B d}{12 b g^5 (a+b x)^3 (b c-a d)}+\frac {B}{16 b g^5 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 46
Rule 2548
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx &=-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {-b c+a d}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5}\\ &=-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \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 g^5}\\ &=\frac {B}{16 b g^5 (a+b x)^4}-\frac {B d}{12 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 166, normalized size = 0.81 \begin {gather*} \frac {\frac {B (-b c+a d) \left (-\frac {3 (b c-a d)^4}{(a+b x)^4}+\frac {4 d (b c-a d)^3}{(a+b x)^3}-\frac {6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac {12 d^3 (b c-a d)}{a+b x}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^5}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^4}}{4 b g^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. \(687\) vs.
\(2(195)=390\).
time = 0.53, size = 688, normalized size = 3.34 Too large to display
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. 649 vs.
\(2 (193) = 386\).
time = 0.33, size = 649, normalized size = 3.15 \begin {gather*} -\frac {1}{48} \, B {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} + \frac {12 \, \log \left (\frac {d x e}{b x + a} + \frac {c e}{b x + a}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {A}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 636 vs.
\(2 (193) = 386\).
time = 0.44, size = 636, normalized size = 3.09 \begin {gather*} -\frac {3 \, {\left (4 \, A - B\right )} b^{4} c^{4} - 16 \, {\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} - 48 \, {\left (A - B\right )} a^{3} b c d^{3} + {\left (12 \, A - 25 \, B\right )} a^{4} d^{4} + 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} 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 )} x^{2} + 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 12 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right )}{48 \, {\left ({\left (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}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \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. 944 vs.
\(2 (178) = 356\).
time = 3.05, size = 944, normalized size = 4.58 \begin {gather*} - \frac {B \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}}{4 a^{4} b g^{5} + 16 a^{3} b^{2} g^{5} x + 24 a^{2} b^{3} g^{5} x^{2} + 16 a b^{4} g^{5} x^{3} + 4 b^{5} g^{5} x^{4}} + \frac {B d^{4} \log {\left (x + \frac {- \frac {B a^{5} d^{9}}{\left (a d - b c\right )^{4}} + \frac {5 B a^{4} b c d^{8}}{\left (a d - b c\right )^{4}} - \frac {10 B a^{3} b^{2} c^{2} d^{7}}{\left (a d - b c\right )^{4}} + \frac {10 B a^{2} b^{3} c^{3} d^{6}}{\left (a d - b c\right )^{4}} - \frac {5 B a b^{4} c^{4} d^{5}}{\left (a d - b c\right )^{4}} + B a d^{5} + \frac {B b^{5} c^{5} d^{4}}{\left (a d - b c\right )^{4}} + B b c d^{4}}{2 B b d^{5}} \right )}}{4 b g^{5} \left (a d - b c\right )^{4}} - \frac {B d^{4} \log {\left (x + \frac {\frac {B a^{5} d^{9}}{\left (a d - b c\right )^{4}} - \frac {5 B a^{4} b c d^{8}}{\left (a d - b c\right )^{4}} + \frac {10 B a^{3} b^{2} c^{2} d^{7}}{\left (a d - b c\right )^{4}} - \frac {10 B a^{2} b^{3} c^{3} d^{6}}{\left (a d - b c\right )^{4}} + \frac {5 B a b^{4} c^{4} d^{5}}{\left (a d - b c\right )^{4}} + B a d^{5} - \frac {B b^{5} c^{5} d^{4}}{\left (a d - b c\right )^{4}} + B b c d^{4}}{2 B b d^{5}} \right )}}{4 b g^{5} \left (a d - b c\right )^{4}} + \frac {- 12 A a^{3} d^{3} + 36 A a^{2} b c d^{2} - 36 A a b^{2} c^{2} d + 12 A b^{3} c^{3} + 25 B a^{3} d^{3} - 23 B a^{2} b c d^{2} + 13 B a b^{2} c^{2} d - 3 B b^{3} c^{3} + 12 B b^{3} d^{3} x^{3} + x^{2} \cdot \left (42 B a b^{2} d^{3} - 6 B b^{3} c d^{2}\right ) + x \left (52 B a^{2} b d^{3} - 20 B a b^{2} c d^{2} + 4 B b^{3} c^{2} d\right )}{48 a^{7} b d^{3} g^{5} - 144 a^{6} b^{2} c d^{2} g^{5} + 144 a^{5} b^{3} c^{2} d g^{5} - 48 a^{4} b^{4} c^{3} g^{5} + x^{4} \cdot \left (48 a^{3} b^{5} d^{3} g^{5} - 144 a^{2} b^{6} c d^{2} g^{5} + 144 a b^{7} c^{2} d g^{5} - 48 b^{8} c^{3} g^{5}\right ) + x^{3} \cdot \left (192 a^{4} b^{4} d^{3} g^{5} - 576 a^{3} b^{5} c d^{2} g^{5} + 576 a^{2} b^{6} c^{2} d g^{5} - 192 a b^{7} c^{3} g^{5}\right ) + x^{2} \cdot \left (288 a^{5} b^{3} d^{3} g^{5} - 864 a^{4} b^{4} c d^{2} g^{5} + 864 a^{3} b^{5} c^{2} d g^{5} - 288 a^{2} b^{6} c^{3} g^{5}\right ) + x \left (192 a^{6} b^{2} d^{3} g^{5} - 576 a^{5} b^{3} c d^{2} g^{5} + 576 a^{4} b^{4} c^{2} d g^{5} - 192 a^{3} b^{5} c^{3} g^{5}\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. 511 vs.
\(2 (193) = 386\).
time = 3.68, size = 511, normalized size = 2.48 \begin {gather*} \frac {{\left (\frac {48 \, {\left (d x e + c e\right )} B d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {72 \, {\left (d x e + c e\right )}^{2} B b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {48 \, {\left (d x e + c e\right )}^{3} B b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} + \frac {48 \, {\left (d x e + c e\right )} A d^{3} e^{3}}{b x + a} - \frac {48 \, {\left (d x e + c e\right )} B d^{3} e^{3}}{b x + a} - \frac {72 \, {\left (d x e + c e\right )}^{2} A b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} + \frac {36 \, {\left (d x e + c e\right )}^{2} B b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} + \frac {48 \, {\left (d x e + c e\right )}^{3} A b^{2} d e}{{\left (b x + a\right )}^{3}} - \frac {16 \, {\left (d x e + c e\right )}^{3} B b^{2} d e}{{\left (b x + a\right )}^{3}} - \frac {12 \, {\left (d x e + c e\right )}^{4} B b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{4}} - \frac {12 \, {\left (d x e + c e\right )}^{4} A b^{3}}{{\left (b x + a\right )}^{4}} + \frac {3 \, {\left (d x e + c e\right )}^{4} B b^{3}}{{\left (b x + a\right )}^{4}}\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 )}}{48 \, {\left (b^{3} c^{3} g^{5} e^{3} - 3 \, a b^{2} c^{2} d g^{5} e^{3} + 3 \, a^{2} b c d^{2} g^{5} e^{3} - a^{3} d^{3} g^{5} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.62, size = 578, normalized size = 2.81 \begin {gather*} \frac {B\,d^4\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4\,g^5+8\,a^3\,b^2\,c\,d^3\,g^5-8\,a\,b^4\,c^3\,d\,g^5+4\,b^5\,c^4\,g^5}{4\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3-25\,B\,a^3\,d^3+3\,B\,b^3\,c^3+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2-13\,B\,a\,b^2\,c^2\,d+23\,B\,a^2\,b\,c\,d^2}{12\,\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 {d^2\,x^2\,\left (B\,b^3\,c-7\,B\,a\,b^2\,d\right )}{2\,\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 {d\,x\,\left (13\,B\,a^2\,b\,d^2-5\,B\,a\,b^2\,c\,d+B\,b^3\,c^2\right )}{3\,\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 {B\,b^3\,d^3\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b\,g^5+16\,a^3\,b^2\,g^5\,x+24\,a^2\,b^3\,g^5\,x^2+16\,a\,b^4\,g^5\,x^3+4\,b^5\,g^5\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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