Optimal. Leaf size=81 \[ \frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d}-\frac {B i (b c-a d)^2 \log (a+b x)}{2 b^2 d}-\frac {B i x (b c-a d)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2525, 12, 43} \[ \frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d}-\frac {B i (b c-a d)^2 \log (a+b x)}{2 b^2 d}-\frac {B i x (b c-a d)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2525
Rubi steps
\begin {align*} \int (4 c+4 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac {2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}-\frac {B \int \frac {16 (b c-a d) (c+d x)}{a+b x} \, dx}{8 d}\\ &=\frac {2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}-\frac {(2 B (b c-a d)) \int \frac {c+d x}{a+b x} \, dx}{d}\\ &=\frac {2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}-\frac {(2 B (b c-a d)) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{d}\\ &=-\frac {2 B (b c-a d) x}{b}-\frac {2 B (b c-a d)^2 \log (a+b x)}{b^2 d}+\frac {2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 0.86 \[ \frac {i \left ((c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {B (b c-a d) ((b c-a d) \log (a+b x)+b d x)}{b^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 127, normalized size = 1.57 \[ \frac {A b^{2} d^{2} i x^{2} - B b^{2} c^{2} i \log \left (d x + c\right ) + {\left ({\left (2 \, A - B\right )} b^{2} c d + B a b d^{2}\right )} i x + {\left (2 \, B a b c d - B a^{2} d^{2}\right )} i \log \left (b x + a\right ) + {\left (B b^{2} d^{2} i x^{2} + 2 \, B b^{2} c d i x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 1395, normalized size = 17.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 940, normalized size = 11.60 \[ -\frac {B \,a^{4} d^{3} 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 (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} \left (d x +c \right )^{2} b^{2}}+\frac {2 B \,a^{3} c \,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 )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} \left (d x +c \right )^{2} b}-\frac {3 B \,a^{2} c^{2} 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 )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} \left (d x +c \right )^{2}}+\frac {2 B a b \,c^{3} e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} \left (d x +c \right )^{2}}-\frac {B \,b^{2} c^{4} 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 (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} \left (d x +c \right )^{2} d}+\frac {B \,a^{2} 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 (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2}}-\frac {B a b c \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2}}+\frac {B \,b^{2} c^{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 (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} d}+\frac {A \,a^{2} d \,e^{2} i}{2 \left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2}}-\frac {A a b c \,e^{2} i}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2}}+\frac {A \,b^{2} c^{2} e^{2} i}{2 \left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right )^{2} d}+\frac {B \,a^{2} d e i}{2 \left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) b}-\frac {B a c e i}{\frac {a d e}{d x +c}-\frac {b c e}{d x +c}}+\frac {B b \,c^{2} e i}{2 \left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) d}+\frac {B \,a^{2} d i \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{2 b^{2}}-\frac {B a c i \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{b}+\frac {B \,c^{2} i \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 144, normalized size = 1.78 \[ \frac {1}{2} \, A d i x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B c i + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B d i + A c i x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.64, size = 126, normalized size = 1.56 \[ x\,\left (\frac {i\,\left (2\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{2\,b}-\frac {A\,i\,\left (2\,a\,d+2\,b\,c\right )}{2\,b}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,d\,i\,x^2}{2}+B\,c\,i\,x\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^2\,d\,i-2\,B\,a\,b\,c\,i\right )}{2\,b^2}+\frac {A\,d\,i\,x^2}{2}-\frac {B\,c^2\,i\,\ln \left (c+d\,x\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.99, size = 253, normalized size = 3.12 \[ \frac {A d i x^{2}}{2} - \frac {B a i \left (a d - 2 b c\right ) \log {\left (x + \frac {B a^{2} c d i + \frac {B a^{2} d i \left (a d - 2 b c\right )}{b} - 3 B a b c^{2} i - B a c i \left (a d - 2 b c\right )}{B a^{2} d^{2} i - 2 B a b c d i - B b^{2} c^{2} i} \right )}}{2 b^{2}} - \frac {B c^{2} i \log {\left (x + \frac {B a^{2} c d i - 2 B a b c^{2} i - \frac {B b^{2} c^{3} i}{d}}{B a^{2} d^{2} i - 2 B a b c d i - B b^{2} c^{2} i} \right )}}{2 d} + x \left (A c i + \frac {B a d i}{2 b} - \frac {B c i}{2}\right ) + \left (B c i x + \frac {B d i x^{2}}{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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