Optimal. Leaf size=173 \[ -\frac {b i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)^2}+\frac {d i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 (a+b x)^2 (b c-a d)^2}-\frac {b B i (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)^2}+\frac {B d i (c+d x)^2}{4 g^4 (a+b x)^2 (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 225, normalized size of antiderivative = 1.30, 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} \[ -\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (a+b x)^3}+\frac {B d^2 i}{6 b^2 g^4 (a+b x) (b c-a d)}+\frac {B d^3 i \log (a+b x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B d^3 i \log (c+d x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B i (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac {B d i}{12 b^2 g^4 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(8 c+8 d x) \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 {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^4 (a+b x)^4}+\frac {8 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^4 (a+b x)^3}\right ) \, dx\\ &=\frac {(8 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^4}+\frac {(8 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^4}\\ &=-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac {(4 B d) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {(8 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac {(4 B d (b c-a d)) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {\left (8 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac {(4 B d (b c-a d)) \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^2 g^4}+\frac {\left (8 B (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}{3 b^2 g^4}\\ &=-\frac {8 B (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac {2 B d}{3 b^2 g^4 (a+b x)^2}+\frac {4 B d^2}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {4 B d^3 \log (a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}-\frac {4 B d^3 \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.39, size = 187, normalized size = 1.08 \[ -\frac {i \left (\frac {12 A b c}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}-\frac {12 a A d}{(a+b x)^3}-\frac {6 B d^3 \log (a+b x)}{(b c-a d)^2}+\frac {6 B d^3 \log (c+d x)}{(b c-a d)^2}-\frac {6 B d^2}{(a+b x) (b c-a d)}+\frac {6 B (a d+2 b c+3 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {4 b B c}{(a+b x)^3}+\frac {3 B d}{(a+b x)^2}-\frac {4 a B d}{(a+b x)^3}\right )}{36 b^2 g^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.08, size = 363, normalized size = 2.10 \[ \frac {6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (6 \, A + B\right )} b^{3} c^{2} d - 6 \, {\left (2 \, A + B\right )} a b^{2} c d^{2} + {\left (6 \, A + 5 \, B\right )} a^{2} b d^{3}\right )} i x - {\left (4 \, {\left (3 \, A + B\right )} b^{3} c^{3} - 9 \, {\left (2 \, A + B\right )} a b^{2} c^{2} d + {\left (6 \, A + 5 \, B\right )} a^{3} d^{3}\right )} i + 6 \, {\left (B b^{3} d^{3} i x^{3} + 3 \, B a b^{2} d^{3} i x^{2} - 3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i x - {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.46, size = 244, normalized size = 1.41 \[ -\frac {{\left (12 \, B b i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {18 \, {\left (b x e + a e\right )} B d i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + 12 \, A b i e^{4} + 4 \, B b i e^{4} - \frac {18 \, {\left (b x e + a e\right )} A d i e^{3}}{d x + c} - \frac {9 \, {\left (b x e + a e\right )} B d i e^{3}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{36 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x e + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 804, normalized size = 4.65 \[ -\frac {B a b d \,e^{3} i \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 )^{3} \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^{2} c \,e^{3} i \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 )^{3} \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 b d \,e^{3} i}{3 \left (a d -b c \right )^{3} \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^{2} c \,e^{3} i}{3 \left (a d -b c \right )^{3} \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 b d \,e^{3} i}{9 \left (a d -b c \right )^{3} \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^{2} e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}+\frac {B \,b^{2} c \,e^{3} i}{9 \left (a d -b c \right )^{3} \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 d \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}+\frac {A a \,d^{2} e^{2} i}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}-\frac {A b c d \,e^{2} i}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}+\frac {B a \,d^{2} e^{2} i}{4 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}-\frac {B b c d \,e^{2} i}{4 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.38, size = 933, normalized size = 5.39 \[ -\frac {1}{36} \, B d i {\left (\frac {6 \, {\left (3 \, b x + a\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}} + \frac {5 \, a b^{2} c^{2} - 22 \, a^{2} b c d + 5 \, a^{3} d^{2} - 6 \, {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} - 16 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x}{{\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}} - \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{4}} + \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{4}}\right )} - \frac {1}{18} \, B c i {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {{\left (3 \, b x + a\right )} A d i}{6 \, {\left (b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}\right )}} - \frac {A c i}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.87, size = 361, normalized size = 2.09 \[ -\frac {\frac {6\,A\,a^2\,d^2\,i-12\,A\,b^2\,c^2\,i+5\,B\,a^2\,d^2\,i-4\,B\,b^2\,c^2\,i+6\,A\,a\,b\,c\,d\,i+5\,B\,a\,b\,c\,d\,i}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (6\,A\,a\,b\,d^2\,i+5\,B\,a\,b\,d^2\,i-6\,A\,b^2\,c\,d\,i-B\,b^2\,c\,d\,i\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,d^2\,i\,x^2}{a\,d-b\,c}}{6\,a^3\,b^2\,g^4+18\,a^2\,b^3\,g^4\,x+18\,a\,b^4\,g^4\,x^2+6\,b^5\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{3\,b^2\,g^4}+\frac {B\,a\,d\,i}{6\,b^3\,g^4}+\frac {B\,d\,i\,x}{2\,b^2\,g^4}\right )}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B\,d^3\,i\,\mathrm {atanh}\left (\frac {6\,b^4\,c^2\,g^4-6\,a^2\,b^2\,d^2\,g^4}{6\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{3\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 11.30, size = 629, normalized size = 3.64 \[ - \frac {B d^{3} i \log {\left (x + \frac {- \frac {B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i + \frac {B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {B d^{3} i \log {\left (x + \frac {\frac {B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i - \frac {B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {\left (- B a d i - 2 B b c i - 3 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{6 a^{3} b^{2} g^{4} + 18 a^{2} b^{3} g^{4} x + 18 a b^{4} g^{4} x^{2} + 6 b^{5} g^{4} x^{3}} + \frac {- 6 A a^{2} d^{2} i - 6 A a b c d i + 12 A b^{2} c^{2} i - 5 B a^{2} d^{2} i - 5 B a b c d i + 4 B b^{2} c^{2} i - 6 B b^{2} d^{2} i x^{2} + x \left (- 18 A a b d^{2} i + 18 A b^{2} c d i - 15 B a b d^{2} i + 3 B b^{2} c d i\right )}{36 a^{4} b^{2} d g^{4} - 36 a^{3} b^{3} c g^{4} + x^{3} \left (36 a b^{5} d g^{4} - 36 b^{6} c g^{4}\right ) + x^{2} \left (108 a^{2} b^{4} d g^{4} - 108 a b^{5} c g^{4}\right ) + x \left (108 a^{3} b^{3} d g^{4} - 108 a^{2} b^{4} c g^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________