Optimal. Leaf size=152 \[ \frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{i^2 (c+d x) (b c-a d)}-\frac {2 A B (a+b x)}{i^2 (c+d x) (b c-a d)}-\frac {2 B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{i^2 (c+d x) (b c-a d)}+\frac {2 B^2 (a+b x)}{i^2 (c+d x) (b c-a d)} \]
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Rubi [C] time = 0.78, antiderivative size = 472, normalized size of antiderivative = 3.11, number of steps used = 26, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac {2 b B^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac {2 b B^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac {2 b B \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (b c-a d)}+\frac {2 B \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (c+d x)}-\frac {2 b B \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (b c-a d)}-\frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d i^2 (c+d x)}-\frac {b B^2 \log ^2(a+b x)}{d i^2 (b c-a d)}-\frac {b B^2 \log ^2(c+d x)}{d i^2 (b c-a d)}-\frac {2 b B^2 \log (a+b x)}{d i^2 (b c-a d)}+\frac {2 b B^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac {2 b B^2 \log (c+d x)}{d i^2 (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}-\frac {2 B^2}{d i^2 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(95 c+95 d x)^2} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {(2 B) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{95 (a+b x) (c+d x)^2} \, dx}{95 d}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {(2 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{9025 d}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {(2 B (b c-a d)) \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9025 d}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {(2 B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{9025}-\frac {(2 b B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{9025 (b c-a d)}+\frac {\left (2 b^2 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {\left (2 B^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{9025 d}-\frac {\left (2 b B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{9025 d (b c-a d)}+\frac {\left (2 b B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{9025 d (b c-a d)}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {\left (2 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{9025 d}-\frac {\left (2 b B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{9025 d (b c-a d) e}+\frac {\left (2 b B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{9025 d (b c-a d) e}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {\left (2 B^2 (b c-a d)\right ) \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}{9025 d}-\frac {\left (2 b B^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{9025 d (b c-a d) e}+\frac {\left (2 b B^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{9025 d (b c-a d) e}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}+\frac {\left (2 b B^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{9025 (b c-a d)}-\frac {\left (2 b B^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{9025 (b c-a d)}-\frac {\left (2 b^2 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{9025 d (b c-a d)}+\frac {\left (2 b^2 B^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{9025 (b c-a d)}-\frac {\left (2 b B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{9025 d (b c-a d)}-\frac {\left (2 b^2 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9025 d (b c-a d)}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}+\frac {2 b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9025 d (b c-a d)}+\frac {2 b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}\\ \end {align*}
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Mathematica [C] time = 0.49, size = 315, normalized size = 2.07 \[ \frac {\frac {B \left (2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{b c-a d}-\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d i^2 (c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 155, normalized size = 1.02 \[ -\frac {{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d - {\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (A B - B^{2}\right )} b d x + {\left (A B - B^{2}\right )} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x + {\left (b c^{2} d - a c d^{2}\right )} i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 179, normalized size = 1.18 \[ -{\left (\frac {{\left (b x e + a e\right )} B^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + \frac {2 \, {\left (b x e + a e\right )} {\left (A B - B^{2}\right )} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {{\left (b x e + a e\right )} {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1236, normalized size = 8.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.30, size = 416, normalized size = 2.74 \[ {\left (2 \, {\left (\frac {1}{d^{2} i^{2} x + c d i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {{\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b c^{2} d i^{2} - a c d^{2} i^{2} + {\left (b c d^{2} i^{2} - a d^{3} i^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {1}{d^{2} i^{2} x + c d i^{2}} - \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} + \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{d^{2} i^{2} x + c d i^{2}} - \frac {A^{2}}{d^{2} i^{2} x + c d i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.62, size = 222, normalized size = 1.46 \[ \frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,B^2}{b\,d^2\,i^2}-\frac {2\,A\,B}{b\,d^2\,i^2}\right )}{\frac {x}{b}+\frac {c}{b\,d}}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{d^2\,i^2\,\left (x+\frac {c}{d}\right )}+\frac {B^2\,b}{d\,i^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{x\,d^2\,i^2+c\,d\,i^2}+\frac {B\,b\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a\,d^2\,i^2+b\,c\,d\,i^2}{d\,i^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,4{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.46, size = 432, normalized size = 2.84 \[ \frac {2 B b \left (A - B\right ) \log {\left (x + \frac {2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c - \frac {2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} + \frac {4 B a b^{2} c d \left (A - B\right )}{a d - b c} - \frac {2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {2 B b \left (A - B\right ) \log {\left (x + \frac {2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c + \frac {2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} - \frac {4 B a b^{2} c d \left (A - B\right )}{a d - b c} + \frac {2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} + \frac {\left (- B^{2} a - B^{2} b x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a c d i^{2} + a d^{2} i^{2} x - b c^{2} i^{2} - b c d i^{2} x} + \frac {- A^{2} + 2 A B - 2 B^{2}}{c d i^{2} + d^{2} i^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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