Optimal. Leaf size=89 \[ -\frac {i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {B i^2 (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)} \]
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Rubi [B] time = 0.49, antiderivative size = 287, normalized size of antiderivative = 3.22, number of steps used = 14, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2528, 2525, 12, 44} \[ -\frac {d^2 i^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^4 (a+b x)}-\frac {d i^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^4 (a+b x)^2}-\frac {i^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^3 g^4 (a+b x)^3}-\frac {B d^3 i^2 \log (a+b x)}{3 b^3 g^4 (b c-a d)}+\frac {B d^3 i^2 \log (c+d x)}{3 b^3 g^4 (b c-a d)}-\frac {B d i^2 (b c-a d)}{3 b^3 g^4 (a+b x)^2}-\frac {B i^2 (b c-a d)^2}{9 b^3 g^4 (a+b x)^3}-\frac {B d^2 i^2}{3 b^3 g^4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(17 c+17 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^4}+\frac {578 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^3}+\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac {\left (289 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^4}+\frac {(578 d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^4}+\frac {\left (289 (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^4}\\ &=-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (289 B d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {(289 B d (b c-a d)) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (289 B (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (289 B d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (289 B d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (289 B (b c-a d)^3\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (289 B d^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac {\left (289 B d (b c-a d)^2\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}{b^3 g^4}+\frac {\left (289 B (b c-a d)^3\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}{3 b^3 g^4}\\ &=-\frac {289 B (b c-a d)^2}{9 b^3 g^4 (a+b x)^3}-\frac {289 B d (b c-a d)}{3 b^3 g^4 (a+b x)^2}-\frac {289 B d^2}{3 b^3 g^4 (a+b x)}-\frac {289 B d^3 \log (a+b x)}{3 b^3 (b c-a d) g^4}-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {289 B d^3 \log (c+d x)}{3 b^3 (b c-a d) g^4}\\ \end {align*}
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Mathematica [B] time = 0.30, size = 315, normalized size = 3.54 \[ -\frac {i^2 \left (-3 a^3 A d^3-3 a^3 B d^3 \log (c+d x)-a^3 B d^3-9 a^2 A b d^3 x+3 B (b c-a d) \left (a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-9 a^2 b B d^3 x \log (c+d x)-3 a^2 b B d^3 x-9 a A b^2 d^3 x^2-9 a b^2 B d^3 x^2 \log (c+d x)-3 a b^2 B d^3 x^2+3 B d^3 (a+b x)^3 \log (a+b x)+3 A b^3 c^3+9 A b^3 c^2 d x+9 A b^3 c d^2 x^2+b^3 B c^3+3 b^3 B c^2 d x-3 b^3 B d^3 x^3 \log (c+d x)+3 b^3 B c d^2 x^2\right )}{9 b^3 g^4 (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 271, normalized size = 3.04 \[ -\frac {3 \, {\left ({\left (3 \, A + B\right )} b^{3} c d^{2} - {\left (3 \, A + B\right )} a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left ({\left (3 \, A + B\right )} b^{3} c^{2} d - {\left (3 \, A + B\right )} a^{2} b d^{3}\right )} i^{2} x + {\left ({\left (3 \, A + B\right )} b^{3} c^{3} - {\left (3 \, A + B\right )} a^{3} d^{3}\right )} i^{2} + 3 \, {\left (B b^{3} d^{3} i^{2} x^{3} + 3 \, B b^{3} c d^{2} i^{2} x^{2} + 3 \, B b^{3} c^{2} d i^{2} x + B b^{3} c^{3} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{9 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.30, size = 114, normalized size = 1.28 \[ \frac {{\left (3 \, B e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 3 \, A e^{4} + B e^{4}\right )} {\left (d x + c\right )}^{3} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{9 \, {\left (b x e + a e\right )}^{3} g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 406, normalized size = 4.56 \[ \frac {B a d \,e^{3} i^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {B b c \,e^{3} i^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {A a d \,e^{3} i^{2}}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {A b c \,e^{3} i^{2}}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {B a d \,e^{3} i^{2}}{9 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {B b c \,e^{3} i^{2}}{9 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.63, size = 1515, normalized size = 17.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.19, size = 423, normalized size = 4.75 \[ -\frac {x^2\,\left (3\,A\,b^2\,d^2\,i^2+B\,b^2\,d^2\,i^2\right )+x\,\left (3\,A\,a\,b\,d^2\,i^2+B\,a\,b\,d^2\,i^2+3\,A\,b^2\,c\,d\,i^2+B\,b^2\,c\,d\,i^2\right )+A\,a^2\,d^2\,i^2+A\,b^2\,c^2\,i^2+\frac {B\,a^2\,d^2\,i^2}{3}+\frac {B\,b^2\,c^2\,i^2}{3}+A\,a\,b\,c\,d\,i^2+\frac {B\,a\,b\,c\,d\,i^2}{3}}{3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^4\,g^4}+\frac {B\,c\,d\,i^2}{3\,b^3\,g^4}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^4\,g^4}+\frac {B\,c\,d\,i^2}{3\,b^3\,g^4}\right )+\frac {2\,B\,a\,d^2\,i^2}{3\,b^3\,g^4}+\frac {2\,B\,c\,d\,i^2}{3\,b^2\,g^4}\right )+\frac {B\,c^2\,i^2}{3\,b^2\,g^4}+\frac {B\,d^2\,i^2\,x^2}{b^2\,g^4}\right )}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B\,d^3\,i^2\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 26.25, size = 614, normalized size = 6.90 \[ - \frac {B d^{3} i^{2} \log {\left (x + \frac {- \frac {B a^{2} d^{5} i^{2}}{a d - b c} + \frac {2 B a b c d^{4} i^{2}}{a d - b c} + B a d^{4} i^{2} - \frac {B b^{2} c^{2} d^{3} i^{2}}{a d - b c} + B b c d^{3} i^{2}}{2 B b d^{4} i^{2}} \right )}}{3 b^{3} g^{4} \left (a d - b c\right )} + \frac {B d^{3} i^{2} \log {\left (x + \frac {\frac {B a^{2} d^{5} i^{2}}{a d - b c} - \frac {2 B a b c d^{4} i^{2}}{a d - b c} + B a d^{4} i^{2} + \frac {B b^{2} c^{2} d^{3} i^{2}}{a d - b c} + B b c d^{3} i^{2}}{2 B b d^{4} i^{2}} \right )}}{3 b^{3} g^{4} \left (a d - b c\right )} + \frac {- 3 A a^{2} d^{2} i^{2} - 3 A a b c d i^{2} - 3 A b^{2} c^{2} i^{2} - B a^{2} d^{2} i^{2} - B a b c d i^{2} - B b^{2} c^{2} i^{2} + x^{2} \left (- 9 A b^{2} d^{2} i^{2} - 3 B b^{2} d^{2} i^{2}\right ) + x \left (- 9 A a b d^{2} i^{2} - 9 A b^{2} c d i^{2} - 3 B a b d^{2} i^{2} - 3 B b^{2} c d i^{2}\right )}{9 a^{3} b^{3} g^{4} + 27 a^{2} b^{4} g^{4} x + 27 a b^{5} g^{4} x^{2} + 9 b^{6} g^{4} x^{3}} + \frac {\left (- B a^{2} d^{2} i^{2} - B a b c d i^{2} - 3 B a b d^{2} i^{2} x - B b^{2} c^{2} i^{2} - 3 B b^{2} c d i^{2} x - 3 B b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{3 a^{3} b^{3} g^{4} + 9 a^{2} b^{4} g^{4} x + 9 a b^{5} g^{4} x^{2} + 3 b^{6} g^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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