Optimal. Leaf size=85 \[ \frac {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 0.29, antiderivative size = 191, normalized size of antiderivative = 2.25, 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} \[ -\frac {b g \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (c+d x)}+\frac {g (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i^3 (c+d x)^2}+\frac {b^2 B g \log (a+b x)}{2 d^2 i^3 (b c-a d)}-\frac {b^2 B g \log (c+d x)}{2 d^2 i^3 (b c-a d)}-\frac {B g (b c-a d)}{4 d^2 i^3 (c+d x)^2}+\frac {b B g}{2 d^2 i^3 (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(49 c+49 d x)^3} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{117649 d (c+d x)^3}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{117649 d (c+d x)^2}\right ) \, dx\\ &=\frac {(b g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{117649 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{117649 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}+\frac {(b B g) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{117649 d^2}-\frac {(B (b c-a d) g) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{235298 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}+\frac {(b B (b c-a d) g) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{117649 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{235298 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}+\frac {(b B (b c-a d) g) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{117649 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{235298 d^2}\\ &=-\frac {B (b c-a d) g}{470596 d^2 (c+d x)^2}+\frac {b B g}{235298 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x)}{235298 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{235298 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{117649 d^2 (c+d x)}-\frac {b^2 B g \log (c+d x)}{235298 d^2 (b c-a d)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.15, size = 207, normalized size = 2.44 \[ \frac {g \left (-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 (c+d x)}+\frac {(b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 (c+d x)^2}-\frac {B \left (\frac {2 b^2 \log (a+b x)}{b c-a d}-\frac {2 b^2 \log (c+d x)}{b c-a d}+\frac {b c-a d}{(c+d x)^2}+\frac {2 b}{c+d x}\right )}{4 d^2}+\frac {b B \left (\frac {b \log (a+b x)}{b c-a d}-\frac {b \log (c+d x)}{b c-a d}+\frac {1}{c+d x}\right )}{d^2}\right )}{i^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.71, size = 185, normalized size = 2.18 \[ -\frac {2 \, {\left ({\left (2 \, A - B\right )} b^{2} c d - {\left (2 \, A - B\right )} a b d^{2}\right )} g x + {\left ({\left (2 \, A - B\right )} b^{2} c^{2} - {\left (2 \, A - B\right )} a^{2} d^{2}\right )} g - 2 \, {\left (B b^{2} d^{2} g x^{2} + 2 \, B a b d^{2} g x + B a^{2} d^{2} g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.07, size = 152, normalized size = 1.79 \[ \frac {1}{4} \, {\left (\frac {2 \, {\left (b x e + a e\right )}^{2} B g i \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {2 \, {\left (b x e + a e\right )}^{2} A g i}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x e + a e\right )}^{2} B g i}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 1049, normalized size = 12.34 \[ -\frac {B \,a^{3} d g \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 )^{2} \left (d x +c \right )^{2} i^{3}}+\frac {3 B \,a^{2} b c g \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 )^{2} \left (d x +c \right )^{2} i^{3}}-\frac {3 B a \,b^{2} c^{2} g \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 )^{2} \left (d x +c \right )^{2} d \,i^{3}}+\frac {B \,b^{3} c^{3} g \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 )^{2} \left (d x +c \right )^{2} d^{2} i^{3}}-\frac {A \,a^{3} d g}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} i^{3}}+\frac {3 A \,a^{2} b c g}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} i^{3}}-\frac {3 A a \,b^{2} c^{2} g}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d \,i^{3}}+\frac {A \,b^{3} c^{3} g}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{2} i^{3}}+\frac {B \,a^{3} d g}{4 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} i^{3}}-\frac {3 B \,a^{2} b c g}{4 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} i^{3}}-\frac {B \,a^{2} b g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{3}}+\frac {3 B a \,b^{2} c^{2} g}{4 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d \,i^{3}}+\frac {2 B a \,b^{2} c g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (d x +c \right ) d \,i^{3}}-\frac {B \,b^{3} c^{3} g}{4 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{2} i^{3}}-\frac {B \,b^{3} c^{2} g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (d x +c \right ) d^{2} i^{3}}-\frac {A \,a^{2} b g}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{3}}+\frac {2 A a \,b^{2} c g}{\left (a d -b c \right )^{2} \left (d x +c \right ) d \,i^{3}}-\frac {A \,b^{3} c^{2} g}{\left (a d -b c \right )^{2} \left (d x +c \right ) d^{2} i^{3}}+\frac {B \,a^{2} b g}{2 \left (a d -b c \right )^{2} \left (d x +c \right ) i^{3}}-\frac {B a \,b^{2} c g}{\left (a d -b c \right )^{2} \left (d x +c \right ) d \,i^{3}}-\frac {B a \,b^{2} g \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 )^{2} d \,i^{3}}+\frac {B \,b^{3} c^{2} g}{2 \left (a d -b c \right )^{2} \left (d x +c \right ) d^{2} i^{3}}+\frac {B \,b^{3} c g \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 )^{2} d^{2} i^{3}}-\frac {A a \,b^{2} g}{2 \left (a d -b c \right )^{2} d \,i^{3}}+\frac {A \,b^{3} c g}{2 \left (a d -b c \right )^{2} d^{2} i^{3}}+\frac {B a \,b^{2} g}{4 \left (a d -b c \right )^{2} d \,i^{3}}-\frac {B \,b^{3} c g}{4 \left (a d -b c \right )^{2} d^{2} i^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.20, size = 567, normalized size = 6.67 \[ -\frac {1}{4} \, B b g {\left (\frac {2 \, {\left (2 \, d x + c\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{4} i^{3} x^{2} + 2 \, c d^{3} i^{3} x + c^{2} d^{2} i^{3}} - \frac {b c^{2} - 3 \, a c d + 2 \, {\left (b c d - 2 \, a d^{2}\right )} x}{{\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}} - \frac {2 \, {\left (b^{2} c - 2 \, a b d\right )} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} i^{3}} + \frac {2 \, {\left (b^{2} c - 2 \, a b d\right )} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} i^{3}}\right )} + \frac {1}{4} \, B a g {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac {{\left (2 \, d x + c\right )} A b g}{2 \, {\left (d^{4} i^{3} x^{2} + 2 \, c d^{3} i^{3} x + c^{2} d^{2} i^{3}\right )}} - \frac {A a g}{2 \, {\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.63, size = 198, normalized size = 2.33 \[ -\frac {x\,\left (2\,A\,b\,d\,g-B\,b\,d\,g\right )+A\,a\,d\,g+A\,b\,c\,g-\frac {B\,a\,d\,g}{2}-\frac {B\,b\,c\,g}{2}}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,a\,g}{2\,d^2\,i^3}+\frac {B\,b\,c\,g}{2\,d^3\,i^3}+\frac {B\,b\,g\,x}{d^2\,i^3}\right )}{2\,c\,x+d\,x^2+\frac {c^2}{d}}+\frac {B\,b^2\,g\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 5.76, size = 382, normalized size = 4.49 \[ \frac {B b^{2} g \log {\left (x + \frac {- \frac {B a^{2} b^{2} d^{2} g}{a d - b c} + \frac {2 B a b^{3} c d g}{a d - b c} + B a b^{2} d g - \frac {B b^{4} c^{2} g}{a d - b c} + B b^{3} c g}{2 B b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac {B b^{2} g \log {\left (x + \frac {\frac {B a^{2} b^{2} d^{2} g}{a d - b c} - \frac {2 B a b^{3} c d g}{a d - b c} + B a b^{2} d g + \frac {B b^{4} c^{2} g}{a d - b c} + B b^{3} c g}{2 B b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} + \frac {- 2 A a d g - 2 A b c g + B a d g + B b c g + x \left (- 4 A b d g + 2 B b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac {\left (- B a d g - B b c g - 2 B b d g x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________