Optimal. Leaf size=275 \[ \frac {B (b c-a d) \log (f+g x) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 g (f+g x)^3}+\frac {b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {B (b c-a d) (-a d g-b c g+2 b d f)}{3 (f+g x) (b f-a g)^2 (d f-c g)^2}-\frac {B (b c-a d)}{6 (f+g x)^2 (b f-a g) (d f-c g)}-\frac {B d^3 \log (c+d x)}{3 g (d f-c g)^3} \]
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Rubi [A] time = 0.40, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2525, 12, 72} \[ \frac {B (b c-a d) \log (f+g x) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 g (f+g x)^3}+\frac {b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {B (b c-a d) (-a d g-b c g+2 b d f)}{3 (f+g x) (b f-a g)^2 (d f-c g)^2}-\frac {B (b c-a d)}{6 (f+g x)^2 (b f-a g) (d f-c g)}-\frac {B d^3 \log (c+d x)}{3 g (d f-c g)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 72
Rule 2525
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 g (f+g x)^3}+\frac {B \int \frac {b c-a d}{(a+b x) (c+d x) (f+g x)^3} \, dx}{3 g}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 g (f+g x)^3}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x) (f+g x)^3} \, dx}{3 g}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 g (f+g x)^3}+\frac {(B (b c-a d)) \int \left (\frac {b^4}{(b c-a d) (b f-a g)^3 (a+b x)}+\frac {d^4}{(b c-a d) (-d f+c g)^3 (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^3}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)^2}+\frac {g^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}\right ) \, dx}{3 g}\\ &=-\frac {B (b c-a d)}{6 (b f-a g) (d f-c g) (f+g x)^2}-\frac {B (b c-a d) (2 b d f-b c g-a d g)}{3 (b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 g (f+g x)^3}-\frac {B d^3 \log (c+d x)}{3 g (d f-c g)^3}+\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^3 (d f-c g)^3}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 260, normalized size = 0.95 \[ \frac {B (b c-a d) \left (\frac {g \log (f+g x) \left (a^2 d^2 g^2+a b d g (c g-3 d f)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{(b f-a g)^3 (d f-c g)^3}+\frac {b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}+\frac {d^3 \log (c+d x)}{(b c-a d) (c g-d f)^3}+\frac {g (a d g+b c g-2 b d f)}{(f+g x) (b f-a g)^2 (d f-c g)^2}-\frac {g}{2 (f+g x)^2 (b f-a g) (d f-c g)}\right )-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(f+g x)^3}}{3 g} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 18285, normalized size = 66.49 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 848, normalized size = 3.08 \[ \frac {1}{6} \, {\left (\frac {2 \, b^{3} \log \left (b x + a\right )}{b^{3} f^{3} g - 3 \, a b^{2} f^{2} g^{2} + 3 \, a^{2} b f g^{3} - a^{3} g^{4}} - \frac {2 \, d^{3} \log \left (d x + c\right )}{d^{3} f^{3} g - 3 \, c d^{2} f^{2} g^{2} + 3 \, c^{2} d f g^{3} - c^{3} g^{4}} + \frac {2 \, {\left (3 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f^{2} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} f g + {\left (b^{3} c^{3} - a^{3} d^{3}\right )} g^{2}\right )} \log \left (g x + f\right )}{b^{3} d^{3} f^{6} + a^{3} c^{3} g^{6} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f^{5} g + 3 \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{4} g^{2} - {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f^{3} g^{3} + 3 \, {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f^{2} g^{4} - 3 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} f g^{5}} - \frac {5 \, {\left (b^{2} c d - a b d^{2}\right )} f^{2} - 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f g + {\left (a b c^{2} - a^{2} c d\right )} g^{2} + 2 \, {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} f g - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} g^{2}\right )} x}{b^{2} d^{2} f^{6} + a^{2} c^{2} f^{2} g^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{5} g + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{4} g^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3} g^{3} + {\left (b^{2} d^{2} f^{4} g^{2} + a^{2} c^{2} g^{6} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{4} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} f^{5} g + a^{2} c^{2} f g^{5} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{4} g^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3} g^{3} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} g^{4}\right )} x} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g}\right )} B - \frac {A}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.67, size = 1154, normalized size = 4.20 \[ \frac {\ln \left (f+g\,x\right )\,\left (g\,\left (3\,B\,a^2\,b\,d^3\,f-3\,B\,b^3\,c^2\,d\,f\right )-g^2\,\left (B\,a^3\,d^3-B\,b^3\,c^3\right )-3\,B\,a\,b^2\,d^3\,f^2+3\,B\,b^3\,c\,d^2\,f^2\right )}{3\,a^3\,c^3\,g^6-9\,a^3\,c^2\,d\,f\,g^5+9\,a^3\,c\,d^2\,f^2\,g^4-3\,a^3\,d^3\,f^3\,g^3-9\,a^2\,b\,c^3\,f\,g^5+27\,a^2\,b\,c^2\,d\,f^2\,g^4-27\,a^2\,b\,c\,d^2\,f^3\,g^3+9\,a^2\,b\,d^3\,f^4\,g^2+9\,a\,b^2\,c^3\,f^2\,g^4-27\,a\,b^2\,c^2\,d\,f^3\,g^3+27\,a\,b^2\,c\,d^2\,f^4\,g^2-9\,a\,b^2\,d^3\,f^5\,g-3\,b^3\,c^3\,f^3\,g^3+9\,b^3\,c^2\,d\,f^4\,g^2-9\,b^3\,c\,d^2\,f^5\,g+3\,b^3\,d^3\,f^6}-\frac {\frac {2\,A\,a^2\,c^2\,g^4+2\,A\,b^2\,d^2\,f^4+2\,A\,a^2\,d^2\,f^2\,g^2+2\,A\,b^2\,c^2\,f^2\,g^2+3\,B\,a^2\,d^2\,f^2\,g^2-3\,B\,b^2\,c^2\,f^2\,g^2-4\,A\,a\,b\,c^2\,f\,g^3-4\,A\,a\,b\,d^2\,f^3\,g+B\,a\,b\,c^2\,f\,g^3-4\,A\,a^2\,c\,d\,f\,g^3-5\,B\,a\,b\,d^2\,f^3\,g-4\,A\,b^2\,c\,d\,f^3\,g-B\,a^2\,c\,d\,f\,g^3+5\,B\,b^2\,c\,d\,f^3\,g+8\,A\,a\,b\,c\,d\,f^2\,g^2}{2\,\left (a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4\right )}+\frac {x^2\,\left (B\,a^2\,d^2\,g^4-2\,B\,f\,a\,b\,d^2\,g^3-B\,b^2\,c^2\,g^4+2\,B\,f\,b^2\,c\,d\,g^3\right )}{a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4}+\frac {x\,\left (-B\,a^2\,c\,d\,g^4+5\,B\,a^2\,d^2\,f\,g^3+B\,a\,b\,c^2\,g^4-9\,B\,a\,b\,d^2\,f^2\,g^2-5\,B\,b^2\,c^2\,f\,g^3+9\,B\,b^2\,c\,d\,f^2\,g^2\right )}{2\,\left (a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4\right )}}{3\,f^3\,g+9\,f^2\,g^2\,x+9\,f\,g^3\,x^2+3\,g^4\,x^3}-\frac {B\,b^3\,\ln \left (a+b\,x\right )}{3\,a^3\,g^4-9\,a^2\,b\,f\,g^3+9\,a\,b^2\,f^2\,g^2-3\,b^3\,f^3\,g}+\frac {B\,d^3\,\ln \left (c+d\,x\right )}{3\,c^3\,g^4-9\,c^2\,d\,f\,g^3+9\,c\,d^2\,f^2\,g^2-3\,d^3\,f^3\,g}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{3\,g\,\left (f^3+3\,f^2\,g\,x+3\,f\,g^2\,x^2+g^3\,x^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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